2004-11-29

A Semi-Nonparametric Model of the Pricing Kernel andInterest Rates

Yuriy KitsulUniversity of North Carolina - Chapel Hill

[email protected]

November 29, 2004

Introduction and Motivation

• Models of the short-term (risk-free) interest rate describe the time-seriesdynamics of the yield of the bond of instantaneous maturity (in the continuoustime set-up), which serves as a discount rate of risk-adjusted cash-flows innumerous financial applications

• Dynamic models of the term structure of interest rates describe:

- how bond yields depend on time to maturity at each moment of time

- how the form of this dependence changes with time

• Need a model, which would produce a good fit of bond yields

Yuriy Kitsul 1

Objectives

• To model the pricing kernel and bond yields in a flexible pure-diffusionframework that allows

- to introduce various levels of model complexity ⇐⇒ semi-nonparametrics

- to have closed-form solutions for bond yields ⇐⇒ eigenfunctions

• Within the offered framework to let the data decide on the level of modelcomplexity/number on semi-nonparametric terms/elements

Yuriy Kitsul 2

Two Major Themes

• Semi-nonparametrics:

- Flexibility (as in nonparametrics) ⇒ 1) information about unknown relationships2) identification of mispriced securities

- Closed form expressions of the stochastic processes of interest (as inparametric models)

• Empirical sufficiency of pure diffusions (motivated by problematic hedgingproperties of jumps, e.g. review and discussion by Jones (2003) in the contextequity options)

Yuriy Kitsul 3

In this presentation

• Methodology:

- Model the unknown functional relationship between the pricing kernel andthe state variables

- Explore the implications for the bond yields, the short risk-free interest rateand the market price of risk.

- Offer two methods of implementing the ”no-arbitrage” condition.

• Empirics:

- Estimate the model with Gallant&Tauchen’s (2004) EMM-MCMC

- Find that one factor model with a sufficient, but not too high, number ofsemi-nonparametric terms can not be rejected by the univariate yields data.

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Current Issues and Some Recent Literature - 1, Modeling 1

• Bt(T ), price at time t of a zero-coupon bond maturing and paying 1 dollarat time T > t:

MtBt(T ) = EPt {MT · 1} (1)

where P is the physical, or real-world, probability measure,Mt is the instantaneous stochastic discount factor, or pricing kernel, whichassigns prices at time t to payoffs at time t + dt.

• The pricing kernel, Mt,T , which assigns prices at time t to payoffs at time

T > t is Mt,T = MTMt

and the price of bond becomes:

Bt(T ) = EPt {Mt,T · 1} (2)

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Current Issues and Some Recent Literature - 1, Modeling 2

• Under risk-neutral probability measure, Q, which exists and is unique under”absence of arbitrage” and completeness:

Bt(T ) = EQt {e−

R Tt rsds · 1} (3)

where rs is instantaneous (short-term) risk-free interest rate, or money marketspot interest rate.

• To switch back to physical measure, P , Girsanov theorem is used:

Bt(T ) = EPt {e−

R Tt rsds · ξt,T · 1} (4)

where Radon-Nikodym derivative ξt,T = e(R Tt λsdWs−1

2

R Tt λ2

sds) depends onλs, known as the market price of risk.

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Current Issues and Review of Some Recent Literature - 2, TermStructure Literature 1.

• Depending on how rs is modeled, the literature can be categorized into thefollowing types of models:

– affine (eg. Dai/Singleton(00))– non-linear (eg. quadratic: Ahn/Dittmar/Gallant(02), Ahn/Dittmar/Gallant/Gao(03))– regime-shift1(eg. Bansal/Zhou(02),Bansal/Tauchen/Zhou(03))– jump-diffusion (eg. Duffie/Pan/Singleton(00))

• Relatively simple specification for underlying factors to obtain closed-formbond pricing formulas. Complexity is usually achieved by adding extra factors,jumps or regime shifts.

