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HEC Montréal
Affiliated to the University of Montreal
Essays on the Valuation of Derivatives on Long Maturity Treasury Bonds
By
Ramzi Ben-Abdallah
Management Sciences Department
HEC Montréal
This thesis is presented in partial fulfillment of the requirements for the
degree of Philosophiae Doctor (Ph.D.) in Business Administration
June, 2008
Copyright © 2008, Ramzi Ben-Abdallah
HEC Montréal
Affiliated to the University of Montreal
This thesis entitled:
Essays on the Valuation of Derivatives on Long Maturity
Treasury Bonds
By
Ramzi Ben-Abdallah
Has been evaluated by the following jury:
President-reviewer: Prof. Geneviève Gauthier
Thesis supervisors: Prof. Michèle Breton
Prof. Hatem Ben-Ameur
Internai examiner: Prof. Jan Ericsson
External examiner: Prof. Simon Benninga
Abstract
In this thesis, we investigate the theoretical and empirical pricing of the Chicago
Board of Trade (CBOT) Treasury-bond futures. This contract is one of the most
traded in the world, largely because of its ability to hedge long-term interest-rate
risk. The difficulty to price it arises from its multiple inter-dependent embedded
delivery options, which can be exercised at various times and dates during the
delivery month. This thesis is composed of three essays.
In the first essay, we propose a numerical method for the theoretical pricing of the
CBOT T-bonds futures by considering a continuous-time model with a continuous
underlying factor (the interest rate), moving according to a Markov diffusion process
consistent with the no-arbitrage principle. We propose a model that can handle
all the delivery rules embedded in the CBOT T-bond futures, interpreted here as
an American-style interest-rate derivative. Our pricing procedure, the first to our
knowledge to tackle ail the complexities of the CBOT futures contract in a stochastic
interest-rate framework, is a backward numerical algorithm combining Dynamic
Programming (DP), approximation by finite elements, and fixed-point evaluation.
Numerical illustrations, provided under the Vasicek (1977) and Cox-Ingesoll-Ross
(CIR) (1985) models, show that the interaction between the quality and timing
options in a stochastic environment makes the delivery strategies complex, and flot
easy to characterize.
The second essay is devoted to the application of the True Notional Bond Sys-
tem (TNBS) proposed by Oviedo (2006) for the theoretical pricing of the CBOT
Treasury-bond futures. This system is proposed as an alternative to the current con-
version factor system (CFS), whose imperfections are well known. Oviedo (2006)
showed that the TNBS outperforms the CFS when interest rates are deterministic.
In this essay, we compare the effectiveness of the two systems in a stochastic envi-
ronment. To do so, we price the CBOT T-bond futures as well as ail its embedded
delivery options under the TNBS in a stochastic interest-rate framework. Our pric-
iv
ing procedure is an adaptation of the Dynamic Programming algorithm described in
the first essay, giving the value of the futures contract under the TNBS as a function
of time and current short-term interest rate. Numerical illustrations, also provided
under the Vasicek and CIR models, show that, in a stochastic framework, TNBS
does flot always outperform the CFS. However, as the long-term mean moves away
from the level of the notional rate, the TNBS performs increasingly better than the
CFS.
The third essay is an empirical investigation of the futures pricing model pro-
posed in the first essay. Comparing market futures prices with theoretical prices, ob-
tained using our pricing algorithm, allows to verify if the delivery options are priced
correctly by the market, and if short traders in futures contracts are exercising the
strategic delivery options skillfully and optimally or if they are under-utilizing them.
To do so, we price the futures contract under the Hull-White (1990) mode!. This
model requires the use of market data to fit the zero-coupon yield curve. We use
the Augmented Nelson and Siegel yield-curve fitting model proposed by Bjork and
Christensen (1999) in order to make the original Nelson and Siegel family consistent
with the Hull-White model. As proxies for the short-term interest rate, we use the
1-month and 3-month US Treasury-Bills yields covering the period 1982-2008. Fu-
tures prices are obtained from the Chicago Board of Trade. Empirical results show
that futures prices are generally undervalued, which means that the market over-
values the embedded delivery options. According to our findings, observed futures
prices are on average 2% lower than theoretical futures prices over the 1990-2008
time period, priced two months prior to the first day of delivery months.
JEL classification: C61, C63, G12, G13.
Mathematics Subject Classification (2000): 90C39, 49M15, 65D05.
Keywords: Futures, asset pricing, dynamic programming, Treasury Bonds,
cheapest-to-deliver, embedded delivery options, interest-rate models.
Résumé
Nous nous intéressons, au niveau de cette thèse, à l'évaluation du contrat à terme sur
les obligations gouvernementales américaines transigé au Chicago Board of Trade
(CBOT). Ce contrat est l'un des plus transigés au monde et il est principalement
utilisé pour la couverture du risque de taux d'intérêt de long terme. Ce contrat est
difficile à évaluer du fait de l'interdépendance de ses nombreuses options implicites de
livraison, qui peuvent être exercées d'une manière prématurée à différents moments
et dates au cours d'un mois de livraison. Cette thèse est composée de trois essais.
Dans le premier essai, nous proposons une méthode numérique pour l'évaluation
de ce contrat. Nous considérons un modèle en temps continu et un sous-jacent
continu (le taux d'intérêt court) dont la dynamique est un processus de diffusion
Markovien satisfaisant le principe d'absence d'opportunité d'arbitrage. Le modèle
que nous proposons tient compte simultanément et explicitement de toutes les op-
tions de livraisons présentes dans ce contrat, qui peut être interprété comme un
dérivé sur taux d'intérêt de type américain. Notre algorithme numérique combine
la programmation dynamique, l'approximation par éléments finis ainsi que la résolu-
tion d'un problème de point fixe. Des illustrations numériques sont proposées pour
les dynamiques Vasicek (1977) et Cox-Ingersoll-Ross (CIR) (1985).
Dans le second essai, nous étudions l'application du système appelé True No-
tional Bond System (TNBS) proposé par Oviedo (2006) comme alternative au sys-
tème de facteur de conversion (CFS) utilisé actuellement par le CBOT pour rendre
les obligations livrables équivalentes à la livraison. Ce dernier système souffre de
plusieurs imperfections, qui peuvent donner lieu à des manipulations du marché
de l'obligation la moins chère à livrer. Nous proposons une procédure numérique
qui permet d'évaluer le contrat ainsi que toutes ses options implicites de livraison
dans un environnement de taux d'intérêts stochastiques si le TNBS est le système
utilisé pour obtenir des obligations comparables à la livraison. Des illustrations
numériques, proposées pour les dynamiques Vasicek et CIR, nous permettent de
vi
comparer les deux systèmes et montrent que le TNBS n'est pas toujours meilleur
que le CFS, contrairement à ce qui a été montré par Oviedo (2006) lorsque les taux
sont déterministes. Cependant, le TNBS performe de mieux en mieux au fur et à
mesure que les taux s'éloignent du niveau de long terme.
Le troisième essai consiste à tester empiriquement le modèle d'évaluation du
contrat à terme que nous proposons au niveau du premier essai. Pour ce faire,
nous évaluons le contrat en supposant que le taux d'intérêt court suit la dynamique
proposée par Hull et White (1990), qui représente une extension du modèle de
Vasicek (1977) en permettant à la moyenne de long terme de dépendre du temps, et
aussi de répliquer la structure à terme initiale des taux. Le calibrage est effectué en
utilisant le modèle augmenté de Nelson et Siegel, proposé par Bjork et Christensen
(1999) et cohérent avec le modèle Hull-White (1990). Les rendements des obligations
zéro-coupons émises par le gouvernement américain de courtes maturités (1 mois et
3 mois) sur la période 1982-2008 sont utilisés comme approximation pour les taux
d'intérêt courts. Les résultats empiriques montrent que les prix des contrats à terme
sont sous-évalués par le marché sur la période 1990-2008, d'en moyenne 2% lorsque
les contrats sont évalués deux mois avant le premier jour du mois de livraison. Ceci
impliquerait que les options implicites de livraison sont sur-évaluées par le marché.
Classification JEL: C61, C63, G12, G13.
Classification Mathematics Subject (2000): 90C39, 49M15, 65D05.
Mots clés: Contrats à terme, tarification d'actifs financiers, programmation
dynamique, obligations gouvernementales américaines, moins chère à livrer, options
implicites de livraison, modèles de taux d'intérêt.
vii
Contents
Abstract iii
Résumé v
List of tables x
List of figures xiii
Notation xiv
Aknowledgements xviii
1
2
Introduction
Review of the literature
1
5
2.1 The CBOT conversion factor system 5
2.2 Valuation of the delivery options embedded in the CBOT T-Bond
futures 8
2.2.1 Valuation of the quality option 9
2.2.2 Valuation of the timing options 17
2.3 Pricing embedded delivery options in other bond futures markets . 23
3 Pricing the CBOT T-Bond Futures 27
3.1 Notation 27
3.2 Price and CTD of an associated T-Bond Forward contract 28
3.3 Model and DP formulation 30
3.3.1 End-of-the-month Period 31
3.3.2 Delivery Month 32
3.3.3 Initial period 33
3.4 Dynamic Programming Procedure 34
3.4.1 Optimization Procedure 34
3.4.2 Interpolation Procedure
3.4.3 Expectations of Interpolation Functions
3.4.4 Root Finding Procedure
3.4.5 Algorithm
viii
35
36
36
37
3.5 Numerical Illustration 38
3.5.1 Convergence 40
3.5.2 Options values 41
3.5.3 Optimal delivery strategy under the CFS 45
3.5.4 Sensitivity of the options values to the input parameters 49
4 An Analysis of the True Notional Bond System Applied to the
CBOT T-Bonds Futures 56
4.1 Notation 56
4.2 A general description of the True Notional Bond System 57
4.3 Model and DP formulation 58
4.3.1 End-of-the-month Period 58
4.3.2 Delivery Month 59
4.3.3 Initial Period 60
4.4 The Dynamic Programming pro cedure 60
4.5 Numerical Illustration 62
4.5.1 Option prices under TNBS 62
4.5.2 Comparison of option prices under TNBS and CFS 65
4.5.3 Sensitivity analysis of the TNBS 71
4.5.4 Optimal delivery strategy under the TNBS 75
5 Prices of the CBOT T-Bonds Futures: An Empirical Investigation 80
5.1 The Hull-White model (extended Vasicek) 80
5.2 The Data 82
5.3 Empirical results 84
ix
6 Conclusion and suggestions for future research 89
Bibliography 91
Appendix xix
Transition parameters xix
The Vasicek model xix
The CIR model XX
The Hull-White model )ocii
Futures pricing algorithm under the CFS xxii
Futures pricing algorithm under the TNBS xxvi
Hull-White input parameters xxix
Observed vs. theoretical futures prices xxxiii
X
List of Tables
I The forward contract CTD 29
H Properties of the deliverable set 39
III Convergence of the DP futures prices (Vasicek) 41
IV Input data 42
V Summary statistics on short rate 83
VI Hull-White input parameters (1) xxx
VII Hull-White input parameters (2) xxxi
VIII Hull-White input parameters (3) =di
IX Observed vs. theoretical futures prices (1) xxxiv
X Observed vs. theoretical futures prices (2) xxxv
List of Figures
xi
1 Timing options embedded in the CBOT T-bonds futures 3
2 Convergence of the DP futures prices (Vasicek) 41
3 Quality option values vs. interest rates at inception (CFS) 43
4 Evolution of the quality option during the delivery month (CFS) 44
5 Timing option values vs. interest rates at inception (CFS) 45
6 Optimal delivery strategy on day 16 of the DM (Vasicek) 47
7 Optimal delivery strategy on day 16 of the DM for possible variations
of interest rates (Vasicek) 48
8 Optimal delivery strategy 3 days before the end of the DM for possible
variations of interest rates (Vasicek) 48
9 Optimal delivery strategy on the last day of the DM for possible
variations of interest rates (Vasicek) 49
10 Quality option sensitivity to rbar and interest rates (kappa=0.2) 52
11 Quality option sensitivity to rbar and interest rates (kappa=0.5) 52
12 Quality option sensitivity to rbar and interest rates (kappa=0.8) 53
13 Quality option sensitivity to rbar and sigma (kappa=0.8) 53
14 Quality option sensitivity to rbar and sigma (kappa=-0.8) 54
15 Impact of the volatility on futures prices 54
16 Quality option sensitivity to kappa and rbar 55
17 Timing option value sensitivity to sigma (kappa=-0.5, rbar-=0.06) 55
18 Quality option values vs. interest rates at inception (TNBS) 63
19 Evolution of the value of the quality option through the DM (TNBS) 64
20 Timing option values vs. interest rates at inception (TNBS) 65
21 Comparison of the quality option values (with timing) under the CFS
and the TNBS at inception for the Vasicek mode!. 66
22 Comparison of the timing options values (with quality) under the
CFS and the TNBS at inception for the Vasicek model. 66
xii
23 Comparison of ail the options (quality and timing) under the CFS
and the TNBS at inception for the Vasicek model. 67
24 Comparison of the quality option values (with timing) under the CFS
and the TNBS at inception for the CIR model. 67
25 Comparison of the timing options values (with quality) under the
CFS and the TNBS at inception for the CIR model. 68
26 Comparison of ail the options (quality and timing) under the CFS
and the TNBS at inception for the CIR model. 68
27 Comparison of the quality option values (with timing) under the CFS
and the TNBS at inception for the CIR model (high volatility). . . 69
28 Comparison of the quality option (with timing) under the CFS and
the TNBS on day 1 of the delivery month for the CIR model (mod-
erate volatility) 71
29 Comparison of the quality option value sensitivity to the long-term
mean (rbar) at inception under TNBS and CFS for the CIR model. 71
30 Options values sensitivities to kappa (Vasicek) 72
31 Options values sensitivities to rbar (Vasicek) 73
32 Options values sensitivities to sigma (Vasicek) 73
33 Options values sensitivities to kappa (CIR) 74
34 Options values sensitivities to rbar (CIR) 74
35 Options values sensitivities to sigma (CIR) 75
36 Optimal decision and CTD on days 15 and 16 of the DM (CIR) . . 77
37 Optimal decision and CTD on days 15 and 16 of the DM for possible
variations of interest rates (CIR, rbar=0.0617) 77
38 Optimal decision and CTD during the end-of-the-month period (days
18 and 19 of the DM) for possible variations of interest rates (CIR,
rbar=0.0617) 78
39 Optimal decision and CTD on days 15 and 16 of the DM for possible
variations of interest rates (CIR, rbar=0.08) 78
40 CTD on the last day of the delivery month for possible variations of
xiii
interest rates (CIR) 79
41 Spot-yield surface. 84
42 Observed vs. theoretical futures prices at inception. 85
43 Observed vs. theoretical futures prices on day 1 of the DM. 86
44 Futures pricing errors at inception 87
45 Futures pricing errors on day 1 of the DM 87
46 Futures pricing errors at inception and on day 1 of the DM 88
47 Yield curves at inception corresponding to extremal futures pricing
differences. 88
xiv
Notation
General
• E H the expectation under the risk-neutral measure Q;
• I (•) the indicator function.
Bond and futures markets
• (c, M) e 0 an eligible T-bond with a principal of 1 dollar, a continuous coupon
rate c, and a maturity M, where the finite set 0 of eligible bonds is known at
the date the contract is written;
• {rd a Markov process for the risk-free short-term interest rate;
• p(r, t, 7- ) the price at t of a zero-coupon bond maturing at T > t when rt = r
under the process {rt }
P(r,t,T) = E [exp( — rudu) I rt = ; (1)
• p (t, c, M, r) the price at t of the eligible T-bond (c, M) when rt = r under the
process {rt }
p (t, c, M,r) = c p(r,t,u)du p(r,t, M); (2)
• PV(t, c, M, r) the price at t of the eligible T-bond (c, M) when its yield to
maturity is r
PV (t, c, M, r) = c exp(—r(u — t))du exp (—r (M — t)) ; (3 )
• c M the minimum (maximum) bond coupon rate among the deliverable bonds;
• M (M) the minimum (maximum) bond maturity among the deliverable bonds;
• z(t) the yield of a zero-coupon bond of maturity t;
XV
• y the yield to maturity of the notional bond;
• f(t) the forward-rate curve;
• g* the settlement price for the T-bond futures;
• g° the price of a forward contract on T-bonds.
Contract
• in the time index;
• to the inception date;
• tn the first day of the delivery month;
• tn the last futures trading date during the delivery month;
• tr, the last date of the delivery month;
• 4,22 the marking-to-market date;
• tn.,5 the delivery notice date;
• t8n, the delivery position date;
• T the maturity of the T-bond futures contract;
• l'al the CBOT conversion factor corresponding to the T-bond (c, M), where
the set {fa,/ : (c, M) E el is known at the date the contract is written:
.fcm = PV (ta , c, M, 6%) . (4)
DP value functions
• vme (•) the expected exercise value for the short trader at t m , as a function of
the state variables;
xvi
• vma (-) the actual exercise value for the short trader at tni , as a function of the
state variables;
• vmh (•) the holding value for the short trader at 4,2 , as a function of the state
variables;
• v,m8 0 the value function for the short trader at t rn8 , as a function of the state
variables;
• um2 (•) the settlement value for the short trader at tn.,2 , as a function of the state
variables;
• st (r) the price of the T-bond futures at t when r t = r;
• F(.) the DP operator.
Interpolation funct ions
• g = {ai, , %} the grid defined on the set of interest rates, with the conven-
tion that ao = —oo and aq+ i =
• q the number of grid points;
• —h (r) the piecewise linear interpolation function of h : Ç —> IR;
• (t r , ak) the expected value at t and ak of a future continuous payoff 'h(•) at
T;
• at the intercept of the ith segment of interpolation function;
• Oi the slope of the ith segment of interpolation function;
• At'T and Bt ' T the transition parameters from k at t to i at r.
Interest-rate models
• {Bt } a standard Brownian motion;
xvii
• f the long-term mean of the short-term interest rate;
• k the mean reversion speed of the short-term interest rate;
• cr the volatility of the short-term interest rate;
• -Yo, l'i, .721 1/3 and -y4 the parameters of the Augmented Nelson and Siegel
model.
xviii
Acknowledgments
It is my pleasure to express here my indebtedness and gratitude to the people who
supported me during these last years.
First of ail, I would like to express my sincere thanks to my supervisors Prof.
Michèle Breton and Prof. Hatem Ben-Ameur for their personal guidance, construc-
tive comments, stimulating suggestions and unrelenting encouragements. They were
always there to listen and help and I really owe them a lot.
I wish also to express my thanks to Prof. Simon Benninga, Prof. Jan Ericsson,
Prof. Jeroen Rombouts and Prof. Rodolfo Oviedo for taking the time to evaluate
my work and for their helpful comments.
I also want to express my appreciation and gratitude to my wonderful parents,
my brothers and my sister in-law for their unconditional support, understanding
and encouragements.
Finally, I would like to thank the Institut de Finance Mathématique de Mon-
tréal (IFM2 ), the Fonds Québécois de la Recherche sur la Nature et les Technolo-
gies (FQRNT), the Natural Sciences and Engineering Research Council of Canada
(NSERC), the Fondation Desjardins, HEC Montréal and the Tunisian Ministry of
Higher Education for their financial support.
To my parents Hayet and Foued,
To my brothers Hassib and Riadh,
To my sister-in-law Sameh,
To my nephews Skander and Yassine.
xix
1 Introduction
A futures contract is an agreement between two investors traded on an exchange
to sell or to buy an underlying asset at some given time in the future, called the
delivery date, for a given price, called the futures price. By convention, at the time
the futures is written (the inception date), the futures price is known and sets the
value for both parties to zero. A futures contract is marked-to-market once a day
to eliminate counterparty risk. Precisely, at the end of each trading day, the futures
contract is rewritten at a new settlement price, that is, the closing futures price,
and the difference with the last settlement futures price is substracted (resp. added)
from the short (resp. long) trader account.
A forward contract is similar to a futures contract in many ways in so far as it
is also an agreement to purchase a specified quantity of some asset at a fixed price,
called the forward price, on a fixed future date, called the delivery date. In addition,
the forward price is also set so that the value for both parties is zero. However,
a forward contract is not standardized since it is flot traded on an exchange, and
requires only one payment at the maturity.
Although futures contracts call for the short trader to deliver the underlying asset
at maturity, they also frequently allow some variation by offering him the option as
to what, when, where and how much is delivered. We refer to these flexibilities as
the quality option, the timing option, the location option, and the quantity option
respectively. If such delivery options are valuable to the short, then he should flot
acquire them costlessly. This value should be reflected in a reduced corresponding
futures price. In fact, because the short never explicitly pays for these options, the
futures price should be bid down to compensate the long for the additional delivery
risk. Thus, the greater the value of the option, the lower would be the futures price.
The Treasury Bond futures traded on the Chicago Board of Trade (the CBOT
T-bond futures in the sequel) is the most actively traded and widely used futures
contract in the United States, largely because of its ability to hedge long-term
2
interest-rate risk. It calls for the delivery of $100,000 of a long-term governmental
bond. The notional or reference bond is a bond with a 6% coupon rate and a
maturity of 20 years. Delivery months are March, June, September and December.
Since the notional bond is a hypothetical bond that is generally flot traded in
the market place, the short has the option to choose which bond to deliver among
a deliverable set fixed by the CBOT. The actual delivery day within the delivery
month is also at the option of the short. These two delivery privileges offered to the
short trader are known as the guality option (or choosing option) and the timing
option. They have been the subject of considerable research.
The quality option allows the delivery of any governmental bond with at least
15 years to maturity or earliest call. To make the delivery fair for both parties,
the price received by the short trader is adjusted according to the quality of the
T-bond delivered. This adjustment is made via a set of conversion factors defined
by the CBOT as the prices of the eligible T-bonds at the first delivery date under
the assumption that interest rates for all maturities equal 6% par annum, com-
pounded semiannually. The T-bond actually delivered by the short trader is called
the cheapest- to - deliver (CTD).
The timing option allows the short trader to deliver early within a delivery month
according to special features, that is, the delivery seguence and the end- of-month
delivery raie. The delivery sequence consists of three consecutive business days: The
position day, the notice day, and the delivery day. During the position day, the short
trader can declare his intention to deliver until up to 8:00 p.m., while the CBOT
closes at 2:00 p.m. (Central Standard Time). On the notice day, the short trader
has until 5:00 p.m. to state which T-bond will be actually delivered. The delivery
then takes place before 10:00 a.m. of the delivery day, against a payment based
on the settlement price of the position day (adjusted according to the conversion
factor). Finally, during the last seven business days before maturity, trading on
the T-bond futures contracts is suspended while delivery, based on the last futures
settlement price, remains possible according to the delivery sequence. The so-called
3
wild card play (or end-of-the day option or six hours option) and the end-of-month
option refer respectively to the timing option during the three day delivery sequence
and to the end-of-month rue. These timing options are summarized in Figure 1.
Deliver)/ Month Position date-
\ ,,' . Lest 7 business deys
t, L—... .A.. P m 4\ r End-of-day option
P.m•
Notice date
Les settlement eate
• I 1
1 "
. ‘ . ,\> ..\\\ \ \ \\\\\., s \ ,
Onliverv date
e e .e b , e
Delivery sequence
End-of-rnonth option
Futures Market open
Bond Market open
Figure 1: Timing options embedded in the CBOT T-bonds futures
Recently, the awareness about the importance and the impact of the embedded
delivery options in the CBOT T-bond futures contracts has substantially increased,
resulting in a considerable volume of research dealing with these options. This
research is becoming more and more complex with the use of recent results about
option pricing theory as well as numerical and empirical methods of estimation.
Most of the papers in the literature for the theoretical pricing of the CBOT T-bond
futures focus on the quality option alone, thus ignoring the joint effect of ail the
delivery options embedded in the futures contract. Other papers considering both
the choosing and the timing options use simplifying assumptions on the dynamics
or the strategies.
To date, no work has been presented regarding the identification of optimal
exercise strategies in the CBOT T-bond futures trading and the pricing of the
8:00
5:00
2:00
10:
4
contract under stochastic interest rates when the interaction of ail the delivery
options is taken into account. In fact, complexity arising from ail the embedded
inter-dependent delivery rules makes the contract computationally and analytically
difficult to price.
In this thesis, we investigate the theoretical and empirical pricing of the Chicago
Board of Trade (CBOT) Treasury-bond futures. This thesis is organized as follows.
Section 2 is a detailed review of the literature dealing with the conversion-factor
risk and the valuation of the futures written on governmental bond (the CBOT T-
bond futures or other major world exchanges governmental bond futures) as well as
its embedded delivery options. Section 3 is devoted to the CBOT T-bond futures
pricing model that we propose. In this section, we first describe an associated
governmental bond forward contract and present the CTD rule for this contract. We
then present the DP formulation, the numerical procedures and algorithm used to
solve the dynamic program as well as the results of numerical illustrations. Section 4
is an analysis of the True Notional Bond System (TNBS) proposed by Oviedo (2006)
applied to the CBOT T-bond futures. In this section, we describe the TNBS and
the dynamic program used to price the contract and its embedded delivery options
under this system. Numerical illustrations are also provided under the TNBS as well
as a comparison of the options values obtained under both systems (TNBS versus
CFS). In Section 5, we investigate empirically the futures pricing model proposed
in Section 3. We also describe the Hull-White (1990) model and present the yield-
curve fitting model used, namely the Augmented Nelson and Siegel model proposed
by Bjork and Christensen (1999). We then describe the short-term interest-rates
and the futures prices data. Finally, Section 6 is a general conclusion where we
summarize our main findings and present some directions for future research.
2 Review of the literature
The pricing of the T-bond futures contract and the modeling and measurement of
its implicit delivery options have been extensively examined in the literature using
different methods and leading to non consensual empirical results.
This section is structured as follows. First, we describe the CBOT conversion
factor system, highlight its imperfections, and report on the literature proposing
performance improvements to it. Second, we examine the quality option valuation
models and approaches. Third, we review the literature dealing with the valuation
of the timing options. Then, we review the literature on the valuation of options
embedded in other international governmental bond futures contracts.
2.1 The CBOT conversion factor system
Since the CBOT T-bond futures contract allows for the delivery of any governmental
bond from the eligible set, the CBOT establishes a conversion factor system relating
the actual bonds to the reference bond. The objective is to make equivalent the
delivery of any bond from the list of eligible issues. Such adjustment factors are
computed as being the clean prices of the deliverable bonds, on the delivery day,
such that the yields to maturity of ah deliverable issues are equal to the reference
coupon rate (see Equation 4). The futures invoice price is computed as the hast
settlement futures price times the conversion factor. However, this method for
computing the adjustment factor is imperfect, as it represents not an exact, but
an approximate, equivalence relation between the bonds in the eligible set, thus
creating what is known in the literature as the conversion-factor risk. In the current
system, the conversion factors for all deliverable bonds are precomputed quantities,
assuming a flat term structure at the level of 6%, constant during the life of the
contract. Thus, the aforesaid objective is only met in the very special case when the
yield to maturity of all bonds is equal to the reference rate of 6%. Because of its
imperfections, the current system can result in losses from delivering bonds other
6
than the CTD. This can cause an increase in the spot price of the CTD due to the
pressure of the short traders (trying to purchase it) or through a market squeeze
of the CTD, when, for instance, the long trader deliberately provokes its shortage.
In spite of its shortcomings, the system adopted by the CBOT is currently the
standard in an major exchanges of the world because of its virtues of being objective,
predictable and requiring limited information, ah these merits contributing to the
smooth functioning of the futures market.
The issue of the poor performance of the current conversion factor system has
been the subject of a substantial volume of research; many papers in that literature
propose alternative systems, aiming at the improvement of the performance of the
T-bond futures market through the reduction of the conversion-factor risk.
Livingston (1984) derives a rule for the identification of the CTD in the T-bond
futures contracts by examining the impact of the coupon level and time to maturity
on the bond to be delivered. He shows that the CTD will be the bond with the
smallest ratio of bond price divided by conversion factor. In addition, the author
investigates the question of shortage of supply of the CTD, since this might allow a
market corner by the long traders. The author shows that shortage of supply will
have a minimal impact if long maturity bonds are CTD. This impact can however
become significant when short maturity bonds are CTD, which will occur when bond
yields are below the reference rate.
Kane and Marcus (1984) examine the impact of the conversion-factor risk on
the variability of futures returns and find that this risk can be a source of profitable
trading opportunities. They suggest a change in the settlement procedure in order
to reduce the conversion-factor risk and increase the efficiency of the T-bond futures
market in hedging interest-rate risk. The authors propose a modified contract that
adjusts the number of delivered bonds instead of rescaling the futures settlement
price. They also propose to replace the current conversion factor by an "ideal"
one. Indeed, the conversion factor as presently computed does flot equal the ratio
of the value of the delivery bonds to the benchmark bond. Furthermore, one cannot
7
observe the price of the benchmark bond which is hypothetical and not traded in
the market place. The conversion factor they propose is the ratio of values of the
delivered bond to the benchmark bond, both computed using the current interest-
rate level instead of the reference rate. A drawback of this ideal factor is that it
requires subjective assumptions about the term structure of interest rates.
Arak, Goodman and Ross (1986) show, using simulations, that biases in the
conversion factor system and the cash market result in a positive value for the quality
option. They propose a rule for the choice of the CTD based on the computation
of an unbiased or "fair" conversion factor which uses the long-term rate instead of
the reference rate of 6% in the conversion factor formula.
Benninga and Wiener (1999) focus on the identification of the CTD when the
term structure of interest rates is flat and the delivery is allowed on a single date.
They derive an optimal delivery rule and show that the CTD has always extremal
duration. The authors argue that their rule holds when the risk-free short-term
interest rate moves according to a mean-reverting stochastic process, such as Vasicek
(1977), and when the corresponding long-run interest rate is used as the determinant
of the CTD in their rule.
More recently, Oviedo (2006) proposes an alternative method for computing
futures invoice prices called the True Notional Bond System (TNBS). The author
shows that this system is equivalent to the Conversion Factor System when interest
rates are constant at 6%, and outperforms the CFS for any other flat term structure.
The TNBS relies on the convergence of the futures price to the spot price of the
underlying bond (the notional bond) at expiration. According to the proposed
system, the futures invoice price is set to be a function of the last settlement futures
price and the bond to be delivered and is computed as the present value of the
remaining cash flows of the bond to be delivered, discounted at the yield to maturity
of the notional bond at the delivery date. Under the TNBS, the short trader is
indifferent in delivering any of the eligible bonds for any level of a flat yield curve.
Since the criterion of closeness of the futures prices to spot market prices is equivalent
8
to the criterion of minimizing the losses of delivering alternative bonds (other than
the CTD), the author compares empirically these losses computed for each of the
systems (CFS vs. TNBS). The results show that the average loss in the CFS is
more than twice the one of the TNBS. Hence, the TNBS improves the design of
the T-bond Futures contracts and consequently provides a tighter bound to the
overpricing of the CTD and the futures price as well. This system also allows to
lessen the incentives to deliberately provoke shortages of the CTD.
2.2 Valuation of the delivery options embedded in the CBOT T-
Bond futures
Models for the valuation of the T-bond futures embedded delivery options are di-
vided into two main classes, namely the ones considering directly a stochastic diffu-
sion process for bond prices, and the ones modeling the interest rates.
Valuing the embedded delivery options through the modeling of bond prices was
the initial approach proposed in the literature, and it relies on the assumption of
a joint lognormal distribution process for the bonds that constitute the deliverable
setl. This approach has been criticized as unsuitable for interest-rate derivatives.
Assuming that the bond prices follow a geometric Brownian motion is deemed un-
satisfactory since interest rates exhibit mean reversion while geometric Brownian
motion does flot. In addition, such models imply constant bond price volatility,
which is unreasonable since price volatility must be decreasing towards zero as the
bond approaches maturity. They are also inconsistent with the bond price equal-
ing one at maturity. Moreover, when considering a realistic number of deliverable
bonds, multidimensional bond price processes become difficult to implement since
the variance-covariance structure amongst ah l bond returns must be estimated. Fi-
nally, while the valuation of the quality option should essentially rely on the vari-
ability of interest rates, this class of models assumes a flat term structure of interest
rates and no marking-to-market, reducing therefore the futures contract to a forward
I See for example Gay and Manaster (1984), Boyle (1989) and Hemler (1990).
9
contract. In fact, Jarrow and Oldfield (1981) show that, under a flat term structure
of interest rates, the price of a standard futures contract on a financial asset equals
the price of its associated hypothetical forward contract.
The second class of models consisting in modeling the interest-rates term struc-
ture overcomes the shortcomings of the former by taking into consideration the
stochastic nature of interest rates in an equilibrium framework 2 . This approach also
presents the advantage of increased tractability since it requires a smaller number
of inputs.
The following sections present papers focusing on the quality and timing options
respectively.
2.2.1 Valuation of the quality option
The valuation of the quality option embedded in T-bond futures contracts has been
the subject of a substantial volume of research. Indeed, because of the complex
nature of the contract, it is a common practice to focus on the quality option alone
and ignore the other delivery options, by assuming that delivery can only occur at a
single predetermined date. Researchers also generally ignore the marking-to-market,
thus valuing futures contracts as if they were forward contracts. Moreover, the qual-
ity option, whose value arises because of the conversion factor system shortcomings,
is generally considered to have the greatest effect of all of the delivery options and
is therefore the most documented.
Several methods for valuing the quality option are proposed in the literature and
most of the empirical evidence suggests that its presence substantially affects futures
prices, putting downward pressure on them prior to the delivery date. Four main
estimation techniques are proposed, namely the exchange option pricing formula 3 ,
the terminal payoff approach4 , the implied option value 5 and the switching option
2 See for example Ritchken and Sankarasubramanian (1992, 1985), Cherubini and Esposito (1995), Bick (1997), Carr and Chen (1997), Lacoste (2002) and Henrard (2006).
3 See for example Gay and Manaster (1984), Boyle (1989) and Hemler (1990). 4 See for example Kane and Marcus (1986a), Hedge (1990) and Hemler (1990). 5 See for example Hedge (1988, 1990) and Hemler (1990).
10
method6 . Each one of these techniques can be applied into one of the two main
categories of models mentioned above (modeling bond prices vs. modeling interest-
rates term structure).
The first is a direct approach which employs an explicit model based on the stock
option valuation model of Black and Scholes (1973). This approach, which relies on
the exchange option pricing formula as developed by Margrabe (1978), Stulz (1982),
and Johnson (1987), treats delivery options as exchange options, i.e., an option to
exchange one asset for another.
The work of Margrabe (1978) is an extension to the Black and Scholes formula
for pricing European options. The author assumes two assets and denotes by xi the
price of the asset 1 and x2 the price of the asset 2, then the value of an option to
exchange the asset 1 for the asset 2 is a solution to a differential equation subject
to boundary and initial conditions on the assets. The European style option that
pays-off max(0; x i — x2) at the expiration date T is simultaneously a call option
on the asset 1 with strike price x2 and a put option on the asset 2 with strike
price x i . The formula has been extended to multidimensional assets (see Margrabe
(1982)). This approach presents some drawbacks. First, Margrabe's formula may
not be completely precise. Moreover, the accuracy of Margrabe's model cannot be
tested directly since no market data on the value of exchange options is available.
Furthermore, for tractability reasons, the model assumes that the underlying asset
follows a geometric Brownian motion which is, as said before, flot convenient when
dealing with interest-rate futures.
The second approach uses historical or simulated terminal payoffs to value the
quality option embedded in CBOT T-bond futures contracts. The method consists
in computing the payoff to the short from delivery of the current cheapest bond at
the expiration of the futures contract, rather than an earlier cheapest bond acquired
at the initiation of a buy-and-hold hedging strategy. This method presents the
advantage of being simple, since it is independent of any pricing model or formula.
6 See for example Barnhill and Seale (1988a, 1988b) and Hedge (1990).
11
However, it only allows pricing the quality option at the expiration date, and cannot
therefore give estimates for this option prior to the delivery date. It thus represents
an ex-post measure of the value of the quality option.
The third method, called the implied option value approach, gives an ex-ante es-
timation of the value of the delivery options by computing the excess of the forward
price of the cheapest-to-deliver bond over its futures invoice price. In fact, a strat-
egy consisting in buying forward one unit of the cheapest bond and selling futures
contracts in proportion to the conversion factor of the cheapest bond is equivalent
to buying forward one unit of the embedded delivery options. The implied option
value approach is an empirical method that allows pricing the quality option prior
to expiration without the need of any complex pricing model. However, since the
estimate obtained using this approach is implied from a pricing equation in terms of
the other observable variables, it can be seen as a residual that could include some
misspecifications present in the pricing formula. Hence, the estimate of the quality
option could represent the joint effect of any other delivery options flot taken into
account by this approach.
The fourth way of estimating the quality option is an ex-post measure provided
by generating a stream of cash fiows, or profits, that the short futures position could
accumulate by continuously rolling over the long forward position from the current
cheapest bond to the next cheapest bond over the life of futures contract. This
so-called switching option method presents the same shortcomings as the terminal
payoff method, as it is an ex-post measure and cannot price the quality option
prior to expiration. The use of this approach can lead to some upward bias in
the estimation of the quality option value that can be due to bad quotes yielding
switching profit opportunities that do not really exist.
Ah the above described approaches rely on distinct assumptions, models and
data making it difficult to compare the differences in the obtained results. Besides,
since the delivery options are not traded in an organized market, one cannot assess
these approaches by simply comparing market prices to theoretical prices. Moreover,
12
as mentioned previously, flot taking into account the presence of other delivery
options may lead to measurement errors since it is hard to extract the incremental
value of a particular delivery option. Nevertheless, estimates for the quality option
are in general around 1 to 2% of the futures price, three months prior to expiration.
Some exceptions exist, such as Kane and Marcus (1986a), who report a high value
of 4.6% for the quality option at the delivery date.
We now review the papers valuing the quality option according to various meth-
ods and models as mentioned above.
The paper of Gay and Manaster (1984) deals with the valuation of the quality
option embedded in a commodity futures contract. The contract considered is the
wheat futures contract traded on the CBOT and is assumed to offer only the quality
option to the short by allowing the delivery of two qualities of the underlying asset.
The contract is assumed to be exercised on the first day of the delivery month and
the location option is ignored. It is also assumed that the term structure of interest
rates is flat, thus making forward and futures contracts equivalent. The authors
develop a futures pricing model that incorporates explicitly the value of the quality
option based on the exchange formula discussed earlier. In order to conduct an
arbitrage free hedging strategy, the short is supposed to buy one of the two assets
that is held until the maturity of the contract and simultaneously sell an exchange
option that gives the right to exchange this asset for the second. At the maturity
of the contract, the option is exercised only if the cheapest-to-deliver asset is not
the one held by the short. All these assumptions together lead to a futures price
that equals the spot price at maturity of the cheapest-to-deliver asset. The model
is then tested empirically. The data required for this investigation is observable,
except for the exchange option prices that are not traded in any market. This value
is estimated using the Margrabe exchange formula mentioned above. The results
show that the quality option in the wheat futures contract has substantial value,
representing about 2.2% of the futures price.
Kane and Marcus (1986a) use the simulated terminal payoff approach. They
13
price the quality option in the CBOT T-bond futures using 10,000 Monte Carlo
simulations of historically estimated yield-curve dynamics. They use estimated term
structures over the period 1978-1982, estimates of the volatility of those term struc-
tures as well as the set of the deliverable bonds available at that time. They find
option values ranging from 1.39 to 4.6 percentage points of par (ppp).
Hedge (1988) uses the implied option value approach and reports small values for
the quality option, three months prior to delivery. The options values are estimated
for 29 quarterly delivery cycles of the nearby contract during August 1977 through
December 1984. The author finds that the average value of delivery options is less
than 0.5% of the mean futures price.
Barnhill and Seale (1988a) evaluate exercise strategies for the switching option by
analyzing the change in the number of switches, mean profit per switch, cumulative
switching profits for three months holding periods and the variability of these profits
due to changes in the hurdle rate and the transaction costs. Empirical results show
that the average quality option value ranges from 0.28 to 1.19 ppp over the December
1977 to December 1984 period. As expected, they also find that the option value
is positively related to interest-rate volatility and negatively related to transaction
costs.
Boyle (1989) generalizes the work of Gay and Manaster (1984) to the n-dimensio-
nal case, considering several deliverable assets. The prices of these assets are as-
sumed to follow a multivariate lognormal distribution. Since the value of the forward
contract at inception is zero, and since it can be expressed as the difference between
a long call option and a short put option on the minimum of the deliverable assets
with strike price equal to the price of the futures contract, the futures price can
be obtained straightforwardly once the call option on the minimum of n assets is
evaluated. The author carnes out several numerical simulations and reports values
of the quality option ranging from 1% to 11% depending on the number of deliv-
erable assets (2 to 50) and the correlation between them. Boyle also shows that
the quality option has significant value, even when the correlation between assets
14
is very high (0.95 to 0.995). Moreover, the author points out that while the joint
lognormal distribution assumption is reasonable for commodities, it is not suitable
for interest-rate derivatives.
Hemler (1990) estimates the quality option embedded in the CBOT T-bond fu-
tures contracts over the sample period 1977-1986 using the first three estimation
techniques mentioned above and compares the results. The use of the n-asset ex-
change option pricing formula gives option values averaging from 0.7 to 1.2 ppp
while the payoff obtained by switching from the CTD bond three month prior to
delivery to the one cheapest-to-deliver at time of delivery is on average less than 0.3
ppp. Finally, the quality option is found to be worth less than 0.2 ppp three months
prior to delivery using the implied value approach.