1Often in a discrete time framework

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Current Issues and Review of Some Recent Literature - 2, TermStructure Literature 2

Some empirical features of the existing term-structure models:

• A test of empirical model is a joint test of specifications for the short-termrisk-free interest rate and the market price of risk, i.e. test of a model forthe pricing kernel.

• The debate on which approach is empirically preferable is unresolved yet (eg.constant sign of risk premium and trade-off between a structure of volatilityand negative correlation in affine models).

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Current Issues and Some Recent Literature - 3, Time Series Dynamicsof the Short Interest Rate - 1

• Use a proxy for the short interest rate, rt, e.g. three-month Treasury bill rate,which may be a problem according to Chapman, Long and Pearson (1997)

• No need for closed-form bond price solutions

• More sophisticated specifications for rt, which are tested separately, notjointly with specification for the market price of risk

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Current Issues and Some Recent Literature - 3, Time Series Dynamicsof the Short Interest Rate -2

• Can categorized into:

- one-factor non-linear diffusions (e.g. Aıt-Sahalia (96))

- stochastic volatility (e.g. Andersen and Lund (97))

- regime-shifts (e.g Ang and Bekaert (00))

- jump-diffusions (e.g. Johannes (04))

• A variety of estimation methods: Aıt-Sahalia (96), Conley, Hansen, Luttmerand Scheinkman (97), Gallant and Tauchen (98), Eraker(01), Durham(03)and others

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In This Work - 1, Questions

• Is a flexible pure diffusion framework empirically sufficient?

• How many factors/state variables are needed?

• Do we really need any proxies for instantaneous risk-free interest rate to studynon-linearities in its dynamics? What do bond yields of longer maturities tellus about its behavior?

• What is the functional form that describes how investor’s risk preferencesreact to the underlying shocks?

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Essence of Approach - 1

• A1: ∃ a factor/state variable xt, which drives the economy and is governedby:

dxt = µ(xt)dt + σ(xt)dWt, l < x < u (5)

• A2: ∃ a pricing kernel, Mt and it is some real-valued and positive function,M(xt) ∈ L2(q):

Mt = M(xt) such that

∫ u

l

M(x)2q(x)dx < ∞ (6)

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• ∀ such function, M(x), can be expanded on an infinite number of orthogonalterms, φi(x), such that

∫ u

lφi(x)φj(x)q(x)dx =

{h2

i , if i = j, where hi is called norm0 otherwise

Therefore,

Mt =∞∑

i=0

aiφi(xt) (7)

where ai = 1h2

i

∫ u

lφi(x)M(x)q(x)dx - an inner product w/r to the weighting

function q(x).

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• φi(x) are eigenfunctions =⇒ closed form bond-prices

• Truncate expansion to a finite number of terms, assume some specific formfor xt, and, thus, for eigenfunctions, and estimate ai from the data

• H0: Given parametric specification for underlying factors and a number ofterms in the expansion, are there parameter values, for which the modeledpricing kernel is an acceptable representation of the true pricing kernel?

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Another Brief Digression into the Literature

• Non-linear empirical pricing kernel literature: Bansal/Viswanathan(1993),Bansal/Hseih/Viswanathan(1993), Chapman(1997), and others.

• What is different in this work:

- Latent state variables, interest rates application and closed form prices

• Papers that discuss eigenfunctions2:

Hansen/Scheinkman/Touzi(1998), Chen/Hansen/Scheinkman(2000), Meddahi(2001-a,b), Florens/Renault/Touzi(1998), Gorovoi/Linetsky(2003) and others

• Rogers (1997) uses eigenfunctions to form the potential and presents bondpricing formulas, but implementation issues are not obvious

2Some of the following discussion will be based on the first three sources

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Eigenfunctions Framework - 1

• A diffusion process that governs xt can also be described by an infinitesimalgenerator, A:

Aφ(x) = µ(x)φ(x)′ + 0.5σ(x)φ(x)′′ (8)

• Define scale function, S(x), and speed density, m(x), (Karlin/Taylor(81))

S(x) =∫ x

s(ξ)dξ (9)

where s(x) = exp(− ∫ x 2µ(ξ)σ2(ξ)

dξ)

m(x) =1

s(x)σ2(x)(10)

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• Density of stationary distribution, defined as q(x) = lims→∞ p(x, s|y, t):

q(x) =m(x)∫ r

lm(ξ)dξ

(11)

• Consider the following equation (l < x < u):

Aφ(x) = −δφ(x) (12)

which is the same as

µ(x)φ(x)′ +12σ2(x)φ(x)′′ + δφ(x) = 0 (13)

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• Under some appropriate boundary protocol there will be a countable numberof functions, φi(x) and constants δi, which solve this equation and areorthogonal w/r to a stationary density q(x).