Hedge (1990) develops three empirical estimates of the value of the quality op-
tion. Three months prior to expiration, the average value of the quality option price
estimate using the buy-and-hold strategy is 0.329 ppp. The results of his study are
thus consistent with the findings of Hemler (1990). When using the implied option
value approach, the average value of the quality option is about 0.464 ppp 12 weeks
prior to delivery. The author also applies the rollover strategy in order to evaluate
the quality delivery option. He obtains a mean payoff of 2.075 ppp. He argues that
the discrepancy observed relatively to his other two estimates obtained using the
previous empirical methods is mainly due to the nonsynchronous spot-futures data.
Ritchken and Sankarasubramanian (1992) use a single-factor Heath-Jarrow-Mor-
ton (1992) (HJM hereafter) arbitrage-free pricing model with a Vasicek (1997)
volatility structure to value the quality option. They provide an exact closed-form
solution for the prices of futures on pure discount bonds. Futures on coupon-bearing
bonds are priced numerically. They also study the impact of the margin require-
ments as well as the volatility level on the quality option value and show that this
option is worth between 0% and 1% of the futures price, three months prior to
expiration.
Ritchken and Sankarasubramanian (1995) extend their earlier work to a two-
15
factor HJM pricing model. The T-bond futures is priced numerically relative to an
initially specified term and volatility structures for the forward rates. The authors
argue that the use of a two-factor model generates higher values for the quality
option relative to the single-factor framework. They report a value for the quality
option ranging from 0% to 5% of the T-bond futures price, three months prior
to expiration, depending on the market conditions specified through the volatility
parameters. The authors highlight the fact that the magnitude of the quality option
value depends critically on the composition of the deliverable set.
Cherubini and Esposito (1995) derive a closed-form solution for the futures con-
tract in the presence of the quality option alone when interest rates move according
to the Cox, Ingersoll and Ross (1985) (hereafter CIR) model. They argue that un-
der the risk-neutral probability measure, futures prices are equal to the discounted
expectation of the final payoff. The authors begin by deriving an analytical solution
for the prices of futures on zero-coupon bonds, since discount bond prices can be
expressed in closed-form and the distribution of interest rates is also known under
the CIR term structure. Then, the authors express the price of a futures on a basket
of coupon bonds as being the expectation under the risk-neutral probability of the
minimum over the equivalent futures prices at the maturity of the contract. Equiv-
alent futures prices are defined as the adjusted prices of the underlying deliverable
bonds at expiration, where the adjustment occurs by dividing each bond price by its
conversion factor. Since the futures prices of coupon bonds are simply a linear com-
bination of futures prices of the discount bonds constituting each single deliverable
bond, the authors partition the interest-rate domain where the CTD is constant
and price the contract on each subset. Consequently, the futures price is the sum of
the expectations of the equivalent prices over the partitioned domain. A numerical
procedure is however necessary to find the "cross-over" points which are the points
where a change in the CTD is observed. Once the futures are priced, the authors
value the quality option as being the futures mispricing relative to the current price,
which is the minimum over the expectations of the equivalent futures prices. The
16
paper studies the behavior of the quality option value with respect to different de-
livery windows (extremal maturities of the deliverable set) and with respect to the
coupon rate.
Bick (1997) derives an analytical futures pricing formula under the single-factor
Vasicek term-structure mode!. The author uses a partial differential equation val-
uation approach and the formula is developed for futures on zero-coupon Treasury
bonds.
Carr and Chen (1997) develop pricing models for the CBOT T-bond futures
in the presence of the quality option under one- and two-factor CIR interest-rate
models. In the first case, the state variable is the short-term interest rate. The state
space is augmented in the second case to take into account a long-term interest-rate
factor since the underlying basket includes long-term bonds. The futures pricing for-
mula in the one-factor model is very similar to the one developed by Cherubini and
Esposito (1995), accounting for multiple possible changes in the CTD. No procedure
is however suggested to identify the points of the interest-rate space at which the
CTD changes. An empirical study is conducted to test for the proposed two-factor
CIR mode!, for which no closed-form prices can be obtained. The identification
of the CTD regions at maturity is performed through the simulation of equivalent
bond prices (adjusted using the conversion factors) yielding the CTD for various
interest-rate regions. Unlike the case of the single-factor model, for which almost
no "cross-over" rates are reported, contracts in the two-factor model have at least
three possible bonds to deliver. Theoretical futures prices are then computed and
are compared with observed prices over the period 1987-1991. Results show that the
magnitude of the quality option is large (consistent with Ritchken and Sankarasub-
ramanian (1995)) and that using the cost-of-carry model results in a considerable
mispricing compared to the two-factor CIR model.
Lacoste (2002) focuses on the quality option alone, and starts by deriving a
rule for the identification of the CTD when the term structure is assumed to be
flat. This rule consists in delivering the highest duration bond when the interest
17
rate is higher than the coupon rate of the notional bond, and lowest duration bond
in the opposite case. The author points out that a change in the CTD occurs
generally when the current interest rate is in the neighborhood of the reference
coupon rate. In the case of stochastic interest rates, the author shows that a bond
with intermediate duration could be optimal to deliver, but however considers only
the two bonds with extremal durations as candidates for delivery and prices the
contract in an exchange option formula fashion. The quality option is found to be
increasingly valuable as the current interest rate approaches the reference coupon
rate. The author also distinguishes between futures and forward contracts depending
on whether marking-to-market is taken into account or flot. An adjustment factor
applicable to the forward price is proposed to obtain the futures price. The author
reports a value of 40 basis points for the in-the-money exchange option.
Henrard (2006) proposes a semi-analytical formula for the futures price under
the HJM one-factor model using the same set-up as Cherubini and Esposito (1995).
The author uses the expression of the zero-coupon bond price under the HJM term
structure to price the coupon bonds, which are finally used to compute the equivalent
futures prices at the expiration of the contract. The futures price is hence the
expectation under the risk-neutral probability measure of the computed equivalent
prices' minimum. Since the futures formula can be expressed for only one bond, one
needs to know what are the CTD bonds over the considered interest-rate space. The
futures price will be simply the sum of the expectations over the subsets where the
CTD is flot the same. As in Cherubini and Esposito (1995), a "cross-over" points
finding numerical procedure is thus needed to identify the CTD bonds. The author
argues that the number of bonds that could be CTD's ranges between two and five,
even if the basket is very large.
2.2.2 Valuation of the timing options
The timing option offered to the short trader of a CBOT T-bond futures contract
allows for the delivery of the CTD bond on any business day during the delivery
18
month, according to the delivery sequence described in the introduction. In the case
of the CBOT T-bond futures contract, exercising or not the timing option depends
on a trade-off between holding the underlying bond to receive the associated coupon,
and delivering the bond to invest the futures price at the risk-free rate. Hence,
postponing delivery is optimal to the short whenever the bond payouts exceed the
risk-free rate, and the larger are these payouts, the lower is the value of the option
to deliver early.
Recall that the CBOT T-bond futures contract offers three timing options to the
short, namely the accrued interest option, the wild card option and the end-of-the-
month option. The accrued interest option allows the delivery of the bond on any
business day of the delivery month (this option is often identified with the timing
option, when the other two are disregarded). The wild card option or the six hours
option recurs on every trading day of the delivery month and arises because the short
has the right to announce delivery on the position day at 8:00 p.m., based on the
invoice price fixed immediately after the futures market closes at 2:00 p.m.. Since the
spot market remains open until 8:00 p.m., the short can use the information about
spot prices between 2:00 and 8:00 p.m. to decide whether to declare his intention
to deliver or flot. The end-of-the-month option allows the short to deliver during
the last seven business days of the delivery month. The reason why this period is
of interest is that the futures market closes seven days before the last delivery day,
while the bond market is still open and the delivery remains possible. Hence, the
invoice price based on the settlement price on the last futures trading day is frozen
for the last seven business days and the short can use the information on spot prices
during that period for making his delivery decision.
One can only observe the joint effect of all the delivery options embedded in
futures contracts and individual effects are hard to disentangle. Few empirical stud-
ies focus on the pricing of timing options. Instead, most of the research examines
various trading policies and finds if these policies can be sources of profits.
Gay and Manaster (1986) derive the optimal delivery strategy (when and what
19
to deliver) for the T-bond futures contract and compute empirically the payoffs
associated with this policy during the period 1977-1983. They find that the optimal
strategy can generate positive and statistically significant returns, implying that
delivery options are undervalued and futures prices are therefore 'toc. high'. Indeed,
if futures prices did take into account ail the implicit options, there would be no
policy that could produce positive expected returns. The authors also find that the
optimal delivery strategy they derive differs substantially from the observed delivery
strategies, which means that the short traders did not exploit this mispricing by
skillfully exercising the delivery options. Recall that the paper examines the current
contract with an its implicit options. The authors derive a general rule for the
valuation of the wild card option, that depends critically on the conversion factor
of the CTD (fc.m.): when spot prices fall below the invoice price during the wild
card period, the short should mark-to-market if the conversion factor of the CTD
is less than 1 and deliver otherwise. The opposite rule must be applied whenever
bond prices rise during the 6 hours wild card period. The authors also examine the
end-of-the-month option, with its two sources of profit, namely the waiting profit
due to delaying delivery and the quality profit due to switching to the CTD bond
during the hast seven business days of the delivery month. There is no quality profit
if the CTD bond remains the same during the 7 days period, and the waiting profit
is zero if the price of the CTD does not change in that period.
Kane and Marcus (1986b) focus on the wild card option alone by developing
a valuation model and computing option value estimates, as well as deriving rules
for its optimal exercise. The authors develop a valuation formula for the wild card
option. They show that this option is similar to an American option, with the
special feature of readjusting the strike price every day to put the option out-of-the
money. Their results indicate that the option value is 20 cents at the first day of
the delivery month and approaches zero at the end-of-the-month. They assume that
the CTD can be identified in advance, which is not the case in a risky environment,
and also assume a lognormal process for the price of an eligible T-bond, which is
20
inconsistent with the bond's terminal value.
Arak and Goodman (1987) use the Black and Scholes (1973) model to examine
the wild card and the end-of-the-month options. Their analysis reports numerica1
estimates for the options averaging less than 0.3 percentage points of par during the
period 1984-1986. Peck and Williams (1990) use logistic regression to explain the
timing of deliveries during the month via some economic variables.
Boyle (1989) defines the timing option to be the value of delayed delivery and
shows that this option has zero value when no other option such as quality and/or
location options is accounted for. If only one asset is eligible for delivery and if only
one deliverable location is considered, then early delivery is always optimal. Only
the joint effect of delivery options can provide some value to postponing delivery.
The analysis assumes non-stochastic interest rates and does not apply directly to
T-bond futures contracts.
Gay and Manaster (1991) is the first among the rare studies that develop an
analytical framework for pricing T-bond futures in the presence of all the implicit
delivery options. Their model accounts simultaneously and explicitly for the quality
option, accrued interest option, wild card option as well as the end-of-the-month
option. Empirical tests are performed in order to validate the model and the nu-
merical methods used for implementation. The authors consider three families of
payoff functions defining the payoffs to the optimal policy, and corresponding to the
end-of-the-month period, the wild card period and prior to the wild card period.
Equilibrium requires that ail expected payoffs, conditional on available information,
equal zero, which brings analytical and numerical complexity to the model. To al-
leviate this problem, the authors propose some simplifying approximations and an
alternative procedure based on empirical tests. First, instead of jointly determining
futures prices and bond spot prices, the last prices are assumed to be exogenous and
equal to the observed bond prices. Second, the authors solve the equilibrium con-
ditions separately rather than simultaneously, thus introducing some bias. On each
futures trading day, a parameter representing an expected futures price is estimated.
21
This estimation is done by backward induction using Dynamic Programming. The
authors assume that deliverable bond prices follow a joint lognormal distribution,
where parameters of the joint density are estimated from the data. For reasons of
computational tractability, the authors restrict the analysis to the four cheapest-
to-deliver bonds. This restriction results in a downward bias in the value of the
delivery options. 10,000 Monte Carlo simulations are then performed to generate
bond prices. The authors distinguish between results obtained using the bid and
ask prices of the deliverable bonds needed for the estimation. Comparison between
estimated futures prices and observed prices is then provided. Despite the fact that
the estimates based on ask bond prices are found to be doser to the observed prices,
the authors suggest that bid-based futures prices should be considered as better
estimates for the equilibrium prices. One likely explanation is that observed futures
prices are upward biased due to an understatement of the delivery options as dis-
cussed earlier in Gay and Manaster (1986). The authors investigate also the impact
of the presence of delivery options on the futures prices by computing the difference
between the forward price of the CTD bond and the estimated futures price. This
difference is found to be positive in spite of the overestimation of futures prices.
Nielsen and Ronn (1997) propose a discrete-time model with two discrete under-
lying factors, the long and short interest rates, and use a bi-trinomial model to price
the contract. They assume that the CTD coincides with the one obtained from a
constant interest-rate model, which is flot the case in a risky environment.
Chen and Yeh (2004) develop an upper and a lower bound for T-bond futures
prices. The authors argue that the accurate valuation of delivery options, and
especially the timing options, are prohibitively expensive, and one cannot avoid the
use of lattice methods for pricing them. The goal of their study is to provide traders
with an efficient range for futures prices. It is therefore of substantial interest
to derive upper bounds for these options, which provide a lower bound for the
futures price. Upper and lower bounds give a crude conservative estimate for the
T-bond futures price. The upper bound is obtained from the closed-form cost-of-
22
carry model. Analytical lower bounds under one- and two-factor CIR models of
the term structure are obtained. Empirical tests are then performed with weekly
futures prices for the period 1987-2000. The proposed upper bound is on average
2% higher than the actual futures price while the lower bound is found to be 2%
lower. Moreover, the price of the contract with only the quality option is found to
be a good approximation and a tighter upper bound to the actual futures price, as
the value of the timing option is quite small (70 basis points).
Hranaoiva, Jarrow and Tomek (2005) study the timing and location options
in the CBOT corn commodity contract during the 1989-1997 period. The timing
option embedded in this contract is similar to the one in the T-bond futures con-
tract, as delivery is allowed any time during the expiration month, while the futures
trading stops one week before the last possible day of delivery. The authors show
theoretically that early delivery is always optimal if the only option in the contract
is the timing option, which is consistent with Boyle (1989). However, this is not
the case when a joint effect of two options is taken into account. When examining
the timing option alone, the authors consider a binomial discrete time model. A
trinomial model is used for analyzing the combined timing and location options.
The futures price of the contract with the timing option is determined by back-
ward induction, recognizing that the value of the contract is reset to zero every day
through marking-to-market. The reported empirical results show that the timing
option, when considered alone, has some positive value, contrary to the results of
Boyle. This option constitutes 0.2% of the average futures price on the first delivery
day. When the location option is added, the reported value of the joint options
is much smaller, representing 0.04% of the futures price. In addition, the authors
study the impact of these delivery options on the basis behavior and find it to be
statistically important.
23
2.3 Pricing embedded delivery options in other bond futures mar-
kets
Lin and Paxon (1995) implement a one-factor Gaussian HJM model through a
discrete-time binomial grid to price the German Bund futures contract traded on
the London international Financial Futures and Options Exchange (LIFFE) from
December 1988 through November 1991. They find that the quality option is only
worth an average of 9 basis points of contract size, three months prior to delivery.
Yu, Theobald and Cadle (1996) examine the quality option embedded in the
Japanese long-term Government Bond futures (JGB futures) traded on the Tokyo
Stock Exchange. The futures contract is based on the 10-year, 6% coupon rate
JGB. Unlike the US Treasury bond futures, which embed both a quality and a
timing option, the JGB futures includes only a quality option. The bonds eligible
for delivery are JGBs with a remaining time to maturity ranging between 7 and 11
years. The authors use a term-structure approach based upon a two-factor HJM
(1990) model. The option value is found to range from 0.12 to 0.2 ppp. The authors
also find that accounting for the quality option can improve the performance of
optimal hedging strategies.
Chen, Chou and Lin (1999) study empirically the quality option impficit in the
JGB futures. The authors use the Hull and White (1990) term-structure model
implemented through a discrete trinomial tree approach to price the quality option.
Their study includes contracts from June 1990 to march 1994. The average value
of the quality option implicit in the JGB futures, three months prior to maturity, is
0.021 ppp, which represents a small value relative to other empirical studies. The au-
thors argue that this non-significant value for the quality option could be attributed
to many factors, such as the term-structure estimating process, the contract's char-
acteristics, the quality option estimating process or the low Japanese interest-rates
volatility compared to the US market.
Schulte and Violi (2001) examine the impact of futures contracts on the European
spot market. They argue that the high level of activity in EUREX contracts has
24
increased the risk of a shortage in the CTD and that such market squeezes can be
avoided when the CTD at expiration is not easily forecastable.
Buhler and Dullmann (2003) study theoretically and empirically the quality
option embedded in multi-issuer bond futures contracts. These contracts differ from
traditional ones in that they allow the delivery of bonds of different issuers, thus
offering to the short what the authors call the issuer option. Their paper proposes a
valuation model for the futures and its embedded option based on a two-factor affine
credit-risk model. In addition, the authors develop two issuer-dependent conversion
factor systems that account explicitly for the credit risk of the deliverable bonds.
They consider a symmetric delivery basket that consists of four default-free and four
credit-risky bonds with coupons of 4% and 8% and time to maturity of 8.5 and 10.5
years. The reported quality option value ranges from 0.22 and 0.44 ppp depending
on the conversion factor system used. Moreover, an empirical study involving futures
on German and Italian government bonds is conducted to analyze the impact of the
proposed conversion factor systems on the hedge efficiency of the futures contracts.
The German bonds are assumed to be the default-free ones.
Merrick, Naik and Yadav (2005) examine the strategic trading behavior of major
market participants during an attempted delivery squeeze in a bond futures contract
traded on the LIFFE. The authors show how squeezes are accompanied by severe
price distortions. They also show how the market differences in the penalties for
settlement failures in the cash and futures markets create conditions that favor
squeezes.
Balbas and Reitchardt (2006) focus on pricing theoretically the quality option
embedded in the EUREX German Treasury bond futures. The authors value the
quality option through the use of a static portfolio that replicates this option. The
authors show that the sale of a futures contract can be replicated via the sale of
n futures contracts and the simultaneous purchase of n quality options, where n is
the number of bonds constituting the deliverable basket. Empirical results involving
the contract maturing in December 2002 show that the quality option three months
25
prior to expiration is worth 2% of the futures contract nominal value.
Ferreira De Oliveira and Vidal Nunes (2007) derive an approximate and quasi-
analytical solution for the valuation of the quality option under the multi-factor
Gaussian HJM (1992) framework. Their work can be considered as extension to
Ritchken and Sankarasubramanian (1992,1995). The authors use an approximation
that allows them to write the futures price as a univariate and deterministic integral.
A Monte-Carlo experiment is carried out in order to test for the accuracy of the
proposed approximate solution. The model is then tested empirically to study the
magnitude of the value of the quality option embedded in the German Bund futures
contract traded in the EUREX. The underlying deliverable assets are bonds issued
by the German government with maturity ranging from 8 to 11 years. This market
uses the same conversion factor system as the CBOT and the reference coupon rate
is also fixed to 6%. An interesting feature of this contract is that it offers to the short
only one embedded delivery option, namely the quality option, as delivery must take
place on the 10th calendar day of each delivery month. Empirical results over the
period 2000-2004 show that the quality option's value is very small, accounting for
less than 5 basis points of the futures prices. The authors argue that such a finding
can be due to the homogeneity and the small number of deliverable bonds in the
eligible set. Another given possible explanation for this small value is the off-set
of the majority of the contracts prior to expiration, which reduces the need for the
short position to put a downward pressure on the futures prices.
The previous studies of the quality and timing options use simplifying assump-
tions on the dynamics or the strategies. Most of the studies focus on the quality
option alone and do not take into account the delivery sequence or the timing option
by assuming a single possible delivery day. Some other studies consider a stochas-
tic diffusion process for bond prices, thus ignoring the stochastic nature of interest
rates. Researchers also generally ignore the marking-to-market. In the next section,
we propose a model for the pricing of the CBOT T-bond futures in a stochastic
interest-rate framework when the interaction of all the embedded delivery rules and
options are taken into account.
26
3 Pricing the CBOT T-Bond Futures
In this section, we propose a model and a pricing algorithm that can handle ex-
plicitly and simultaneously ail the delivery rules embedded in the CBOT T-bond
futures, in a stochastic interest-rate environment. To do so, we consider a continu-
ous time model with a continuous underlying factor, the risk-free short-term interest
rate. We assume that this rate moves according to a Markov diffusion process that
is consistent with the no-arbitrage principle. Our pricing procedure is a backward
numerical algorithm combining Dynamic Programming (DP), approximation by fi-
nite elements, and fixed-point evaluation. In this context, the DP value function
is the value of the contract for the short trader (which is reset to 0 at the set-
tlement dates) and the DP recursion is given by no-arbitrage pricing (Elliott and
Kopp, 1999). Under a given assumption about the stochastic evolution of interest
rates, the numerical procedure output may be summarized into tluee results. The
first gives the theoretical futures prices at settlement dates. The second gives the
delivery position strategy (deliver or flot) and the third identifies the CTD on the
notice day, given the futures price at the last settlement date. Ail three results are
functions of time and current interest rate.