φi(x) - i’th eigenfunction of A and δi - corresponding i’th eigenvalue.

• Crucial property:E(φi(xT )|Ft) = e−δi(T−t)φi(xt) (14)

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Eigenfunctions Framework - 4, Some Examples

• O-U process of a form below has q(x) = e−x2

2 with eigenfunctions of itsgenerator equal to Hermite polynomials and eigenvalues δi = κi

dxt = −κxtdt +√

2κdWt

H0(xt) = 1,H1(xt) = xt,Hi(xt) = 1√i(xtHi−1 −

√i− 1Hi−2(xt))

• CIR process of a form below has q(x) = xαe−x with eigenfunctions of itsgenerator equal to Laguerre polynomials and eigenvalues δi = κi

dxt = κ(α + 1− xt)dt +√

2κ√

xtdWt

Lα0 (xt) = 1, L1(xt) = 1+α−xt√

1+α

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Bond Pricing - 1

• Recall that we represent a stochastic discount factor, Mt, as

Mt =∑n

i=0 aiφi(xt)

and Bt(T ), price at time t of a zero-coupon bond maturing at T > t:

MtBt(T ) = Et(MT · 1)

• Substituting an expression for the pricing kernel, we obtain:

(n∑

i=0

aiφi(xt))Bt(T ) = Et(n∑

i=0

aiφi(xT )) (15)

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Bond Pricing - 2

• Using the property of eigenfunctions, Et(φi(xT )) = e−δi(T−t)φi(xt),

Bt(T ) =∑n

i=0 aie−δi(T−t)φi(xt)∑n

i=0 aiφi(xt)(16)

=n∑

i=0

aiφi(xt)∑nj=0 ajφj(xt)

e−δi(T−t)

• Instantaneous risk-free interest rate:

rt = lim(T−t)→0

−ln(Bt(T ))T − t

=∑n

i=0 aiδiφi(xt)∑ni=0 aiφi(xt)

(17)

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Market Price of Risk - 1

• Duffie (2001): assume dMt = µM(t)dt + σM(t)dWt

and consider some arbitrary security, with cumulative-return process,

dRt = dStSt

= µS(t)St

dt + σS(t)St

dWt

Then, the Sharpe ratio, or the market price of risk, −λt:

−λt = µR(t)−rtσR(t) = −σM(t)

Mt

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Market Price of Risk - 2

Using Ito lemma:

dMt = {A(n∑

i=0

aiφi(xt))}dt + σ(xt)(n∑

i=0

aiφi(xt))′dWt (18)

Therefore,

λt =σ(xt)(

∑ni=0 aiφi(xt)′)∑n

i=0 aiφi(xt)(19)

Risk premium is very general and can change sign.

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Positivity of Pricing Kernel

• Hansen/Richard(87) and Cochrane(2001): M > 0 ⇐⇒ no-arbitrage.

• Example of how to impose positivity numerically:

- Consider: Mt = a0 + a1H1(xt) + a2H2(xt)

- Find a map between the coefficients (a0, a1, a2) of this combination ofHermite polynomials and coefficients (b, c) of a regular polynomial of thesecond order that has only complex roots and, thus, never crosses zero:

a0 + a1H1(x) + a2H2(x) = (x− b− ic)(x− b + ic)

- Mt is either positive or negative on the whole state space of xt. If negative,multiply it by −1.

• Impose a prior in the process of MCMC estimation.