This section is organized as follows. Section 3.1 defines our notation. In Section
3.2, we describe an associated T-bond forward contract. Section 3.3 presents the
model and the Dynamic Programming formulation for the value of the contract.
Section 3.4 describes in details the numerical procedure. In Section 3.5, we report
on numerical results obtained under both the Vasicek (1977) and CIR (1985) models
for the short-rate process.
3.1 Notation
We consider a frictionless cash and T-bond futures market in which trading takes
place continuously. Denote
E H the expectation under the risk neutral measure Q;
28
• (c, M) E 0 an eligible T-bond with a principal of 1 dollar, a continuous coupon
rate c, and a maturity M, where the set 0 of eligible bonds is known at the
date the contract is written;
• fa,' the CBOT conversion factor corresponding to the T-bond (c, M), where
the set {fcm : (c, M) G el is known at the date the contract is written:
f cm = PV (tu , c, M, 6%) ; (5 )
• {rd a Markov process for the risk-free short-term interest rate;
• p(r,t, 7- ) the price at t of a zero-coupon bond maturing at T > t when Tt = r
under the process {Tt}
T
Pfr,t,T) = E [exp(— I rudu) I ri = 1; t
• p (t, c, M, r) the price at t of the eligible T-bond (c, M) when rt = r under the
process {rd m
p (t, c, M, r) = c f p(r, t, u)du + p(r,t, M); t ( 7)
• gi (r) the price of the T-bond futures at t when rt = r;
• g;` (r) the fair settlement price of the T-bond futures at t when r t = r.
3.2 Price and CTD of an associated T-Bond Forward contract
It is convenient to compute first the theoretical settlement price of a forward contract
on T-bonds and characterize its CTD. The contract modeled is not the CBOT T-
bond futures but an associated forward contract, where only the embedded quality
option is considered, the marking-to-market is ignored and delivery occurs at a
single delivery date T. This type of contract is widely documented in the literature.
Examples include Livingston (1987) and Benninga and Wiener (1999).
(6)
29
For a contract written at to, the CTD (c*, Mt) verifies
gto (r)P(r,to ,T)fc. m. — JT
p(r,to,u)du p(r,to, M*)
= max {gt% (r) p(r, to, T) — c p(r,to, u)du p(r, to, M)} =- 0, (8) (c,m)ee 7'
since the theoretical price of the hypothetical T-bond forward price sets the value
of both parties to O at to. This yields the hypothetical T-bond forward price
go (r) min c rt udu+ rt M (c,m)Ee p(r, , to , T) fM
Minimizing (9) over the set of eligible T-bonds yields the CTD
CfA1 , p(r,to, u)du p(r,to, M) (c* , M*) E arg min
(c,m)ee p(r,to,T) f cm
Notice that the set of conversion factors do not make the eligible T-bonds equal
for delivery, unless rt = 6% for all t > to. Notice also that the CTD computed at to
and r in order to determine the price of the forward contract is not necessarily the
same as the one which will be delivered, unless r is deterministic. It is easy to show
that, for that contract, the CTD always bears an extremal coupon.
We give here the theoretical forward price and we characterize in Table I the
CTD under a flat term structure of interest rates. It should be noted that, in this
case, the optimal maturity can be determined analytically.
(1 —)
e (—r(m—T+to)) min
(c,m)Ee urk ± (1
Table I: The forward contract CTD Case (c*, Mt)
r < 6% M r > 6% and c > 6% c Ni r > 6% and c < 6% c e [M, 117
(9)
(10)
30
This forward contract will be used to obtain a first approximation of the futures
price.
3.3 Model and DP formulation
To be consistent with the CBOT delivery rules, we consider a sequence of monitoring
dates t the lower index m = 0, ..., ï is computed in days from the date the
contract is written and the upper index h E {2, 5, 8} indicates the time in hours
within that day. Assuming that the contract is written at t0 = t , we denote the
marking-to-market dates by t n.„2 for m = 0, , n, where tn represents the last futures
trading date during the delivery month. We denote the delivery position dates by
t8 for m = n, ,ï , where tn and tw, are respectively the first and the last date of
the delivery month, 0 <n <n < Finally, the delivery notice dates are denoted
tui5 for m = n + 1, , i + 1. Our choice of monitoring dates is justified by the fact
that, within the delivery month, it is better for the short trader to wait until 8:00
p.m. each day to decide whether to take or flot a delivery position. Moreover, we
assume that the delivery notice date coincides with the actual delivery, since this
does flot change the (expected) value of the contract.
Our DP model determines the value of the contract for the short trader at
each monitoring date, as a function of the interest rate at the current and last
settlement dates, assuming that the short trader behaves optimally. We obtain the
fair settlement price by making the value of the contract null for both parties at the
settlement dates.
The contract is evaluated by backward recursion in three distinct periods: The
end-of-the-month period, where no trading takes place, but delivery is still possible
(in = n, the beginning of the delivery month where trading and delivery are
both possible (m = n, n), and the period before the delivery month, where no
action is taken by the short trader, but the settlement price is adjusted every day
(m -= 0,
31
3.3.1 End-of-the-month Period
Recall that during the last seven business days before maturity, trading on the T-
bond futures contracts stops while delivery remains possible, based on the settlement
price at the last settlement date, indexed by m'. If an intention to deliver is issued
at the delivery position date t 8m , for m = n, we define the expected exercise value
vine (r', r) and the actual exercise value v ma (r', r) for the short trader, as a function
of the interest rate at the last settlement date, denoted r', and at the current date,
denoted r, as follows:
ft5m+ 1 r du) vme (r', r) =- E [(vina (r' , r tm5 +D e te,, ' I rte =-- ri , (12)
vina (r' , r) = max {gm,(r') f cm — p (t m5 +1 , c, M , r)} , (13) (c,m)ee
where = n is the last settlement date and gm, (r') is the price of the futures
settled at m' when rm, = r'.
Otherwise, if the short trader decides not to deliver at t m8 , for m = n,
we define the holding value vmh (r', r), which is computed by no-arbitrage to be the
expected value of the future potentialities of the contract and given by (16) below.
The short trader will of course issue an intention to deliver at (t 8m , r', r) if and only
if
vine(r', r) > vmh (r', r). (14)
The value function for the short trader at t8m , for m = n, is thus defined
recursively by:
vm8 (r', r) = max {vme (r', r) , vmh (r', r)}
[ ftM-I-1 r i u clu ,
vmh V, r) = E vni8 +1 (r', re ±i ) e— tL ren, = r
(15)
(16)
(r', r) = v7-,e (r', r) , (17)
32
and the settlement value for the short trader at (t 2 „ r 2 ) is the discounted expected
Tri t
value at tm8
v2 , (r') = E v8 , e I rtm2 = , rn mi
- forni r„du
t8
(18)
where m' = n. Notice that equations (12) - (18) define a mapping from the space of
functions : R—>1R to the space of functions vm2 , : R-->I11, but we did not make this
dependency on gin, explicit to alleviate the notation. Moreover, the settlement value
at m' and r' is defined for any settlement price function (ri), constant during the
end-of-the-month period, which can be written
v m2 , (rt) Frn, (g) (7.1 (19)
where Fini(g) : 1[8—>IR is a function defined by the Dynamic Program (12) - (18) with
g =- gini(r') and m' = n.
However, the settlement price at t 2m, should be selected so that the value to both
parties is 0, taking into account the timing and quality options. Consequently, the
fair settlement price at t 2m„ denoted gm* is a function of the settlement date
interest rate such that:
vm2 , (r') = Fm, (‹,)(r1 ) = 0 for all r', (20)
where m' = n.
3.3.2 Delivery Month
During the delivery month (m = n, ...,n — 1), the value of the contract for the
short trader may be evaluated in the same manner as in the previous section, but
the holding value must account for the interim payments at the marking-to-market
dates. Thus, the exercise value functions v me (r', r) and vma (r', r) at respectively t8m
and tm5 , for m = n, ... , n — 1 are given by (12)-(13), where m' = m. The holding
33
value at t, however, accounts for the interim payment at the next marking-to-
market date, that is,
yr% (r', r) =- E [ (gm (r') - en+1 (rtL+, )) e m
qr,
- f 8 - ru clu , + v, 1+1 (rtzn+i ) e tm I rte„ = r
[
t 2n., .. — f 8 ±' ru du
= E (gm (ri) - gm* +1 (rom+i )) e t.
since the settlement value at m 1 is the null function for a fair settlement price.
The value function at t8,2 and tn.,2 is then given by (15) and (18), with m' = m.
Finally, for each marking-to-market date t 2rn , in = n - 1, ..., ri - 1, the settlement
price function gin* ,(r) is such that (20) is verified, with m' = m.
3.3.3 Initial period
Within the time period [tc,, t2n_ 1 ], delivery is not possible, so that the value of the
contract for the short trader only involves taking into account the interim payments
in the marking-to-market account. The value function at t ni2 , for m = 0, n - 1, is
thus given by
tL — f 2 v2 (r) = E[(gm* (r) - gm* +1 (ro )) e
+, 7-clu t. 7n rn+ 1
Ir2 = = 0, (22)
where g,* is such that (20) is satisfied for m' = m, m = 0, n - 1. Therefore, the
successive settlement prices can be obtained by the recursive relation
f 2
E[e - -2mn+i (rtm2 +1 ) e f+1 rudu
t. I ro
p(r, t72n , t2m+1 ) = for all r, m = 0, n - 1. (23)
— rdu
rtr, = (21)
34
3.4 Dynarnic Programming Procedure
Equations (12)-(23) define a dynamic program which can be used to find the fair
settlement prices and the optimal timing and choosing strategies for the short trader
by backward induction. This dynamic program is defined on the state space {(r', r) :
r' > 0, r > 01 and does flot admit a closed-form solution, even for the most simple
case where the interest rate is assumed to be constant. In this section, we describe
a numerical procedure for the solution of this dynamic program. Three specific
numerical problems must be addressed:
• the optimization in (13) which involves the price of the eligible bonds according
to the underlying interest-rate model,
• the computation of the expectations in (12), (16), (18), (21) and (23) of func-
tions which are analytically intractable,
• the determination of the root of (20).
The numerical procedures consists in
• finding the CTD by an appropriate search over the eligible set according to
the properties of the bond prices,
• approximating the functions vn,8 (g, -) and gm* (•) by expectations of linear finite
elements interpolation functions,
• finding the fair settlement price as a fixed point of a contraction mapping.
3.4.1 Optimization Procedure
Finding the CTD at m, r', r consists in solving the following:
vina V, = max (ri ) f cm — c (c,M)E0 m+1
(24)
35
where the finite set of eligible bonds and their conversion factors are fixed of the
signature of the contract and gm, is given. The function to be maximized is
linear in c, so that the optimal coupon is extremal and given by either c -z min c
or max c in the set of eligible bonds. If an analytic expression for p (r,t,r)
is known, it is straightforward to check the properties of the projections of the
function to optimize on c = and c = C. If these are convex, simple inspection of
the extremal maturities will yield the CTD. If either one is concave, a line search for
fixed and/or c can be performed. Otherwise, since the number of eligible bonds
is fixed, an enumeration of ah l eligible bonds with extremal coupons will yield the
CTD and the value of vma (r', r). Notice that, while the value of vnia (r', r) can be
obtained with as much precision as the price of a given bond for any r', r and 9m',
it cannot be expressed in closed-form.
3.4.2 Interpolation Procedure
The interpolation procedure consists in approximating an analytically intractable
function, the value of which is known at a finite number of points, by a piecewise
linear continuous function.
Let g = {ai, ,aq } be a grid defined on the set of interest rates, with the
convention that ao -= —co and aq+1 = +co. Given a function h : g -› R, the
interpolation function : R --> R is given by:
r < ai+i) , for all r R, (25) i=o
where I is the indicator function and the coefficients a i and 13, are obtained by
matching h and h on g and extrapolating outside of Ç, that is
ai±j h(ai) — aih(ai+i ) =
— ai h(a i ) — h(ai)
ez = h( = 1, ...q — I,
— ai
(26)
(27)
36
and ao = al, eo = /31, 0q = aq-1, f3q =
3.4.3 Expectations of Interpolation Functions
Define the transition paramet ers
and
B
E
E
[I (ai <TT < ai+i) r,du
tr [r, I (ai < < ai+i) e — f ru du =
rt = ak ] (28)
, (29)
where to < t< T, k= 1, . . . , q, and i= 0, , q.
We assume that these transition parameters and the discount factor p(r,t,r)
can be obtained with precision from the dynamics of Ir t , t > tol. Notice that for
several dynamics of the interest rates, closed-form solutions exist for the transition
parameters and discount factor, as discussed in Ben-Ameur et al. (2007). Exam-
ples include Vasicek (1977), CIR (1985), and Hull and White (1990). Closed-form
formulas for the transition parameters and discount factor in the Vasicek and in the
CIR model are recalled in the Appendix.
Given an interpolation function h : R —> R, the expected value at t and rt = ak
of a future payoff h,(•) at T is given by:
Ti (t,T,ak) :_=_ E VI (r y ) e — f:- rudu I rt _ ak]
q = E [ E (ai + fi r) I (ai < r, < az+i )
i=o — ak] e — fir rudu 1 1 rt _
9 = E cEiAti[i + )3,Bkt 'Ti for ail ak c g and 0 <t < r. (30)
i=o
3.4.4 Root Finding Procedure
At a given r' and m', the root finding procedure consists in finding a constant g
such that v m2 , (ri) Fm, (g ) (ri)
37
where Fm, (g) is defined by the Dynamic Program (12) - (18), (21) with g = 9m1 (T')
Consider two settlement prices 91 and 92 such that 9 1 < 92• Since
max {91 fcm —p (tm,5 +1 , c, M, r)} < max {92 — p (tm5 +1 , c, M, r)} (31)
(c,M)E0 (c,M)Ee
for ail r and m, it is easy to show that Fm, is strictly monotone in g. Moreover,
lim Fm, (g) (r') = — oo (32)
lim (g) (r') = +Do. (33) 9--"1-00
Therefore, (•) (r') admits a unique root.
Consider the following successive approximation scheme:
g i = g° — (e) (r')
(34)
k+1 k k-1
gk — (.9k ) (11 Fm, (gk) (rt) _ Fm, (gk-1 ) (r')' k > 1, (35)
where go is given.
Now, since (•) (r') is strictly increasing, this Quasi-Newton successive ap-
proximation scheme will converge to the unique root from any starting point; for
example, a good starting point is the price at t m, of a forward contract maturing at
an appropriately chosen T (either T = 1 day during the delivery month or T = 7
days during the end-of-the-month) with a choosing option on the same basket 0
(see Equation 9). Moreover, since the number of exercise strategies is finite, it can
be shown that there exists a neighborhood of the root where (•) (?) is linear in
g, so that this approximation scheme will converge in a finite number of steps.
3.4.5 Algorithm
The algorithm consists in solving the dynamic program (12)-(23) by backward in-
duction from the last delivery position date tl on the grid Ç.
In each of the three periods spanning the contract, the main loop of the algorithm
38
consists in iteratively finding, from an initial guess, the fair settlement price at the
settlement dates, as a function of the current spot interest rate, considering ail the
delivery options. This is realized by applying, at a given marking-to-market date,
the root finding procedure on ail points of g, and then applying the interpolation
procedure to obtain the settlement price as a continuous function of the interest
rate.
The inner loop of the algorithm consists in obtaining, for a given settlement
price function, the value of the contract for the short trader at a given position
date, considering ail the delivery options, as a function of the interest rate at the
last settlement date and the current interest rate. This is realized by applying, on ail
points of Ç x g, the optimization procedure to find the CTD and the actual exercise
value on the grid. The interpolation procedure is then applied to obtain a continuous
function, and the expectation procedure is applied on the time interval between
the position and the notice dates, yielding the exercise value at the position date.
This is compared with the holding value, which is known on the grid points. The
optimal value function at the position date is then interpolated and the expectation
procedure is applied between either two successive position dates (during the end-
of-the-month period) or the last settlement and current position dates. This yields
the value for the short seller at the settlement date as a function of the interest rate,
which is null if the settlement price is fair.
The detailed algorithm is provided in the Appendix.
3.5 Numerical Illustration
In our numerical experiments, the finite set of deliverable bonds contains 62 bonds
with maturity ranging from M = 15 years to M = 30 years in steps of 6 months.
Since only the bonds with extremal coupon rates are optimal to deliver, we consider
only two coupon rates corresponding to the highest and lowest coupon rates in the
current CBOT set of deliverable bonds, namely = 7.625% and c = 4.5%. The
inception date is chosen to be three months prior to the first day of the delivery
39
month. These properties of the set of eligible bonds are summarized in Table II.
Table II: Properties of the deliverable set Min Max Step Notional
Coupon (%) 4.5 7.625 - 6 Maturity (years) 15 30 0.5 20
In our numerical illustrations, we use both the Vasicek and CIR interest-rate
process models given respectively in (56) and (63), using the closed-form formulas
(59)-(61) or (68)-(70) for the discount factor and transition parameters.
The interest-rates grid points a l , aq are selected to be equally spaced with
ao = —oo, ai -= — 8o- dv (T), aq -=- ± 8o-dy (T) for the Vasicek model, while
ai = max(0; T- — 8o-cic (T)) and aq = + 8adc (T) for the CIR model, where
dv 1 (T)= u (1— exp (-2Ts,)), (36)
dc (T) , J r—(exp(—Tk) — exp(-2Tk)) + —2k (1 — exp(—TK)) 2 (37)
and T- is the long-term mean reversion level, K is the mean reversion speed, o- is
the volatility of the short-term interest rate and T is the horizon in years (for an
inception date of three months before the first day of the delivery month, T = 1/3).
We disentangle the individual effects of each implicit option by pricing four
futures contracts embedding different combinations of these options, namely
• F1 : the straight futures contract offering no options at all and corresponding
to the case where the short trader declares his intention to deliver on the first
position day of the delivery month and delivers the notional bond,
• F2 : the contract offering the quality option alone, where the short trader
chooses on day n+1 the bond to be delivered among the deliverable basket,
• F3: the contract offering only the timing option, allowing the short trader to
deliver the notional bond anytime during the delivery month according to the
delivery sequence, and
40
• F4 : the full contract offering ail the embedded delivery options to the short
trader.
The computation of these four prices allows us to price each option alone as
well as in the presence of the other option. For instance, we compute the following
differences
• F1-F4 is the value of ail the embedded options,
• F3-F4 gives the value of the quality option in the presence of the timing option,
• F1-F2 is the value of the quality option without timing,
• F1-F3 is the value of the timing option without quality and
• F2-F4 is the value of the timing option when the quality option is offered to
the short trader.
Notice that definitions of implicit delivery options are not uniform throughout
the literature, and one must be cautious in comparing results across studies. Ac-
cording to our definition, the timing option gives the short trader the right to deliver
late on any day during the delivery month. Some papers define the timing option
as the option to deliver early in the delivery month. The small value they obtain
can be explained by the fact that delaying delivery is often optimal.
3.5.1 Convergence
We first examine the convergence properties of the DP procedure. To do so, we
record the relative error in the price of the futures contract F4 at the inception date
as the number of grid points is doubled from 10 to 1280. We report on the error
with respect to the price obtained for the best precision level (g = 1280).
Table III gives an example of futures prices, under the Vasicek model, corre-
sponding to the interest rate ro = 4% when T. = 0.05, = 0.2 and u = 0.05. Figure
2 below represents the log of the relative error as a function of the log of the distance
41
between grid points in the Vasicek model for various combinations of the input pa-
rameters. The average rate of convergence is 1.2. Notice that in many cases good
precision levels of futures prices can be reached for a relatively small number of grid
points.
Table III: Convergence of the DP futures prices (Vasicek) q Futures price Relative error CPU (sec.) 10 1.393001762 0.052609564 1 20 1.353692561 0.025098768 2 40 1.331981229 0.009207850 3 80 1.322748743 0.002292345 7 160 1.320638214 0.000697896 23 320 1.319952069 0.000178433 125 640 1.319771148 4.1372e-05 603 1280 1.319716546 5081
-4 -3.5 -3 2.5ineeP-2 -1.5 -1 -OE5
• vi
%g • ne • • • •
•
Figure 2: Convergence of the DP futures prices (Vasicek)
3.5.2 Options values
We now report on the values of the quality and timing options at the inception date
for levels of interest rates ranging from 3.5% to 8.5%. In this numerical experiment,
we use the (risk neutral) parameter values of Cases 1 and 2 given in Table IV below,
q = 600 and ro = 6%. These parameters are those of Shoji and Osaki (1996) who
estimate these models using the 1-month U.S. Treasury-Bill rate over the period
-2
-4
.s
6
42
1964-1992.
Table IV: Input data Case 1 Case 2 Case 3
Model Vasicek CIR Vasicek T. 0.062098 0.061677 0.059 K, a
0.565888 0.025416
0.545788 0.091471
0.5 0.02
It is worthwhile mentioning that the aim of this section is to give a numerical
illustration of the results obtained by the DP procedure for a chosen set of para-
meters. Therefore, the reported results are not representative and one could obtain
different values and shapes for the embedded delivery options as a function of the
interest rate if some other combinations of the input parameters or different prop-
erties of the deliverable set are considered. The sensitivity of the implicit delivery
options with respect to the input parameters is addressed in section 3.5.4.
Figure 3 compares, at the inception date, the values of the embedded quality
option (with and without the timing option) as a function of the current interest
rate for the Vasicek and CIR dynamics. For this example, we find that without the
timing option, the quality option is worth an average 0.33 percentage points of par
(ppp) for the Vasicek model while this value is 0.26 ppp for the CIR model. When
the timing option is embedded in the contract, we notice a slight increase in the value
of the quality option which is then worth an average 0.36 ppp (Vasicek) or 0.28 ppp
(CIR). These values are consistent with the literature about the valuation of the
quality option. The fact that the quality option is more valuable in the presence
of the timing option is due to the interaction and the interdependence that exist
between these two options. In fact, if the short trader enjoys both the quality and
the timing options, then at each decision time during the delivery month, he has
the opportunity to choose the cheapest-to-deliver among the deliverable basket and
benefit from bond price movements as well as switches in the CTD that can occur
during the delivery month. So, in the presence of timing, the short is offered the
quality option repeatedly, and can choose the best time to exercise it. The quality
• ,,
— guality «th timing --•- guet} wenn' neng
004 0.045 0.05 0 055 01:6 0.065 0.07 0.075 003 interna rate
165
16
3.55
3.5
3.45
3.4
3.35
3.3
3.25
43
option hos more value when combined to the timing option to reflect the price that
the short should pay to benefit more thon once from having the privilege of choosing
the CTD among the deliverable set.
Quality option at inception (Vasicek) Quality option at inception (CIR)
CIR 29
2.85
28
175
165
16
155
2.5 — gu•lity ne, timing ---• guality eithout teing
145 0.04 0.045 005 0055. 0.06 0045 07 0.075 aie
interest tate
Figure 3: Quality option values vs. interest rates at inception (CFS)
In Figure 4, we report on the evolution, during the delivery month, of the quality
option values (with timing) as a function of the interest rate at the settlement time
under both the Vasicek and the CIR dynamics. We observe that the quality option
does flot exhibit a specific behavior with respect to time to maturity or interest
rates. For example, we can see for the Vasicek dynamics that the relation of the
quality option with elapsed time is nearly flat for low levels of interest rates while this
relation becomes negative for higher levels. Also, we can see for the CIR dynamics
that the shape of the quality option with respect to interest rates differ significantly
on day 1 and day 15 of the delivery month, especially when interest rates are high.
Q.113 *Ornera rocrét ■ dey
0
5 Nive/ norlh dey
15 lb . 0 ffl
0 CE
m.,. raté 004
0.1 0,05
etre,. nit C6 C
IO
I 2
5
44
Evolution of the quality option Evolution of the quality option
during the DM (Vasicek)
during the DM (CIR)
Erolullon of Ne toratity !arion *ring ch. DN (Vadeek)
Frolutioe of the guette leen dodo, Ne DM (Ce)
Figure 4: Evolution of the quality option during the delivery month (CFS)
Figure 5 plots the value of the timing option at the inception date (with and
without the quality option) for both dynamics considered here.
Without the quality option, the timing option is worth an average 0.072 ppp
(Vasicek) and 0.12 ppp (CIR) while when the quality option is offered to the seller,
the timing option is more valuable and is worth an average 0.1 ppp (Vasicek) and
0.15 ppp (CIR). Again, because of the interaction between the options, the timing
option is more valuable in the presence of the quality option. In fact, the short can
benefit, not only from changes in the price of the CTD (always the reference bond
without the quality option), but also from switches in the CTD. It is interesting to
notice here that even if the quality option is on average more valuable, the value
of the timing option can exceed the value of the quality option, especially for low
interest rates. The timing option is observed to be always negatively related to
interest rates and this can be easily explained by the fact that, since we are valuing
the option to deliver late, when interest rates increase, the short trader can invest
the proceeds of early exercise at higher rates. Also, when interest rates are lower
than the long-term mean, one would expect them to go up and consequently lower
the cash bond prices which make late delivery optimal. The opposite effect will lead
the seller to exercise the timing option in order to avoid a general increase in the
cash price of the cheapest-to-deliver.
Timing option at inception (Vasicek)
Timing option at inception (CIR)
25 • 10 ' Vasirek
OR
45
2.5
2
Z 1.5
05
— timing ‘Rth gualgy ••• • treng enthout quakty
2
1.5
0
004 0.045 0.05 0055 0.05 0.065 001 0.005 006 interest rate
004 0045 0.04 0 055 0.06 0.055 007 0.075 004 alerter rate
Figure 5: Timing option values vs. interest rates at inception (CFS)
3.5.3 Optimal delivery strategy under the CFS
We present here some examples of the optimal delivery strategy and the associated
change in the CTD during the delivery month under the Vasicek model. We use
two sets of parameters for this illustration, namely those corresponding to Case 1
and Case 3 given in Table IV in order to compare the delivery strategies as the
position of with respect to the reference coupon rate changes. We choose to report
on the optimal strategy on days 16, 19 and 22 of the delivery month (we assume
22 business days in the delivery month). Notice that the optimal decision depends
not only on the level of interest rates at the time the decision is made (8:00 p.m.),
but also on the last settlement futures price, which is directly related to the level of
interest rates at the settlement date. We report on the optimal strategy for various
combinations of the two state variables r 2 and r8 , corresponding respectively to
the levels of interest rates at the last settlement date (2:00 p.m.) and the current
position date (8:00 p.m.). Figure 6 presents the optimal strategy on day 16 of the
delivery month (the decision to deliver or not is made on day 15 (the last futures
trading day)) for possible values of interest rates according to Case 1, assuming a
rate of 6% at inception. Notice that only a small area around the diagonal of Figure
46
6 is likely to be observed, corresponding to possible variations of the rate in the 6
hours period separating settlement and position times. This area is presented in
Figure 7 for Cases 1 and 3. We notice from this figure that early exercise can be
optimal during the delivery month and that the CTD is either the longest (Case
1) or shortest duration bond (Case 3). Early exercise during the delivery month is
driven essentially by the differential in interest rates during the wild card period. If
during this six hours period, a large move in interest rates occurs, such a move could
make it worth the short's while to deliver early, even at the cost of giving up the
valuable remaining strategic delivery options. In Case 1, early delivery is optimal
when interest rates decrease substantially during the six hours period, and the CTD
is the longest duration bond (c; M) corresponding to a conversion factor less than
one. This is consistent with the rule for exercising early during the delivery month
proposed in the literature, requiring an increase in the issue's price for bonds with
conversion factors less than one to make the wild card profitable (see for example
Kane and Marcus (1986b) and Gay and Manaster (1986)). Notice that, in Case 3,
an increase in interest rates is required to make the early delivery of the shortest
duration bond (; M) optimal.
In that context, it is interesting to compute the value of the so-called wild card
option. To do so, we price a futures contract where the short is hypothetically forced
to make the decision to deliver or not at 2:00 p.m. instead of 8:00 p.m. on each
futures trading day during the delivery month. The value of the wild card option is
computed as the difference between the prices of the full contract and the contract
without it, both in the presence of the quality option. We find that the wild card
option is worth an average 0.007 ppp when the short enjoys the quality option.
This result is consistent with the very small value reported in the literature for this
option. We also verify that, under Case 1, the optimal decision at 2:00 p.m. if the
short is not allowed to play the wild card is found on the diagonal in Figure 6.
Figures 8 and 9 present the optimal delivery strategy for both sets of the input
parameters considered in this section on days 19 and 22 of the delivery month,
3 .6% r 8 (d'Y 150f 87° .,686 r0±2crd(3months)
44
g 6%
8.4%
47
respectively (end-of-the-month). Is is worthwhile observing from Figure 9 that the
CTD is not necessarily the bond with the longest or shortest duration ( (c; M) or
( c ; M)), as is often suggested in the literature, and bonds (c; M) or (c ; M) can be
optimal to deliver. Finally, we price the so-called end-of-month option by computing
the difference between the full contract and the hypothetical contract where the last
possible delivery day and the last futures trading day coincide. We find that this
option is worth an average 0.061 ppp.
Similar results about the delivery strategy are obtained under the CIR dynamics.
Optimal delivery strategy on day 16
of the DM (Case 1)
Case 1
Figure 6: Optimal delivery strategy on day 16 of the DM (Vasicek)
Case Case 3
Case 3 Case 1
Optimal delivery strategy on day
16 of the DM (Case 1)
Optimal delivery strategy on day
16 of the DM (Case 3)
Figure 7: Optimal delivery strategy on day 16 of the DM for possible variations of interest rates (Vasicek)
Optimal delivery strategy on day
19 of the DM (Case 1)
te (3 deys Ware the end 0116e Mit y2±2ad(4days)
-61 3.6%
Optimal delivery strategy on day
19 of the DM (Case 3)
ia (38000 berce* the end t1±2ad(4days)
4.2./
Figure 8: Optimal delivery strategy 3 days before the end of the DM for possible variations of interest rates (Vasicek)
48
ye (dey 15 0116e r2±20(6hours) r9 (dey 15 0( 16e DM) r2±2gd(6hours)
4.2% ±2a
d(3m
on
ths)
î
2ad(3
mon
ths)
r2
(da
y15
of t
he
OM
) =
r
d.r) =
ro±
2ad(
3mon
= r
,±2ad
(3m
onth
s)
3.8%
6%
Exercise (c2,A71)
8.4%
r„±2ad(
3mon
ths)
Optimal delivery strategy on day
22 of the DM (Case 1)
re Ose DM = rlecredays)
Case 1
Optimal delivery strategy on day
22 of the DM (Case 3)
(Iast DM dey) = r2±2crd7mays)
Case 3
49
Figure 9: Optimal delivery strategy on the last day of the DM for possible variations of interest rates (Vasicek)
3.5.4 Sensitivity of the options values to the input parameters
In this last section, we study the impact of a change in the interest-rate model input
parameters on the quality and timing option values. The base case parameter values
are 7- = 0.06, k = 0.5 and cr = 0.02.
Notice that, for both interest-rate dynamics considered in this work, the timing
option is always downward sloping with respect to the short-term interest rate.
However, the quality option does not exhibit such a specific relation (see for instance
Figure 3).
We find that the quality option in the presence of timing is increasing with the
distance between the long-run mean reversion level T. and some rate around the
reference coupon rate. Therefore, we study the sensitivity of the quality option
value to the parameters of the interest-rate process when the long-term mean is in
the neighborhood of the coupon rate of the reference bond (the analysis is carried
out for a reference rate of 6% but we verified that the same qualitative results are
obtained for other reference rates).
50
In Figures 10, 11 and 12, corresponding to a mean reversion speed of 0.2, 0.5
and 0.8 respectively and a volatility of 0.01 for the Vasicek model and 0.02 for the
CIR model, we report on the quality option value when both the long-run mean
reversion level and the short-term interest rate vary in the neighborhood of 6%.
These figures show that for both interest-rate dynamics considered here, the quality
option value has a minimum with respect to the long-term mean when the value
of this parameter is around 6%. We can see from these figures that increasing the
speed of adjustment makes this minimum move towards the level of 6%. The same
effect is obtained as the volatility level is decreased and is illustrated in Figure 13
which represents the sensitivity of the quality option with respect to the long-term
mean for different levels of the volatility and for a high level of the mean reversion
speed (kappa=0.8) for the Vasicek and CIR dynamics. So, for simultaneous high
levels of the mean reversion speed and low levels of the interest-rate volatility, the
quality option shows a minimum around the level of 6% for the long-run mean.
This behavior of the quality option could be explained by the fact that low levels
of volatility as well as high mean reversion speed contribute to obtaining a nearly
flat term structure at 7-. In addition, for the specific case of a flat term structure
at the reference coupon level, all eligible bonds are equal for delivery and therefore
the quality option is worthless. The minimum observed in the stochastic case may
similarly be explained by the fact that the deliverable basket is the most homogenous
for that given combination of the parameters, without however being all equal for
delivery.
Figure 13 also shows that as we move away from the level of the long-term mean
for which the minimum is achieved, the value of the quality option increases since
the deliverable basket becomes more and more heterogenous. Furthermore, we see
from this figure that the relation between the quality option and the volatility could
be either negative or positive depending on the level of the long-term mean. These
positive/negative relations of the quality option with respect to the volatility can be
better seen in Figure 14. Such relations can be explained by the impact of volatility
51
changes on futures and bonds prices. If the CTD is the bond with the highest
maturity (as it is generally the case when T- is larger than the reference rate), an
increase in the level of the volatility will increase the price of the futures embedding
the quality option more than the price of the straight contract, which consequently
lowers the value of the quality option. The opposite effect of an increase in the
volatility level on futures prices is observed for levels of i less than the reference
coupon rate since, in this case, shortest maturity bonds are cheapest-to-deliver,
resulting in a positive relation between the quality option and the volatility. This is
illustrated in Figure 15 where some examples of the impact of changes in volatility
on futures prices are presented.
We also study the simultaneous effect of a variation in the mean reversion speed
and the long-term mean on the value of the quality option for a given level of
volatility. Results are presented in Figure 16 and we notice that the relation between
the quality option and the mean reversion speed is negative for levels of the long-
term mean less than the reference coupon rate while this relation becomes positive
in the opposite case. This result is the opposite of the relation we find between the
quality option and the volatility, and is consistent since an increase in the volatility
could be balanced by a reduction in the mean reversion speed.
Finally, Figure 17 illustrates that the impact of the volatility of interest rates on
the timing option differs according to parameters and interest-rate models.
CIR . ..... •
se
5.8 59 6 6.1 62 5.2 intereu rate rte
aR ..... . •
...
5
\ \ \ _4. ,\ \\„ \> s, ,
\
............
■ ..........
..•••••• .. .... ..••'—' ........ .... ....... .. , .. .•-•' -. .........../›.- 8 8 . ..... :: . « ... •
rbe
52
Quality option (Vasicek)
Quality option (CIR)
Figure 10: Quality option sensitivity to rbar and interest rates (kappa=0.2)
Quality option (Vasicek) Vasliek
Quality option (CIR)
Figure 11: Quality option sensitivity to rbar and interest rates (kappa=0.5)
tiasios6
62 6
Mame rate
53
Quality option (Vasicek)
Quality option (CIR)
Figure 12: Quality option sensitivity to rbar and interest rates (kappa=0.8)
Quality option (Vasicek)
Quality option (CIR)
Onality option (Vaeacekl
Onality eptlote (CIR1 11012
001
0.006
0.0126
0.001
0c0?
2
8 585 5.9 5.95 6 695 6.1 6.15 6.2 8 595 5.9 695 6 695 6.1 6.15 62 longaerm mem (rbar)
long tem mean (rbar)
Figure 13: Quality option sensitivity to rbar and sigma (kappa=0.8)
— rbar4 058 — rbar4.1:E2
------
0.12 0.02 0.01 0.66 005
Dl interest rate
— with qualey, sigma=0.02 — without quasty, sigma.° 02 ••• • weh qualtty, sigme3.06 - • without qualtty, sigma=0.C6
1.22
1.2
1.18
1.16
1 1.11
1.12
1.1
1.08
1.080
54
Quality option (Vasicek) Oualhy option ensieekt
/01 0015 002 0.025 003 0035 804 0015 005 0.055 0.05 MN,
Quality option (CIR) Ou lhy option ICIRe
— rttarql 093 — rbarq1 0E2
••••••,..
N. - \
004 0.06 0 05 0:1 012 011 sigma
3
002
65
6
55
4.
3.5
Figure 14: Quality option sensitivity to rbar and sigma (kappa=0.8)
0.913
0.96
094
0.92
09
0 88
0660
Futures prices (rbar=- 0.05,
kappa=0.8, Vasicek) trhat-0.05. kappn-0.8, Vasieeln
Futures prices (rbar=0.07,
kappa=0.8, Vasicek) (tha,0.07. kappa-0.8. VasIceli)
— vents quatey, si9ena4 CQ — ■inthout quality, sigma41 Ce • • • • enth qualdy, hgma41 CE - • *95012 quattly, sigma4 06 -
002 001 016 6.08
0.12 interest tate
Figure 15: Impact of the volatility on futures prices
0na6ty option Plasigeki
7
Tl 5
4
3
05 kappa
0.4 06 03 kappa
00 07
— sig=0.01 — sig=0.03
sig=008
Quality option (Vasicek) Quality option (CIR)
55
Figure 16: Quality option sensitivity to kappa and rbar
2
0°
Timing option (Vasicek)
Timing option (Vokicek)
002 0.04 006 0( 0.1 072 0.11 interagi rate
Timing option (CIR) Timing option (C1R)
3
2.5
2
1.5
01
802 002
0.04 0.05 006 007 004 009 07 roteras: rate
Figure 17: Timing option value sensitivity to sigma (kappa=0.5, rbar=0.06)
4 An Analysis of the True Notional Bond System Ap-
plied to the CBOT T-Bonds Futures
The modeling and measurement of the delivery options implicit in T-bond futures
contracts has been extensively examined in the literature. In particular, the issue
of the poor performance of the current conversion factor system (which is directly
related to the value of the quality option) bas been the subject of a substantial
volume of research. Oviedo (2006) proposed an alternative method for computing
futures invoice prices, called the True Notional Bond System (TNBS), aiming to
improve the design of the T-bond futures by better achieving the objective of de-
livering same quality bonds. The main purpose of this section is to obtain precise
values for the quality and timing options under the TNBS and to compare them
with the corresponding under the CFS, in a stochastic interest-rate framework. Our
pricing procedure is an adaptation of the Dynamic Programming (DP) algorithm
described in Section 3, giving the value of the futures contract under the TNBS as
a function of time and current short-term interest rate.
This section is organized as follows. Section 4.1 defines our notation. Section 4.2
is a general description of the TNBS. In Section 4.3, we present the pricing model
and the DP formulation. Section 4.4 describes the numerical procedure. In Section
4.5, we report on some numerical results about the comparison of options values
under the two systems for the Vasicek (1977) and CIR (1985) dynamics.
4.1 Notation
We consider a frictionless cash and T-bond futures market in which trading takes
place continuously. Denote
• PV(t, c, M, r) the price at t of the eligible T-bond (c, M) when its yield to
maturity is r
tvt PV (t, c, M, r) = c 1 exp(—r(u — t))du exp (—r (M — t)) ; (38)
57
• y the yield to maturity of the notional bond.
4.2 A general description of the True Notional Bond System
We recall here the description of the TNBS as proposed by Oviedo (2006). This
system relies on the criterion of closeness of the futures invoice prices to spot market
prices at expiration. In the TNBS, the futures invoice price is set as the present
value of the remaining cash flows of the bond to be delivered, discounted at the yield
to maturity implied by the settlement price.
In the TNBS, upon delivery at t, if the level of interest rates is r t = r, the futures
invoice price of the bond (c, M) is computed in two steps.
First, given the futures settlement price g* for the T-bond futures, one can
compute the yield to maturity of the notional bond y that makes its price equal to
g* , that is
g* = PV (t, 6%, 20, y) . (39)
Then, the futures invoice price of the bond (c, M) is given by PV(t, c, M, y), obtained
by using rate y to discount its future cash flows.
The CFS and the TNBS use the same inputs to compute the futures invoice
price, namely the futures settlement price as well as the characteristics of the bond
to be delivered. However, the functional form of the futures invoice price in the
TNBS makes ail deliverable bonds equal for any level of flat yield curves, while in
the CFS this is only achieved at the specific level of 6%. Because of that, the TNBS
clearly outperforms the CFS for flat yield curves. However, using flat yield curves
to evaluate interest-rate derivatives is of limited practical interest. The Dynamic
Program presented in the next section models the value of the CBOT T-bond futures
under the TNBS. The results will allow us to compare the performance of both
systems when the interest rate is assumed stochastic.
58
4.3 Model and DP formulation
Since under the TNBS equation (39) defines an equivalence between the futures
settlement price and the implied yield y, we use this yield as a state variable. Our
dynamic program is defined on the state space {(r, y) : r > 0, y > 0} and determines
the value of the contract for the short trader at each monitoring date, as a function
of the spot interest rate at the current date and the yield to maturity of the notional
bond as implied by the last settlement price, assuming that the short trader behaves
optimally. A fair settlement price makes the value of the contract null for both
parties at the settlement dates.
The contract is evaluated by backward recursion in three distinct periods: The
end-of-the-month period, where no trading takes place, but delivery is still possible
(m = n, ..., Ti), the beginning of the delivery month where trading and delivery are
both possible (m = n, ..., n), and the period before the delivery month, where no
action is taken by the short trader, but the settlement price is adjusted every day
(m = 0, ...,71).