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Multi-Factor Extension

• Independent factors, x1t and x2t, with eigenfunctions φ1i and φ2j,respectively.

Mt =n1∑

i=0

n2∑

j=0

aijφ1i(x1t)φ2j(x2t) (20)

• Define φij(xt) = φ1i(x1t)φ2j(x2t), where xt = [x1t, x2t] The followingproperty3 and all the previous results hold:

Et{φij(xT )} = e−δij(T−t)φij(xt) (21)

where δij = δ1i + δ2j

3See Meddahi(2001 a) for details

Yuriy Kitsul 25

Data and Estimation Method

• Original and updated versions of a data set of Ahn/Dittmar/Gallant/Gao(03)who combine data of McCulloch and Kwon(93) and data provided by DanielWagooner and obtained by methods described in Bliss(97)

• Range: 01/52 - 12/99 (extended data is available till 12/02). Maturity: 6months.

• Method of estimation: Efficient Method of Moments proposed by Gallant andTauchen(96) and extended in Gallant and Tauchen(04) with Markov ChainMonte Carlo methodology along the lines of Chernozhukov and Hong(2003).

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Overview of Efficient Method of Moments (EMM) - 1

• Advantages:

- Used when likelihood methods are not practical.

- Avoids an ad hoc selection of moment conditions when compared to othermethods of moments.

- Perfect for unobservable/latent factor structures and continuous-timemodels.

• Simulation-based method (related to Duffie and Singleton(93) andGourieroux, Monfort and Renault(93)):

Choose the parameters, ρ, of a model of interest (let us call it the mainmodel) in such a manner that the data simulated from the model is as closeto the observed data as possible.

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• Step 1: Projection: Summarize the data using the so-called auxiliary model,which is not the true model, but which approximates the data sufficientlywell and has a readily computable likelihood function f(·) in a closed form.

• Estimate the parameters of the auxiliary model, θ, by maximizing its likelihoodand and using the observed data, yt, xt−1:

θn = arg maxθ∈Θ1n

∑nt=1 log[f(yt|xt−1, θ)]

where n is the number of observations. Therefore, by construction

∂∂θ

1n

∑nt=1 log[f(yt|xt−1, θ)] = 0

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• Step 2: Estimation: The obtained score vector is used to generate momentconditions by simulating {yt, xt−1} from the main model:

m(ρ, θ) = 1N

∑Nt=1

∂∂θlog[f(yt|xt−1, θ)]

• We choose the parameters of the main model, ρ so that the generatedmoment conditions are as close to zero as possible:

ρn = arg minρ m′(ρ, θn)(In)−1m(ρ, θn)

where In is a quasimaximum likelihood information matrix and

nsn(ρ) = nm′(ρ, θn)(In)−1m(ρ, θn) ∼ χ2dim(ρ)−dim(θ)

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Overview of Efficient Method of Moments (EMM) - 4, MCMCExtention

• Construct the analog to the likelihood in the Bayesian Markov Chain MonteCarlo (MCMC) methods:

L(ρ) = e−nsn(ρ)

• Use the standard Metropolis-Hastings algorithm to simulate a chain (and useits mode as the estimate):

{ρ(1), ...., ρ(Nch)}.

• 1) Easy to impose prior restrictions on non-linear functionals ψ(ρ)

2) Much better optimizer than the traditional methods (eg. a hill-climber)

3) Can use an analog of Bayesian posterior for econometric inference.

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Metropolis-Hastings algorithm

• A candidate for the next value in the chain, ρnew is drawn from a proposaldensity q(ρnew|ρold), which, among other things, is easy to simulate from.

• Using the candidate value, ρnew the data simulation of length N is obtainedand the objective function, sn(ρnew), the functional of the parameters, ψnew,the prior, π(ρnew, ψnew), and the likelihood, L(ρnew) = e−nsn(ρnew) arecomputed.

• The chain moves from ρold to ρnew with probability

min{L(ρnew)π(ρnew,ψnew)q(ρnew|ρold)L(ρold)π(ρold,ψold)q(ρold|ρnew) , 1}

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Data and Projected Conditional Mean

53 58 63 68 73 78 83 88 93 980

2

4

6

8

10

12

14

16

18Data and projected conditional mean, Six−month Treasury Bill, 03/53 − 12/99

time

percen

t

projecteddata

Figure 1: 6-month Treasure Bill: Data, 03/53-12/99 and conditional mean obtained with

11s1s0s1s1400000 SNP score .

Yuriy Kitsul 32

Gaussian Factor Case: Empirical Model and Observation Equation

• Factor/state variable dynamics:

dxt = κ(θ − xt)dt + σdWt

• Pricing kernel within a model with n Hermite polynomials, H(n)-model:

Mt =∑n

i=0 aiHi(xt)

• Yield:

yldt(T ) = − 1T−t{ln(

∑ni=0 aie

−κi(T−t)Hi(xt))− ln(∑n

i=0 aiHi(xt))}

• Estimated parameters: {κ, θ, σ, a1, ...an}.

Yuriy Kitsul 33

Estimation Results: 1 Gaussian factor, 2, 3 and 4 Hermite polynomials,Score: 11s1s0s1s1400000

Coefficient/Model CIR H(2) H(3) H(4)

κ 0.12294 0.40527 0.73877 0.604( 0.072865) (0.048579) (0.051739) (0.015484)

θ 5.1059 0.081055 0.19678 -0.14795(0.092615) (0.036123) (0.049139) (0.026694)

σ 0.19717 0.16699 0.2915 0.17139(0.0081534) (0.021523) (0.021654) (0.010689)

a0 1, fixed 1, fixed 1, fixeda1 1, fixed 1, fixed 1, fixeda2 -0.2373 -0.10693 -0.23877

(0.042031) (0.011367) (0.0021196)a3 0.31494 0.36616

(0.016185) (0.0017625)a4 -0.12256

(0.0016904)

χ2 21.323 21.716 9.9722 7.308p− value 0.0033 0.0014 0.0760 0.1205

df 7 6 5 4

Yuriy Kitsul 34

0 10000 20000 30000 40000 50000

0.20.4

0.60.8

1.0

0 10000 20000 30000 40000 50000

4.85.0

5.25.4

0 10000 20000 30000 40000 50000

0.180.20

0.220.24

0.26−32

−30−28

−26−24

−22

Figure 2: CIR model. Chain of each of the parameters, {κ, θ, σ}. Every 50th point is plotted.

.

Yuriy Kitsul 35

0 20 40 60 80 100

0.00.2

0.40.6

0.81.0

0 20 40 60 80 100

0.00.2

0.40.6

0.81.0

0.00.2

0.40.6

0.81.0

Figure 3: CIR model. Auto-correlation function for each of the parameters, {κ, θ, σ}..

Yuriy Kitsul 36

0.0 0.2 0.4 0.6 0.8 1.0 1.2

4.5 5.0 5.5

Figure 4: CIR model. Kernel density for each of the parameters, {κ, θ, σ}..

Yuriy Kitsul 37

V1

V2

V3

Figure 5: CIR model. Scatter plots of pairs of parameters {κ, θ, σ}. Every 50th point is

plotted. .

Yuriy Kitsul 38

0 10000 20000 30000 40000 50000

0.20.5

0.8

0 10000 20000 30000 40000 50000

−0.5−0.2

0.1

0 10000 20000 30000 40000 50000

0.050.15

0 10000 20000 30000 40000 50000

0.61.0

1.4

0 10000 20000 30000 40000 50000

0.61.0

1.4

0 10000 20000 30000 40000 50000

−0.8−0.4

0 10000 20000 30000 40000 50000

−30−26

−22

Figure 6: H(2) model. Chain of each of the parameters, {κ, θ, σ, a0, ..., a2}. Note: a0 and

a1 are fixed for the identification purposes. Every 50th point is plotted..