4.3.1 End-of-the-month Period
If the short trader decides to deliver at a given position date, the expected exercise
value v y) at the delivery position date t n,8 and the actual exercise value v y)
for the short trader, for m = n, ..., rti, are expressed as follows:
t 5 ,
v% (r, y) = E[(42 (rts nu du)
+i , y) e tm i rte --- r , (40)
vrna (r, y) -= max {PV (tm5 +1 , c, M, y) — p (tm5 +1 ,c, M, r)} , (41) (c,N)ee
where (40) represents expectation, discounted over the time period between the
position and notice dates, of the exercise value (41) for the short trader, that is, the
difference between the futures invoice price and the price of the cheapest-to-deliver
bond.
59
Otherwise, if the short trader decides not fo deliver at t 8m , for m -= n, Ti, the
holding value v y) is computed by no-arbitrage to be the expected value of the
future potentialities of the contract and given by (44) below. The short trader will
issue an intention to deliver at (t 8m , r, y) if and only if
vme (r, y) > vmh (r, y). (42)
The value function for the short trader at t 8m , for m = n, is thus defined
recursively by:
v y) = max tvme (r, y) , v mh (r, y)} , (43) t8 - f p
y mh (r, y) = E [ +1 (rt8m±i , y) e t+1 rdu
m I rtt, = r , (44)
v-T, (r, y) = /4 (r, y) , (45)
and the settlement value for the short trader at (t 7,2 , rt2 , y) is the expected discounted
value at t8m„ where 777: = n:
2 vm, \ y) = E 8 vm, (ro t8 , - ftr r,,du
, y) e I rt2 =r . (46)
The fair settlement price gn* (r) at t such that the value to both parties is
0, taking into account the timing and quality options. To obtain it, a rate y* is
selected such that v, (r, y*) -= 0 for all r, thus obtaining a function y* (r) at t n2 ; the
fair settlement price gn* (r) at t2n is the present value of the notional bond (6%, 20)
when its future cash flows are discounted at the rate y*(r).
4.3.2 Delivery Month
During the delivery month, the exercise value functions v y) and v y) of
respectively t8in and tm5 , for ni = n, n — 1 remain the same as in the end-of-the-
month period and are given by (40)-(41), while the holding value at tm8 accounts for
60
the interim payment at the next marking-to-market date, that is,
[ tL+, \ \ _48 r.,,du
vmh (r, y) = E (PV (trn2 , 6%, 20, y) - Yr,* +1 (rtL+1 )) e m
c?,..,. +1 _ f 8 ru du + vm2 +1 (rtL+1 ) e tm i rtan, =r
t2 [
\ \ _ ftp±1 r 1
„du , = E (PV (t2m , 6%, 20, Y) - e+i (r ) tLi_, ) e rtt. . (47)
The value function at t 8m and t2m, is then given by (43) and by (46), with m' = m.
4.3.3 Initial Period
Within the time period [to, t_ 1 ], delivery is flot possible, so that the value of the
contract for the short trader only involves taking into account the interim payments
in the marking-to-market account. The value function at tm2 , for m = 0, n - 1, is
thus given by
t 2 f 2".+1 ru cht v2 (.;\ — E (g:n (r) - et+1 (rtL+1 )) e tn.̀ ) I = r = 0. (48)
Therefore, the successive settlement prices can be obtained by the recursive relation
(r) =
for ail r, m = 0, n - 1. (49) E [gm* +1 (rt 2 ) e- ft% m+i
rdu
p(r,t,tm2 +1 )
t2n+1 rqn =
4.4 The Dynamic Programming procedure
Equations (40)-(49) define a dynamic program that can be used to find the fair
settlement prices and the optimal timing and choosing strategies for the short trader
by backward induction. In this section, we describe a numerical procedure to solve
this dynamic program, which does not admit a closed-form solution, even for the
most simple case where the interest rate for ail maturities is assumed to be constant.
The optimization in (41) consists in finding the CTD by solving the following
61
expression at tn.,5 +1
max U ivi (e
— Y (u—t) — p (r,t,u)) du) + e —Y (m —t) — p (r,t, M)} . (50) (c,m)Ee t e t
The function to be maximized in (50) is linear in c, so that the optimal coupon
is extremal and given by either c min c or é a- max c in the set of eligible bonds,
according to the sign of the integral ftm (e—y(u—t) ___ p (r,t,u)) du. Since the set of
eligible bonds is fixed, a simple enumeration of eligible bonds with extremal coupons
will yield the CTD and the value of v y).
The computation of the expectations in (40), (44), (46), (47) and (49) of func-
tions which are analytically intractable is done by using interpolation functions over
a finite discretization grid over r as described in Sections 3.4.2 and 3.4.3.
We also define the grid g2 = {Yl, • • • , Yq} on the set of yields to maturity of
the notional bond, where Yi and yq are computed such that p(M, 6%, 20, ai) =
PV(t(2) , 6%, 20, y') and p(t(1 , 6%, 20, aq ) = PV(tM, 6%, 20, Y ci) -
The algorithm consists in solving the dynamic program (40)-(49) by backward
induction from the last delivery position date tl on the grid g, x g2.
We start by finding the CTD and the actual exercise values for the short trader
at the notice dates on ail the points of g, x g2, that is, for all possible interest rates
and for ah l possible futures prices represented by the yield to maturity of the no-
tional bond. For each yield to maturity of the notional bond, a linear interpolation
produces the actual exercise value as a continuous function of the interest rate. The
expected exercise values are then obtained at the position date. These are compared
with the holding values, which are known on the two-dimensional grid. Again, for a
given yield to maturity, the optimal value function of the position date is interpo-
lated and the expectation is computed between either two successive position dates
(during the end-of-the-month period) or the hast settlement and current position
dates (during the delivery month). This produces the value for the short trader at
the settlement date as a function of the interest rate and the yield to maturity of
62
the notional bond. This value should be null for a fair settlement price; at every
settlement date, a simple search on the grid g2 for a given interest rate gives, using
linear interpolation, the yield to maturity of the notional bond y*(r) that makes the
settlement value null for that interest rate. This rate is then used to discount the
notional bond's cash flows, thus obtaining the fair futures price as a function of spot
interest rate. These are then interpolated, and their expected values computed, in
order to obtain the holding value of the futures contract.
The detailed algorithm is provided in the Appendix.
4.5 Numerical Illustration
In these numerical experiments, the properties of the set of eligible bonds are those
summarized in Table II.
We apply our dynamic programming procedure to obtain futures prices at the
inception date under both the Vasicek and CIR term-structure models given respec-
tively in (56) and (63), using the closed-form formulas (59)-(61) or (68)-(70) for
the discount factor and transition parameters. We also use the input parameters of
Cases 1 and 2 given in Table IV.
The interest-rates grid points are selected as in Section 3.5. The number of grid
points q is 600.
We disentangle the individual effects of each implicit option as described in
Section 3.5 via the pricing of the four futures contracts Fi, F2, F3 and F4.
4.5.1 Option prices under TNBS
We now report on the prices of the quality and timing options at the inception date
for levels of interest rates ranging from 3.5% to 8.5% and for Case 1 and 2.
Figure 18 compares the values of the embedded quality option at the inception
date (with and without the timing option) as a function of the interest rate under
the two systems. We notice from Figure 18 that the value of the quality option shows
a minimum when the spot interest rate is at the level of the long-run mean. Recall
63
that, under the TNBS, bonds are equal for delivery for any level of flat yield curves.
Under stochastic interest rates evolving according to a mean-reverting process such
as the Vasicek and CIR dynamics, the yield curve will be doser to a nearly flat one
if the current spot interest rate is in the neighborhood of the long-term mean level.
Therefore, at this level of interest rates, homogeneity of the basket of deliverable
bonds reaches its best level in the TNBS. This is not the case for the CFS, where
the impact of the short-term interest rate on the value of the quality option varies
according to the value of the other parameters.
Quality option at inception (Vasicek)
Quality option at inception (CIR)
Vaskek la"
OR
0.01 0645 005 0.055 0.05 0.066 067 0675 08 0.085 iffieret1 rie
gler— quihy .4e eming • • • euaii *54 eming
2 e.64 0 005 005 0.055 0.06 0 065 007 0.07 1 08 0. 85 imeeem
6.5
5.5
Figure 18: Quality option values vs. interest rates at inception (TNBS)
Without the timing option, the quality option is worth an average 0.47 per-
centage points of par (ppp) for the Vasicek model while this value is 0.42 ppp for
the CIR model. When the timing option is embedded in the contract, we notice a
slight increase in the value of the quality option, which is worth an average 0.5 ppp
(Vasicek) and 0.45 ppp (CIR). The fact that the quality option is more valuable in
the presence of the timing option is due to the interaction and the interdependence
that exist between these two options. We also notice from Figure 18 that the dif-
ference between the quality option with and without the timing option reaches its
maximum at the level of the long-term mean, indicating that the interaction is the
most important at this level.
41e
Evolution of the quality option
during the DM (Vasicek)
Evalution Oualltyopdoo Ne**
0 07 0 rc 5 01e 003 15 IC
DM day W1111,n1 ufir
8 —
64
In Figure 19, we report on the evolution, during the delivery month, of the
quality option values under the Vasicek and the CIR dynamics. We observe that
the value of the quality option decreases with elapsed time, especially for levels of
interest rates in the neighborhood of the long-term mean. This is expected since
uncertainty about the CTD is reduced as we approach the last day of the delivery
month, and so should be the value of the quality option.
Evolution of the quality option
during the DM (CM) Fuoludon Quality option KIR)
eœ 007 004 cos «l'end tate day
Figure 19: Evolution of the value of the quality option through the DM (TNBS)
Figure 20 plots the value of the timing option at the inception date (with and
without the quality option) for both dynamics considered here.
Temng i•dlt ■eaLly • • • 1888n eithoul quaky
0•55 004 0045 0.06 0.058 006 0 068 0.07 0.076 0. '05 aeregfge
CIR
Timing option at inception (Vasicek)
Vasico k
Timing option at inception (CIR)
65
Figure 20: Timing option values vs. interest rates at inception (TNBS)
Without the quality option, the timing option is worth an average 0.24 ppp
(Vasicek) and 0.29 ppp (CIR) while when the quality option is offered to the seller,
the timing option is more valuable and is worth an average 0.28 ppp (Vasicek) and
0.32 ppp (CIR). Again, because of the interaction between the options, the timing
option is generally more valuable in the presence of the quality option. Notice that
the timing option in the presence of the quality option under TNBS is no longer
downward sloping.
4.5.2 Comparison of option prices under TNBS and CFS
We now move to a comparison of the values of the quality and timing options
obtained under the CFS and the TNBS.
Using the 1-month U.S. Treasury-Bill rate over the period 31/07/2001- 31/01/2008,
we find a long-term mean around 3%. To illustrate the impact of the long-term mean
on our findings, Figures 21-26 compare the quality option, the timing option as well
as all the options together for the Vasicek and CIR models under both systems at
inception when the long-term mean is at the level of 3% or 6%; the volatility and
the mean reversion speed are those of Cases 1 and 2 given in Table IV.
— CFS •• • 71.50'
66
Comparison of the quality option Comparison of the quality option
at inception (=3%) (Vasicek) at inception (f=6%) (Vasicek)
Quality option {with timing) VatIcelt ta. Quality option {with timing) Vasicok
0.12 CFS —
•• • Ille cl
0.08
0.05
o.c4L
002 ....... ..............................................
0 0 05 001 O015 002 0 025 0.03 0.035 0.01 0-245 -3050,045
intorest rate 04 9.015 0.05 0.05 5.* 3.055 0.07 0.075 0.08
inauest rite
Figure 21: Comparison of the quality option values (with timing) under the CFS and the TNBS at inception for the Vasicek mode!.
Comparison of the timing option
at inception (7=3%) (Vasicek)
0005 0.01 0.015 002 0 325 0.03 00.r 0.04 0245 3.05 0.055 :matou rale
Comparison of the timing option
at inception (f=6%) (Vasicek)
3 %log option {with qoolity)
.................. ... ....
... ....
2 5
0.5
8.04 0.5.45 005 0.055 0.00 0.055 0.37 0.075 0.06 entas' tato
— CFS Q•52
Figure 22: Comparison of the timing options values (with quality) under the CFS and the TNBS at inception for the Vasicek mode!.
— CES • •• TN86
d 4
I
.e 3.51- ".. .......
3F
0.045 005 0.005 0.06 9.955 007 0.0'75
0..8 wn,lasi tg,
67
Comparison of ail the options Comparison of ail the options
at inception (-=3%) (Vasicek)
at inception ('7=6%) (Vasicek)
3.14 All *plions Veskek options ehnitek)
CFS — CFS ••• TISS • 1183
012 OF k.
0.1
• .03
................
0.06
3.04 5.
3,02 ...................................................
03.005 001 03'15 0.02 0.320 013 0.30 imerest rate
3. '04 01240 0:05 0.050 d.00 0.645 3.05 0.055 0.06 0.056 irterest rite
0.07 0.075 0.28
Figure 23: Comparison of ail the options (quality and timing) under the CFS and the TNBS at inception for the Vasicek model.
Comparison of the quality option Comparison of the quality option
at inception (T"=-3%) (CIR) at inception (f=6%) (CIR)
0.12
0.1
0 08
3.05
0.01
00?
.......................................... &0l 0.015 0.02 0.e5 0.03 0.030 1154 0.045 0.06
imefts: :eu
— CFS ••• '7086
Figure 24: Comparison of the quality option values (with timing) under the CFS and the TNBS at inception for the CIR model.
3. — CFS • • • 10i50
.....................
33
Comparison of the timing option
at inception (f=3%) (CIR)
e0I 0.005 0.0 0.525 000 0 033 0.04 ' 045 0 5 Ores: !et
Comparison of the timing option
at inception (i=6%) (CIR)
— CFS •• • TESS
.......................... ...... .....
0043 005 0055 005 0.003 007 0005 0. ,8 rr,n1 rat8
68
Figure 25: Comparison of the timing options values (with quality) under the CFS and the TNBS at inception for the CIR model.
0.12
o 08
I 005
0.84
002
Comparison of ail the options
at inception (f=3%) (CIR)
..........................................
01 0013 302 0025 053 0025 0 24 5 042 305 intarte.! rate
Comparison of ail the options
at inception ('7=6%) (CIR)
ont...1 sel,
Figure 26: Comparison of an the options (quality and timing) under the CFS and the TNBS at inception for the CIR model.
We notice from these figures that all the embedded options are more valuable
under the TNBS at inception when the long-term mean is 6%, which corresponds to
the situation where the CFS achieves its best performance in terms of relating the
deliverable bonds to the reference bond. However, even at that level, the TNBS is
not always outperformed by the CFS, as illustrated for the CIR model in Figure 27:
69
as volatility increases, the value of the quality option also increases in both systems,
but more rapidly in the CFS than in the TNBS. When volatility is very high, the
TNBS outperforms the CFS at the level of 6%. In general, for various parameter
values, we verified that the quality option becomes more valuable under CFS than
TNBS when the long-term mean moves away from the notional rate, and that the
timing option can be either higher or lower under TNBS than under CFS. However,
when the long-run mean is around the notional rate, TNBS is no longer better.
Comparison of the quality option at
inception (high volatility and 7=6%)
••• :353. —
........ ................ ........
0.04 0.00 0.02 207 020 nl 31131e
Figure 27: Comparison of the quality option values (with timing) under the CFS and the TNBS at inception for the CIR model (high volatility).
Recall that, for a flat yield curve at the level of 6%, TNBS and CFS are equivalent
while for any other level of constant interest rates, the CFS compares poorly to the
TNBS, for which ah eligible bonds are equal for delivery. However, in a stochastic
interest-rates framework, this is no longer the case. This is essentially due to the
way futures invoice prices are computed under each system. Under the TNBS,
the futures invoice price is the present value of the cash-flows of the bond to be
delivered, computed assuming constant interest rates. This may be very far from
the value of the bond when interest rates move away from the level of the long-term
mean, and the exercise value and quality option will therefore show extreme values
in that case. This is illustrated in Figure 28 presenting a comparison, under the CIR
0.022
eu
ül0
70
mode!, of the quality option values in the presence of timing under both systems
on day 1 of the delivery month, when the long-term reversion mean is close to the
reference rate. Situations where the TNBS compares poorly to the CFS can be
observed whenever short-term interest rates are low (high) and expected to increase
(decrease) to adjust to a level close to the reference rate. Figure 28 also shows that
the quality option is much less affected by the position of the interest-rate value
relatively to the long-term mean under the CFS. On the other hand, the quality
option value is very sensitive to the long-term mean under the CFS, while this is
flot the case for the TNBS. Figure 29 illustrates the impact of both the long-term
mean and the short-term interest rate on the value of the quality option for the CIR
model, under both systems, and shows that the sensitivity to f under the CFS is
much more important than the sensitivity to r under the TNBS, and that the TNBS
is outperformed by the CFS in a small region where the reference rate is close to the
long-term mean. These situations are likely to happen, because the reference rate
in the CFS is designed to be close to the long-term mean (actually, it was changed
from 8% to 6% in 2000 for that reason). Currently, short-term interest rates are low
while the long-term level is around 3%. Under such a situation, TNBS outperforms
CFS. The CBOT would need to lower again the reference coupon rate to account
for the low interest-rates environment and improve the performance of the CFS.
71
Comparison of the quality option on day 1 of
the DM (moderate volatility and =6%)
9 1
0045 005 CI
006 0 007 0075 008 Warest rat,
— CF-3 ••• 1030
Comparison of the quality option sensitivity
to rbar at inception under TNBS and CFS (1=6.17%)
TnEiS CD CF.
0 04 0 00 0 06 0 97 0.00 0.03
rbat
O 09
006.
O 07
0.36
0.05,
04
0.03
0.02
001
interest rai 0.03
Figure 28: Comparison of the quality option (with timing) under the CFS and the TNBS on day 1 of the delivery month for the CIR mode! (moderate volatility).
Figure 29: Comparison of the quality option value sensitivity to the long-term mean (rbar) at inception under TNBS and CFS for the CIR model.
4.5.3 Sensitivity analysis of the TNBS
We present in figures 30-35 the results for the sensitivity of the option values to the
parameters of the interest-rate models at inception and for both dynamics considered
in this work. The base case parameters values are T- =- 0.06, 1--‘; = 0.5 and o- = 0.02
0018
0 016
0 014
0 012
r. 001
DOM
0004
904 0.045 01 0065 DU 0066 007 interest rite
— Irappa4.1.2 — happa.° 4 •• • • Irappe4 6 - • kappa.° 8
0.075 004
72
for the Vasicek dynamics and = 0.06, K = 0.5 and a- = 0.09 for the CIR dynamics.
From these figures, we observe that both the quality and timing options values
are negatively affected by the mean reversion speed. This can be explained by
the fact that an increase in the mean reversion rate, which determines the relative
volatilities of long and short rates, dampens out short-term rate movements quickly
and therefore reduces the long-term volatility, which is positively related to the
quality option value. We also observe that the relation between the long-term mean
and the quality or timing options could be either positive or negative depending on
the level of spot interest rates. The timing option is however observed to be always
positively affected by the long-term mean. Furthermore, the relation between the
quality option value and the volatility is observed to be positive. In addition, the
hump in the curve representing the timing option observed in the neighborhood
of the long-run mean is more noticeable for curves associated with small levels of
long-term mean or speed of adjustment.
It is worthwhile mentioning here that the behavior of the embedded delivery
options under the TNBS does not show irregularities with respect to interest rates
or unexpected relations with respect to the input parameters of the interest-rate
process as it is observed under the CFS.
Quality option sensitivity to
kappa at inception (Vasicek)
U846I4 •Plien
Timing option sensitivity to
kappa at inception (Vasicek)
10' Tinrinq eptierb 34
16
— rrappa4 2 — 4,8647 4
-• kappa4 6 kappa4
3 4 -
26.
2.6 -
2.4 -
'804 0 045 fr 0-5 1055 0.06 1065 007 0 075 004 interest nt,
Figure 30: Options values sensitivities to kappa (Vasicek)
— rbar43 03 — rliar43136
- • rtar43 09
6
4
nterest rate .803 0 002 004 0.06 006 01 012 014 .8 02 0 0.02 004 0.65 004
interest rale 0.1 012 014
0035
003
0025
002
1
0 015
001
0045
— sig4.101 — sig4 03 • • - - sig=0 05 - - sig4) 06
Quality option sensitivity to
rbar at inception (Vasicek) Ouality option
0 014
8.01 002 0.03 0.04 004 0.04 0.07 0.11 0.33 0.1 011 internat tale
Timing option sensitivity to
rbar at inception (Vasicek)
Timing option
1701 0 '02 003 004 005 006 0.07 0013 063 01 011 etterest rate
73
0 012
Dol
0020
0 CŒ
CI C04
00212
35
1, 2.5
16
Figure 31: Options values sensitivities to rbar (Vasicek)
Quality option sensitivity to Timing option sensitivity to
sigma at inception (Vasicek) sigma at inception (Vasicek)
Onality option
Tintin.] option
Figure 32: Options values sensitivities to sigma (Vasicek)
.3 Timing option 41
4 ?
3 8 »
3.6 -
3.4 -
32-
3 -
28 -
26 -
204035 004 0045 005 0.045 0.06 0055 007 0 075 008 0.085
aune rate
74
Quality option sensitivity to
kappa at inception (CIR)
• 10' Onality opdon
Timing option sensitivity to
kappa at inception (CIR)
— kappa4 2 — kappa4 4 --• icappa41 — • icappa4) 8
— kappa=0.2 — kappa43.4
kappa=0.6 kappa=0.8
14
l o
-
.....................
02836 004 0 045 0E6 0.055 006 00E6 007 0.075 interne rate
008
0.12 002 004 004 005
01
interest rate
7
5
4
3
20
— Mar-40 03 — r04r4.05
— - rbard3.09
— rbal 03 — rbat) 05
rbar41 07 — • t5ar4:109
45
35
3
2.5
00? 0.04 0.06 005 unerest rate
1.50 0.1
0.12
Figure 33: Options values sensitivities to kappa (CIR)
Quality option sensitivity to Timing option sensitivity to
rbar at inception (CIR) rbar at inception (CIR)
finality option Timing option
Figure 34: Options values sensitivities to rbar (CIR)
202
0E0 0.04 002 0.60
007
002 003 arterest rate
10
734 7
6
2
1.5 0.02 003 0.04 005 006 007 0.03 0130 01
imeree rare
as
3
2.5
Quality option sensitivity to
sigma at inception (CIR) 08414 olpTiOP
Timing option sensitivity to
sigma at inception (CIR) Timln9 option
75
Figure 35: Options values sensitivities to sigma (CIR)
4.5.4 Optimal delivery strategy under the TNBS
In this section, we report on the optimal delivery strategy during the delivery month
under the CIR model; the results under the Vasicek model are qualitatively similar.
Since the decision whether to deliver or not is made on the position day at 8:00
p.m. while the choice of the bond to be delivered is made the next day at 5:00 p.m.,
results about the decision and the CTD are presented separately. We assume 22
business days in the delivery month and we choose to report on the optimal decision
(resp. the CTD) on days 15 (resp. 16), 18 (resp. 19) and 21 (resp. 22). For this
illustration, we use the Case 2 parameters given in Table IV as a base case.
The optimal decision on the position day as well as the identification of the CTD
on the notice day depend on the two state variables, namely the current short-term
interest rate (denoted r8 or r5 on the graphical representations) and the yield to
maturity of the notional bond y implied by the last futures settlement price, which
is in turn a function of the spot rate at the last settlement date (denoted r 2 on the
graphical representations).
Figure 36 presents the optimal decisions on days 15-16 of the delivery month
(the last futures trading day) for al possible values of the two state variables. No-
76
tice that only a small area around the diagonal of Figure 36 (the black area) is
likely to be observed, corresponding to possible variations of the rate in the 6 hours
period separating settlement and position times and the 27 hours period separating
settlement and notice times. This area is presented in Figure 37, showing that early
exercise can be optimal during the delivery month. We find that it is better to wait
when the spot rate at the last settlement date is relatively low, and more so when
it has not increased since. We also find that the CTD in the exercise area is a bond
with an intermediate duration, namely the bond with the lowest coupon and lowest
maturity in the eligible basket. This is representative of the optimal strategy during
the first part of the delivery month; the exercise region is increasing very slightly
with time from day 1 to day 15.
Figure 38 represents the optimal decision 4 days before the end of the delivery
month, that is, during the period where trading on the futures is no longer possi-
ble. During the end-of-the-month period, the exercise region is much smaller, and
increases slightly with time. Still, exercising the end-of-the-month option can be
optimal.
We observed that the general configuration of the exercise and CTD regions is
not affected by the parameter values: the relative positions of the exercise and CTD
areas are flot changed, only their shape and relative surface change smoothly with
parameter values; when or jç increases, or when a decreases, the boundary sep-
arating optimal delivery of maximal/minimal coupons moves downwards, and the
boundaries separating maximal/minimal maturities move counter-clockwise, some-
times with the appearance of an additional transition region where bonds with
intermediate maturities are optimal to deliver, but this does not change the delivery
strategy inside the small area where rates are likely to be observed, so that only
bonds (c, M) and (c, M) may be CTD when exercise is optimal, that is, when inter-
est rates are not very low during the first part of the delivery month, or when they
are relatively high during the end-of-the-month period.
Figures 39 and 40 are examples where the CTD may be the bond with the highest
6.2%
7%
r9 é:MY 19) = r2ead(6hours) 3.5% 3.8% 4.1%
Watt .8%
8.5% 8.2%
1.5 (dey 1) = r2±2ad(27hours)
6.2%
7% 8.2% 8.5% 8.8T
8%
8.5%
Exercise
77
duration. Figure 40 represents the cheapest-to-deliver on the last day of the delivery
month, while Figure 39 is an example where the long-term mean is very high and
where the short trader should exercise in ail situations. It is worthwhile mentioning
that the CTD's under the CFS are bonds with extreme durations, which is rarely
the case under the TNBS.
Optimal decision on day 15
of the DM (i=6.17%)
Spot rote rei
CTD on day 16 of the DM
(7—=-6.17%)
Spot rate r5
Yiel
d to
mat
urity
«th
e no
tion&
bond
y
Yiel
d to
mat
urity
of t
he n
otio
n al
bond
y
Figure 36: Optimal decision and CTD on days 15 and 16 of the DM (CIR)
Optimal decision on day 15
CTD on day 16 of the DM
of the DM (7-=6.17%)
(T"=-6.17%)
Figure 37: Optimal decision and CTD on days 15 and 16 of the DM for possible variations of interest rates (CIR, rbar=0.0617).
8% 9.5% 9% 8%
8.5%
9%
Optimal decision on day 15
of the DM (1'=8%)
re (d8y15) = r2±2ad(Ghours)
5.5%
5.6% 5.7% 7.9%
5.6%
›, Exercise 8%
CTD on day 16 of the DM
('7=8%)
IS MaY 18) = r2ead(27hours)
.6%
10.3 9° 10.5/,
8.95
10.2% 10.3% 10%
10.3%
10.6%
78
Optimal decision on day 18 CTD on day 19 of the DM
of the DM (i'=6.17%)
ra (das. , E) = r2ead(3days+6hours)
3.3% 3.8% 4.3%
6.2% 3.8%
>,
7%
(7=6.17%)
14 (deY 19) r2i2ad(4days+3hours)
Figure 38: Optimal decision and CTD during the end-of-the-month period (days 18 and 19 of the DM) for possible variations of interest rates (CIR, rbar=0.0617).
Figure 39: Optimal decision and CTD on days 15 and 16 of the DM for possible variations of interest rates (CIR, rbar=0.08).
79
CTD on day 22 of the DM
(7"=6.17%)
e De d'Y) r2ead(7days+3hours)
7.8%
8.5%
9.2
Figure 40: CTD on the last day of the delivery month for possible variations of interest rates (CIR).
r2 (
da
y15
of
the
DM
), r
et2ad
(3m
onth
s)
5 Prices of the CBOT T-Bonds Futures: An Empirical
Investigation
This section is devoted to an empirical investigation of the CBOT T-bonds futures
pricing model proposed in Section 3 and given by the dynamic program (12)-(23)
when the instantaneous spot interest rate moves according to the Hull-White (1990)
model. This model has been used in the literature dealing with the pricing of futures
on governmental bonds but in its trinomial discrete version by Chen, Chou and Lin
(1999) and applied to value the Japanese long-term Government Bond futures. In
this section, this model is used in its continuous time version for the valuation of
the CBOT T-bonds futures. Under this model, the transition parameters in (28)
and (29) are time-dependant but can still be expressed in closed-form.
This section is organized as follows. In Section 5.1, we describe the Hull-White
(1990) model and present the yield-curve fitting model used here, namely the Aug-
mented Nelson and Siegel model proposed by Bjork and Christensen (1999). Section
5.2 describes the short-term interest rate, the yield-curves and the futures data. In
Section 5.3, we report on the empirical results.
5.1 The Hull-White model (extended Vasicek)
The Hull-White model, also called the extended Vasicek model, was introduced by
Hull and White (1990). This model assumes that the instantaneous short-term
interest-rate process evolves under the risk-neutral probability measure according
to
drt = (t) — rt )dt o-dBt , for t ? 0, (51)
where {Bt ,t > 0} is a standard Brownian motion, K is the mean reversion speed,
(t) is the long-term mean and a- is the volatility. As in the original Vasicek model,
the process has a constant positive volatility and exhibits mean reversion with a
constant positive speed of adjustment, but in the extended version, the long-term
81
level is a deterministic function of time.
Given the current (time 0) term structure and a differentiable function t 1-->
f (t) representing the associated instantaneous fitted forward-rate curve, the term
structure of interest rates in the Hull-White model with
7 (t) = f (t) ± K at
+ —2k2
(1 - e(-2-t)) 1 0 f (t) cr2
(52)
will be identical to the current term structure of interest rates.
This model requires the use of market data to obtain a fitted zero-coupon yield
curve. Several non-parametric fitting techniques can be used to model the yield
curve. These are general curve-fitting families including, for example, B-splines
and Nelson and Siegel (1985) (henceforth NS) curves that do not derive from an
interest-rate model. In this paper, we choose to use the Augmented Nelson and
Siegel yield-curve fitting model proposed by Bjiirk and Christensen (1999), which
extends the NS family curves by the addition of an exponential decay term. These
authors study the question as to when a given parametrized family of forward-rate
curves is consistent with the dynamics of a given arbitrage-free interest-rate model,
in the sense that the model actually will produce forward-rate curves belonging to
the considered family. Bjôrk and Christensen (1999) show that one needs to add
an exponential decay term in order to make the original NS family consistent with
the extended Vasicek model. The main reason for this consistency requirement, as
mentioned by the authors, is that if a given interest-rate model is subject to daily
recalibration, it is important that, on each day, the parameterized family of forward-
rate curves, which is fitted to bond market data, be general enough to be invariant
under the dynamics of the term-structure model; otherwise, the marking-to-market
of an interest-rate contingent daim would result in value changes attributable not
to interest-rate movements, but rather to model inconsistencies.
With five parameters (-yo ,-y i , 'Y2, .Y3, 'Y4), the Augmented NS forward-rate curves
are
f (t) — Ri + (-yi + 72 1) e t/74) ± ,y3e(-2t/-y4) . 74
The resulting equation for the zero-coupon yield curve is then
fi — e ( — t /74 ) ) f i — *4/1'4 ) — e ( — 2t /14 ) ) Z (t) = 70 ± 'Yi
(t/74) + 72
(t/74) e(—t ( 1 h4)) ± .73
(2t/114) (51)
where z (t) is the yield of a zero-coupon bond of maturity t.
Closed-form formulas for the transition parameters and discount factor in the
extended Vasicek model are given in the Appendix.
5.2 The Data
Our interest rates data consist of 3-month maturity Treasury-Bill rates covering
the period from January 1, 1982 to September 30, 2001 and 1-month maturity
Treasury-Bill rates covering the period from October 1, 2001 to March 31, 2008 (315
observations) obtained from the Federal Reserve Statistical Release. The frequency
is monthly. The interest rates are given in percentage and annualized form. We
interpret these rates as proxies for the instantaneous riskless interest rate. Using
1-month or 3-month maturity T-Bill yields as proxies for instantaneous short rates is
unlikely to create a significant proxy bias as shown in Chapman, Long and Pearson
(1999).
In Table V, we presents summary descriptive statistics for the short rate rt ,
the short-rate changes Art = rt — rt— i and squared changes (Art ) 2 . In this table,
ACF(s) denotes the value of the autocorrelation function of order s.
82
(53)
83
Table V: Summary statistics on short rate Variable rt Art (Art) 2 Mean 5.3599 -0.0376 0.106390 Standard deviation 2.5984 0.3245 0.495339 Skewness 0.4989 -2.4606 14.188107 Kurtosis 0.4669 19.3280 226.953564 Minimum 0.85 -2.86 0 lst Quartile 3.6575 -0.1400 0.002500 Median 5.1950 0.0000 0.019600 3rd Quartile 6.9500 0.1400 0.078400 Maximum 14.28 1.36 8.1796 ACF (1) 0.9744 0.3602 0.2492 ACF (2) 0.9386 0.1326 0.0267 ACF (3) 0.9059 0.1055 0.0613 ACF (10) 0.7143 -0.0907 0.0330 ACF (30) 0.3439 -0.0398 0.0321 ACF (50) 0.1671 -0.0542 0.0000
We consider 73 futures contracts traded in the period between January 1, 1990
and March 31, 2008 representing the quarterly delivery cycles of the nearby fu-
tures contract. Futures prices are obtained from the Chicago Board of Trade. It is
worthwhile mentioning that, prior to March 2000, the coupon of the notional bond
was equal to 8%. The basket of the deliverable bonds is determined based on the
information about the issue dates of the 30-year US Treasury bonds available on
the CBOT web site. The properties of the deliverable basket are provided in the
Appendix in Tables VI-VIII for each futures contract over the period of study.
In order to estimate the parameters of the Augmented Nelson and Siegel model,
we use yield curves (Treasury yields for maturities ranging between 3 months and
30 years) for the period between January 1, 1990 and March 31, 2008. Figure 41
shows the estimated spot-yield surface computed 2 months prior to the first day
of the delivery month of the nearest expiring futures contract during the period of
study. Throughout almost the entire sample period, the spot-yield curve presented
a positive slope, although it was approximately flat for some periods. Nevertheless,
it is clear that the period under analysis includes a wide variety of term-structure
shapes.
84
Spot yield surface .
.
20
rnaturity (years) 1
Figure 41: Spot-yield surface.
5.3 Empirical results
In the Hull-White model, the mean reversion speed and the volatility are obtained
from the calibration of the Vasicek model to the short-term interest-rate data, using
the maximum likelihood estimation technique, over the period from January 1, 1982
to inception (two months prior to the nearby futures contract) for each contract.
These estimated values are provided in the Appendix in Tables VI-VIII for each
futures contract over the period of study.
Figures 42 and 43 plot, at inception and on day 1 of the delivery month respec-
tively, a comparison of the observed and theoretical futures prices obtained using
our DP pricing algorithm. These figures show a very good correlation between theo-
retical and observed prices; they also show that observed futures prices are generally
lower than theoretical prices. According to our empirical findings, market futures
prices are on average 2% lower than theoretical futures prices over the 1990-2008
time period, priced two month prior to the first day of the nearby delivery month.
If the Hull-White model accurately describes the movements of the short rate, the
market overvalues the embedded strategic delivery options. This is consistent with
815990 1992 1994 1996 1998 2000 2002 2004 2006 2008
— obssr,ed thecretizal
135
130
125
120
115
110
106
100
96
85
some empirical studies arguing that market futures prices are lower than what they
should be when the term structure of interest rates is upward sloping. Arak and
Goodman (1987) conclude that T-bond futures prices are too low and therefore
believe that the market overvalues the embedded delivery options. Barnhill (1990)
provides empirical evidence that futures prices are more often too low than too high.
Most of the instances in which futures were too high occurred during a period in
which the term structure of interest rates was downward sloping. He argues that
during this period, the expected risk and cost of financing daily resettlement cash
flows may have affected futures prices. For example, Gay and Manaster (1986) find
that futures prices are too high over the period 1977-1983 and conclude that prices
do not adequately value the seller strategic delivery options. They also find that
short traders do not behave optimally in exercising their options, suggesting that
the high futures prices reflect shorts' actual behavior, not the way they should op-
timally behave. Recall that over our period of study, the spot yield curve presented
generally a positive slope as shown in Figure 41.
Observed vs. theoretical futures puces et Inception
Figure 42: Observed vs. theoretical futures prices at inception.
86
Observed vs. theoretical futures prices on dey 1 of the DM 135
130
125
120
115
tia 110
2 105
100
95
90
86 1990 1952 1994 1996 1598 2000 2002 2004
— obser,ed th.coretical
2006 2008
Figure 43: Observed vs. theoretical futures prices on day 1 of the DM.
We present in Figures 44-46 the differences between market and theoretical fu-
tures prices at inception and on the first day of the delivery month, separately and
on the same figure. These figures show that pricing differences can be positive or
negative and are generally small (for some contracts less than 0.05), especially at
inception. Therefore, the model can be used to forecast the futures prices with
a good level of accuracy. Notice that theoretical futures prices on day 1 of the
delivery month are obtained using inception date yield curve, which explains the
higher differences on the first day of the delivery month. Ail futures prices are re-
ported in Tables IX and X given in the Appendix. We also present in Figure 47
the yield curves at inception corresponding to the highest undervaluation of futures
prices (maximum negative difference), obtained for December 1992 contract, and
the highest overvaluation of futures prices (maximum positive difference), obtained
for September 1998 contract. We verified that the highest undervaluation corre-
sponds to the yield curve with the highest positive slope among ail the yield curves
considered in this study. However, the highest overvaluation is associated with a
nearly flat yield curve as shown in Figure 47. These results are consistent with the
findings of Barnhill (1990).
P090 1952 1994 1996 1598 2000 2002 2004 2006 2008
Futures pricing differences et inception
11°990 1992 1994 1996 1998 2000 2002 2004 2006 2008
Figure 44: Futures pricing errors at inception.
Futures pricing differences on day 1 of the DM
87
4
2
o
,
-4
-6
Figure 45: Futures pricing errors on day 1 of the DM.
Futures pricing differences et inception and on dey 1 of the DM 10
1
-•—••• alcepton day 1
.5
-10
.16
211990 1992 1994 1996 1998 2000 2002 2004 2006 2008
Figure 46: Futures pricing errors at inception and on day 1 of the DM.
Yield curves at inception corresponding to extremal futures pricing differences
0 08
88
0.07
0.06
10,81;11992 — 07001/1990
...... ..................
......
30
0.03 /
0.02
0.01
o 5
10
15
20
25 maturity
Figure 47: Yield curves at inception corresponding to extremal futures pricing dif-ferences.
6 Conclusion and suggestions for future research
In this thesis, we investigate the theoretical and empirical pricing of the Chicago
Board of Trade (CBOT) Treasury-bond futures. The difficulty to price it arises from
its multiple inter-dependent embedded delivery options, which can be exercised at
various times and dates during the delivery month. This thesis is composed of three
essays.
In the first essay, we propose a numerical method for the theoretical pricing of
CBOT T-bonds futures by considering a continuous time model with a continuous
underlying factor (the interest rate), moving according to a Markov diffusion process
consistent with the no-arbitrage principle. We propose a model that can handle ahl
the delivery rules embedded in the CBOT T-bond futures, interpreted here as an
American-style interest-rate derivative. Numerical illustrations, provided under the
Vasicek (1977) and Cox-Ingesoll-Ross (1985) models, show that the interaction be-
tween the quality and timing options in a stochastic environment makes the delivery
strategies complex, and not easy to characterize.
The second essay is devoted to the application of the True Notional Bond System
proposed by Oviedo (2006) for the theoretical pricing of the CBOT Treasury-bond
futures. This system is proposed as an alternative to the current conversion fac-
tor system, whose imperfections are well known. Oviedo (2006) showed that the
TNBS outperforms the CFS when interest rates are deterministic. In this essay, we
compare the effectiveness of the two systems in a stochastic environment. Numeri-
cal illustrations, also provided under the Vasicek and CIR models, show that, in a
stochastic framework, TNBS does not always outperform the CFS. However, as the
long-term mean moves away from the level of the notional rate, the TNBS performs
increasingly better than the CFS.
The third essay is an empirical investigation of the futures pricing model pro-
posed in the first essay. To do so, we price the futures contract under the Hull-White
(1990) model. Empirical results show that futures prices are generally undervalued,
90
which means that the market overvalues the embedded delivery options. According
to our findings, observed futures prices are on average 2% lower than theoretical
futures prices over the 1990-2008 time period, priced two months prior to the first
day of delivery months.
The numerical and empirical illustrations are provided here under the one-factor
Vasicek, CIR and Hull-White models, but the models we propose are flexible and
can be used with any specification for multi-factor interest-rate dynamics, provided
the transition parameters and discount factor can be obtained in closed-form or
approximated efficiently (see for example Ben-Ameur et al. (2008)).
Another research direction would be to price the call and put American options
written on T-bond futures, which are traded on the Chicago Board of Trade. In
fact, our DP algorithm could easily be adapted to price these options.
It would also be interesting to investigate the pricing of options embedded in
other futures contracts such as the location option and the quantity option, which
are embedded in commodity futures.
Future research could also include proposing new systems to improve the per-
formance of futures contacts written on a basket of assets and relying on conversion
factor systems to make the short indifferent in delivering any underlying asset.
Investigating the implications of the embedded delivery options and the conver-
sion factor system on the hedging effectiveness of the CBOT T-bond futures would
also be a very interesting research direction since this contract is essentially used to
hedge long-term interest-rate risk.
One can also examine the use of other numerical methods for the pricing of the
CBOT T-bond futures, such as the use of Monte-Carlo simulations, finite differences
and lattice methods.
91
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xix
Appendix
Transition parameters
We give, for the Vasicek (1977), CIR (1985) and Hull-White (1990) models, the
closed-form formulas for the transition parameters A tti-Es 211 and Bkt„tifs
defined respectively in (28) and (29) as well as for the discount factor p(r,t,t + Ô)
defined in (1). For all models, the derivation of these closed-forms starts from the
distribution of the random vector
(rt+8, ruc/u) t-l-b
(55)
conditional on the value of r t , for 0 < t < t + S. For proofs and more details about
the derivation of these closed-forms we refer to Ben-Ameur et al. (2007).
The Vasicek mode!
Under the risk-neutral probability measure, the interest-rate process is the solution
to the following stochastic differential equation
drt = rç,(T- — r t )dt o-dBt, for t > 0, (56)
where {Bt , t > 0} is a standard Brownian motion, h; is the mean reversion speed, is
the long-term mean and u is the volatility. For the Vasicek model, the distribution
of the random vector (55) conditional on Tt = r is bivariate normal with mean
8) = (r, d), tt2(r, (5)) = (7' + e —"(r — 7-(5 + 1 e " (r (57)
and covariance matrix
E (8) = u 1 2 (à) c
2= ( 1 — e-2") 2K Ê2,- (1 — 2e —" e-2")
(6) _L 0-21 2
( — 3 + 2K6 4e—ks e-2K(5) (58)
The discount factor and the transition parameters are then given by
p(r, t, t b) = exp( ft2 (r, 5) + (6) /2), (59)
A t,t+s = e -012(a k ,6)+4(6) / 2 ) {(1)(Xk , i) — (60)
and
Bkt 't±s = e —(122 (ak ' .5)±4(6)/ 2) [(Pl (ak, 5 ) — 0- 12 (6))( 4) (Xk,i)
(5) (61) — ,
where
(ai — mi(ak,(5) + cri2 (6))/cri for i = 0, ..., q,
xk,-1 = —00 (62)
and cl) is the standard normal distribution function.
The CIR mode!
Under the risk-neutral probability measure, the interest-rate process is the solution
to the following stochastic differential equation
drt = tç(i. — rt )dt + fr•tdBt , for t > 0. (63)
For the CIR model, the distribution of the random vector (55) conditional on
rt = r is characterized by its Laplace transform:
t+5 E [exp(—w ruclu — vrt+6) I rt
exp(X(6, cu, v) — (64)
xxi
where
le 2-y(w)e ( w ) ±' )8/2 X(8,w,v) = 1
(72 °g [(vo-2 + -y(w) + k)(e7(w)(5 - 1) + 2-y(w)] ' (65)
vey(w) + k ± el (w) 5 (-y(w) - K)) + 2w(e-Y ( ')(5 - 1) Y(8, w, v) = and (66)
(vo-2 + -y(w) + k)(e-Y()6 - 1) + 2-y(w) = \/k2 2w0-2. (67)
For the CIR model, the discount factor and the transition parameters are given
by
p(r,t,t + 6) = exp(X(8, 1,0) - rY(8, 1, 0)), (68)
A t ,t+6 oo
= P(ak,t,t 8)Ee-A./2(A./2).“ Fd+2.( ai+1 )
u=0 Fd -1- 2u( —ai) '
(69)
and
Bkt,t.+ s 00
-2 (ai-Fifd+2u(—ai+1 ) - aifd+2u(--"ai) p(ak, t 6) 77E e A,/2(À,,u7,2).“
u=o Ti
- d+2u( +(d + 2u)(Fd+2.(—ai+1 ) - F 11 ijj
(70)
where Fd±2u and fd+2u are the distribution and the density functions of a chi-square
random variable with d + 2u degrees of freedom,
= \/K2 + 2o-2 , (71) u2 (e^yb 1)
2 ((-y + k)(e-Yê - 1) + 2-y) (72)
d = 0_2 and (73)
8-y2e6 ak Ak = (74) u2 [(-y + K)(e-Y6 - 1) + 2-y] (e -Yê -
The Hull-White model
For the extended Vasicek model, the distribution of the random vector (55) condi-
tional on rt = r is bivariate normal with mean
/2*(r, t, 6) (r, t, 6), /4(r, t, 6)), (75)
where
0.2 /21(r, t, 8) = e(-") (r f (t + (5) - f (t))+— ((1 - e) + e(- kt) (e(-K ('+6)) - e(- kt) ))
2n2 (76)
/4(r, t, 6) = (r - f (t)) (1 - )
+ (t + 8) y (t 6) - ty (t)
0.2 0.2 2 2
+W (5. (1 e(-")) (e(-K(t+6)) e(-kt) ) • (77)
The covariance matrix, the discount factor and the transition parameters are given
by the same expressions as in the Vasicek model, but with ,u, replaced with p* It
is worthwhile mentioning that, unlike the Vasicek and CIR models, the transition
parameters under the Hull-White model are not only function of the time interval
6, but also function of time t and therefore need to be updated at each time t.
Futures pricing algorithm under the CFS
1. Initialization:
Define Ç. Define E. Set 7i)t-i (aw, ak) =- 0 for ail aw, ak C Ç.
2. Step 1: (end-of-the-month, m =
2.1 Set m = h-. Set e = 1.
2.2 Set gn (aki) using (9) with T = t1+1 and m' = tn2 .
2.3 Compute (aie):
2.3.1 Apply the optimization procedure at (m, ak, , ak) for ail ak E g yield-
ing
Un% (aie, ak) = v ak) .
2.3.2 Apply the interpolation procedure, setting h(ak) =- -17 (ak , , ak), ak c
Ç, yielding
(ak, , =
and apply the expectation procedure to Ti(r) = (ak, , r) at t =
and T =tm5 +1 for ail ak e g, yielding
—e (a k, , ak) = h ( m+i , ak) for ail ak e Ç.
2.3.3 Compute
(ak, , ak) = max [Ume (ak,, ak) , îjmh (aki, ak)] for ail ak e g
and apply the interpolation procedure, setting h(a k) = `67,-„(ak, , ak),
ak c g, yielding
(ak, , r) =
2.3.4 While m > n, apply the expectation procedure to h(r) = îJm (ak,, r)
at t = t _ T = t8 for ail ak E Ç, yielding
Unih _ 1 (ak, , ak) = —h (t,28 _ 1 , tms , a k) for ail ak c g,
set m = m — 1 and go to step 2.3.1,
Else, apply the expectation procedure to h(r) = (ak, ,r) at t = tn2
and T = t8„, yielding 76;2, (ak,) = (t272 , t7,8 , aie).
2.3.5 While I (ak, )1 > e, apply the root finding procedure to update
gri (ak,) and go to step 2.3.1.
Else, set -e.,̀ (ak,) = gr, (ak,).
xxiv
2.4 While k' < q, set k' = k' + 1 and go to step 2.2.
2.5 Apply the interpolation procedure, setting h(ak) = (ah), ak E g, yield-
ing
(r) =Ît(r),
and apply the expectation procedure to ii(r) at t = and T -=- tn2 , for
ail ak e g, yielding
—8 / 8 2 (ak) = h ak) for ail ak E Ç.
3. Step 2 (delivery month, m = n, n — 1)
3.1 Set m -= n — 1.
3.2 Set k' =- 1.
3.3 Set gm (ak, ) using (9) with T = tm5 +l and m' = tm2 .
3.4 Compute (a, , ):
3.4.1 Apply the optimization procedure at (m, ak, , ak) for ail ak E g as in
step 2.3.1, yielding îima (aie, ak) = vma (ak, ak) •
3.4.2 Apply the interpolation and expectation procedures at t = t
T = t5m+1 as in step 2.3.2, setting h(ak) = ak), ak e g,
yielding Ume (ak, ak) = (t8m ,t,n5 +1 , ak) for ail ak C Ç.
3.4.3 Using (21), compute
Umh (ak, , ak) = (ak , ) p (ak, t7n8 , t m2 +1 ) — m8 (ak) for ail ak C g•
3.4.4 Compute
76m (ak, , ak) = max [Ume (ak, , a k) (ak, , ak)1 for ail ak e g
and apply the interpolation procedure as in step 2.3.3, setting h(ak) =
Um(aki, ak), ak E g, yielding iîn, (a k', r) =
XXV
3.4.5 Apply the expectation procedure to h(r) = 5m (aki, r) at t = t
T = tm8 as in step 2.3.4, yielding '752m (ak , ) = (0, t8m, ak')
3.4.6 While Im (ak,)1 > e, apply the root finding procedure to update
(ak,) and go to step 3.4.1,
Else set (ak, ) = gm (aki)-
3.5 While k' < q , set k' = k' + 1 and go to step 3.3.
3.6 Apply the interpolation procedure as in step 2.5, setting h(ak) =9,* (ak),
ak e Ç, yielding (r) =Ît(r).
3.7 While m > n, apply the expectation procedure to h(r) as in step 2.5 at
t = C.,„ 1 and r = t2m , for an ak E g, yielding --9-8,72_ 1 (ak) =Tt (t8m_1, tni2 ak)
for all ak E g, set m = m - 1 and go to step 3-2.
Else, apply the expectation procedure to h(r) as in step 2.5 at t = t 2m_ 1
and T = tm2 , for ah l ak e g, yielding 92m-i (ak) = (tm2 --1, tm2 ak) for all
ak E Ç.
4. Step 3 (before the delivery month, m = -1, ,n - 1)
4.1 Set m n - 1.
4.2 Using (23), compute
7-em (ak) P(ak,q,,,,t2m+i) •
4.3 Apply the interpolation procedure as in step 3.6, setting h(ak) = 9m* (ak),
ak c g, yielding gm (r) = h(r).
Apply the expectation procedure to h(r) as in step 3.7 at t = t m2 _ 1 and
T = tm2 , for all ak E g, yielding (ak) = (tm2 -1, t, for all
ak E Ç.
4.4 While m > -1, set m -= m - 1 and go to step 4.2.
xxvi
Futures pricing algorithm under the TNBS
1. Initialization:
Define Çi and g2. Set il (r, y) = 0 for ail r, y G Ç1 X g2.
2. Step 1: (end-of-the-month, m =
2.1 Set j = 1.
2.1.1 Set m= ï.
2.1.2 Compute v,̀1,2 (r, y3 ) at (m, r, yi) for ail r C
2.1.3 Interpolate 42 (r, y3 ), setting h(r) = uma (r, yi ), r e Çi , yielding
"ÎY'lLn. ( 1', Yi) = îl(r),
and compute the expectation of —h(r) = l (r, yi ) at t = 4,28 and
T = 4,25 +1 for ail r e Ç1, yielding
Ume (r, y3 ) = (t8 t5,2+1 , r) for ail r E
2.1.4 Compute
for all r E
and interpolate h(r) = r E Ç1 , yielding
vYm (r,yj) = Ti(r).
2.1.5 While m > n, compute the expectation of h(r) = VJm (r, y3 ) at t =
t8m_ i and T = tm8 for ail r e Çi , yielding
'772 v 1 sr, 3 — 8 h t t8 r) for all r E gi, - - — m-1, m)
xxvii
set m = m — 1 and go to step 2.1.2,
Else, compute the expectation of h(r) = (r, yi ) at t = t2n and
T = t8n , yielding (r, y_7 ) =
2.1.6 While j < q, set j = j +1 and go to step 2.1.1.
2.2 Find using linear interpolation the function r*(r) such that /-5;2,.(r,r*) = 0
and compute, for ail r E Ç 1 , the futures price gn* (r) as the price of the
notional bond if its yield to maturity is r*(r).
2.3 Interpolate h(r) = (r), r E Ç1 , yielding
(r) -=
and compute the expectation of h(r) at t = t n8 _ 1 and T = tn2 , for ail
r E gl , yielding
for ail r gi.
3. Step 2 (delivery month, m =- n, ...,n — 1)
3.1 Set m = n — 1.
3.2 Set j =- 1.
3.2.1 Compute dyin (r, yi ) for ail r E g1
3.2.2 Interpolate and compute expectations at t = t 7n8 and T = trn8 +1 as
in step 2.1.3, setting h(r) = 713-"tn (r,yi), r E gl, yielding ( 7', Y3) =
(t,,n8 , trn8 +1 , r) for ail r E gi.
3.2.3 Using (47), compute
Umh (r, y3) = P (tm2 , 6%, 20, y j) p (r,t7n8 , trn2 +1 ) — j;n8 (r) for ail r E
3.2.4 Compute
for all r E
and interpolate as in step 2.1.4, setting h(r) = y3 ), r e yielding î)n.„ (r, y3 ) =
3.2.5 Compute the expectation of h(r) = Vm (r, y3 ) at t = t T = tm8
yielding (r, y3 ) =- (t2m , t,L r) , for ail r E gi.
3.2.6 While j < q, set j = j 1 and go to step 3.2.1.
3.3 Find as in step 2.2 the function r*(r) such that ii2m (r, r*) = 0 and compute
the futures price gm* (r) , for ail r E gi , as the price of the notional bond
if its yield to maturity is r*(r).
3.4 Interpolate as in step 2.3, setting h(r) = -en (r), r G Çi, yielding g*m (r) =
h(r).
3.5 While m > n, compute the expectation of h(r) at t = em_ i and T = t2m ,
for ail r E gi, yielding (r) -= 7/, (t_ 1 , tm2 , r) for ail r E gl , set
m = m — 1 and go to step 3.2.
Else, compute the expectation of h(r) at t = t_ ' T tm2 , for ail
r E Çi , yielding (r) = 7 (tm2 _ 1 , tm2 , r) for all r E gi.
4. Step 3 (before the delivery month, m = —1, ...,n — 1)
4.1 Set m = n — 1.
4.2 Using (49), compute
-2, ( (r) =
4.3 Interpolate setting h(r) r e gi, yielding (r) = h(r).
Compute the expectation of h(r) at t = t_ T =- t2m , for all r E
yielding (r) -= h (tm2 _ 1 , tm2 , r) for ail r E
4.4 While m > —1, set m = m — 1 and go to step 4.2.
Hull-White input parameters
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Observed vs. theoretical futures prices
Table IX: Observed vs. theoretical futures prices (1) Contract Inception
Observed Theoretical Difference Day 1 of the DM
Observed Theoretical Difference Mar-90 98.40625 100.329807 -1.923557 92.65625 100.05593 -7.39968 Jun-90 91.90625 92.67514 -0.76889 94.375 93.056412 1.318588 Sep-90 94.28125 95.199369 -0.918119 88.75 95.617884 -6.867884 Dec-90 90.40625 91.701602 -1.295352 95.28125 92.387567 2.893683 Mar-91 96.8125 99.967477 -3.154977 95.25 100.534853 -5.284853 Jun-91 95.3125 98.675985 -3.363485 95.0625 98.93197 -3.86947 Sep-91 93.5 96.253235 -2.753235 98.15625 97.104595 1.051655 Dec-91 100.03125 104.719791 -4.688541 100.09375 106.118183 -6.024433 Mar-92 104.09375 110.596185 -6.502435 100.3125 110.491767 -10.179267 Jun-92 99.40625 104.912949 -5.506699 100.09375 105.631667 -5.537917 Sep-92 100.96875 107.687302 -6.718552 105.15625 109.185747 -4.029497 Dec-92 106.25 114.847873 -8.597873 103.40625 113.85578 -10.44953 Mar-93 105.53125 111.698345 -6.167095 111.875 112.527052 -0.652052 Jun-93 109.21875 116.370489 -7.151739 112 116.455788 -4.455788 Sep-93 113.90625 118.749529 -4.843279 119.53125 119.402965 0.128285 Dec-93 119.40625 121.677572 -2.271322 115.84375 121.953267 -6.109517 Mar-94 113.625 116.305612 -2.680612 110.96875 116.169335 -5.200585 Jun-94 104.625 104.963668 -0.338668 104.28125 105.79824 -1.51699 Sep-94 101.375 102.851986 -1.476986 103.59375 104.508871 -0.915121 Dec-94 98.59375 99.202812 -0.609062 98.375 101.544409 -3.169409 Mar-95 98.71875 97.561444 1.157306 103.9375 100.798213 3.139287 Jun-95 104.5625 105.494974 -0.932474 113.875 107.078705 6.796295 Sep-95 113.5625 114.104653 -0.542153 113.5625 114.82787 -1.26537 Dec-95 114.3125 114.630663 -0.318163 119.9375 115.224198 4.713302 Mar-96 121.34375 120.297266 1.046484 116.03125 120.981639 -4.950389 Jun-96 111.78125 112.847489 -1.066239 107.96875 113.342746 -5.373996 Sep-96 109.5625 109.546977 0.015523 107.90625 110.330758 -2.424508 Dec-96 109.90625 110.024529 -0.118279 116.25 111.167576 5.082424 Mar-97 111.375 111.235002 0.139998 110.65625 111.984539 -1.328289 Jun-97 107.46875 107.556113 -0.087363 110.15625 109.342355 0.813895 Sep-97 111.6875 111.730518 -0.043018 113.59375 112.535728 1.058022 Dec-97 116.34375 115.613803 0.729947 119.28125 115.948731 3.332519 Mar-98 121.46875 119.798768 1.669982 119.65625 120.38474 -0.72849 Jun-98 120.75 119.667794 1.082206 122.03125 120.407564 1.623686 Sep-98 123.53125 121.606848 1.924402 126.84375 122.818804 4.024946 Dec-98 132.65625 132.572808 0.083442 130.1875 132.418695 -2.231195
Table X: Observed vs. theoretical futures prices (2) Contract Inception
Observed Theoretical Difference Day 1 of the DM
Observed Theoretical Difference Mar-99 127.46875 128.587439 -1.118689 120.59375 128.408532 -7.814782 Jun-99 119.9375 122.3878 -2.4503 116.5625 122.220973 -5.658473 Sep-99 115.53125 117.189526 -1.658276 114.1875 117.495451 -3.307951 Dec-99 112.84375 115.517199 -2.673449 111.875 115.412156 -3.537156 Mar-00 89.875 89.313724 0.561276 94.875 89.854988 5.020012 Jun-00 97.8125 98.395572 -0.583072 96.5 99.558029 -3.058029 Sep-00 97.6875 98.953069 -1.265569 100.5625 98.724865 1.837635 Dec-00 98.09375 97.319612 0.774138 102.0625 101.558027 0.504473 Mar-01 106.28125 109.032667 -2.751417 106.25 110.40985 -4.15985 Jun-01 103.875 107.554017 -3.679017 100.96875 108.792633 -7.823883 Sep-01 100.90625 104.820956 -3.914706 104.4375 105.342928 -0.905428 Dec-01 103.875 110.900584 -7.025584 105.15625 112.666146 -7.509896 Mar-02 100.125 105.928179 -5.803179 103 106.125005 -3.125005 Jun-02 98.25 102.272102 -4.022102 102.34375 103.376459 -1.032709 Sep-02 102.9375 108.283813 -5.346313 112.71875 108.80916 3.90959 Dec-02 113.3125 119.411898 -6.099398 109.34375 120.747683 -11.403933 Mar-03 110.15625 115.8982 -5.74195 115.875 116.32504 -0.45004 Jun-03 112.75 118.582212 -5.832212 119.5625 118.521818 1.040682 Sep-03 117.09375 121.634924 -4.541174 105.6875 121.851594 -16.164094 Dec-03 112.0625 116.822155 -4.759655 108.84375 117.052693 -8.208943 Mar-04 107.90625 113.170733 -5.264483 113.8125 113.559115 0.253385 Jun-04 113.46875 118.349956 -4.881206 105.84375 118.812429 -12.968679 Sep-04 106.8125 110.50898 -3.69648 112.625 111.444562 1.180438 Dec-04 111.46875 114.543795 -3.075045 110.96875 114.368071 -3.399321 Mar-05 112.53125 114.471798 -1.940548 112.90625 115.062629 -2.156379 Jun-05 111.78125 113.070535 -1.289285 118.40625 114.290928 4.115322 Sep-05 117.625 118.211076 -0.586076 118.25 118.326621 -0.076621 Dec-05 113.75 113.693861 0.056139 112.0625 114.364261 -2.301761 Mar-06 114.375 114.589959 -0.214959 112.5625 114.463788 -1.901288 Jun-06 109.03125 109.944224 -0.912974 106.46875 110.308925 -3.840175 Sep-06 105.75 106.11142 -0.36142 110.75 107.22811 3.52189 Dec-06 112.53125 112.425885 0.105365 114.625 112.32355 2.30145 Mar-07 111.71875 111.905101 -0.186351 112.875 113.04965 -0.17465 Jun-07 111.34375 112.482169 -1.138419 108.53125 113.612338 -5.081088 Sep-07 108.03125 108.31816 -0.28691 111.40625 111.507787 -0.101537 Dec-07 111.71875 112.631735 -0.912985 118 115.856367 2.143633 Mar-08 117.90625 120.165775 -2.259525 119.671875 126.34062 -6.668745