Yuriy Kitsul 39

0 20 40 60 80 100

0.00.4

0.8

0 20 40 60 80 100

0.00.4

0.8

0 20 40 60 80 100

0.00.4

0.8

0 20 40 60 80 100

0.00.4

0.8

0 20 40 60 80 100

0.00.4

0.8

0 20 40 60 80 100

0.00.4

0.8

Figure 7: H(2) model. Auto-correlation function for each of the parameters,

{κ, θ, σ, a0, ..., a2}. Note: a0 and a1 are fixed for the identification purposes..

Yuriy Kitsul 40

0.2 0.4 0.6 0.8 1.0

−0.5 0.0 0.5

−0.1 0.0 0.1 0.2 0.3

−1 0 1 2 3

−1 0 1 2 3

−1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2

Figure 8: H(2) model. Kernel density for each of the parameters, {κ, θ, σ, a0, ..., a2}. Note:

a0 and a1 are fixed for the identification purposes..

Yuriy Kitsul 41

V1

V2

V3

V4

V5

V6

Figure 9: H(2) model. Scatter plots of pairs of parameters {κ, θ, σ, a0, ..., a2}. Note: a0

and a1 are fixed for the identification purposes. Every 50th point is plotted..

Yuriy Kitsul 42

0 10000 20000 30000 40000 50000

0.500.60

0.70

0 10000 20000 30000 40000 50000

−0.20

−0.05

0 10000 20000 30000 40000 50000

0.140.18

0.22

0 10000 20000 30000 40000 50000

0.61.0

1.4

0 10000 20000 30000 40000 50000

0.61.0

1.4

0 10000 20000 30000 40000 50000

−0.245

−0.230

0 10000 20000 30000 40000 50000

0.360

0.370

0 10000 20000 30000 40000 50000

−0.130

−0.115

−20−14

−8

Figure 10: H(4) model. Chain of each of the parameters, {κ, θ, σ, a0, ..., a4}. Note: a0 and

a1 are fixed for the identification purposes. Every 50th point is plotted..

Yuriy Kitsul 43

0 20 40 60 80 100

0.00.4

0.8

0 20 40 60 80 100

0.00.4

0.8

0 20 40 60 80 100

0.00.4

0.8

0 20 40 60 80 100

0.00.4

0.8

0 20 40 60 80 100

0.00.4

0.8

0 20 40 60 80 100

0.00.4

0.8

0 20 40 60 80 100

0.00.4

0.80.0

0.40.8

Figure 11: H(4) model. Auto-correlation function for each of the parameters,

{κ, θ, σ, a0, ..., a4}. Note: a0 and a1 are fixed for the identification purposes..

Yuriy Kitsul 44

0.4 0.5 0.6 0.7

−0.3 −0.2 −0.1 0.0 0.1

0.10 0.15 0.20 0.25

−1 0 1 2 3

−1 0 1 2 3

−0.25 −0.24 −0.23 −0.22

0.355 0.360 0.365 0.370 0.375 0.380

Figure 12: H(4) model. Kernel density for each of the parameters, {κ, θ, σ, a0, ..., a4}. Note:

a0 and a1 are fixed for the identification purposes..

Yuriy Kitsul 45

V1

V2

V3

V4

V5

V6

V7

V8

Figure 13: H(4) model. Scatter plots of pairs of parameters {κ, θ, σ, a0, ..., a4}. Note: a0

and a1 are fixed for the identification purposes. Every 50th point is plotted..

Yuriy Kitsul 46

53 58 63 68 73 78 83 88 93 980

2

4

6

8

10

12

14

16

18Projected and reprojected conditional means, Six−month Treasury Bill, 03/53 − 12/99

time

percent

projectedreprojected

53 58 63 68 73 78 83 88 93 98−4

−3

−2

−1

0

1

2

3Relative difference between projected and reprojected conditional mean, Six−month Treasury Bill, 03/53 − 12/99

time

percent

Figure 14: Conditional projected and reprojected first moments and their difference; ”1

Gausssian factor - 4 Hermite polynomials” model, estimated using 11s1s0s1s1400000 SNP

score.Yuriy Kitsul 47

53 58 63 68 73 78 83 88 93 980

1

2

3

4

5

6

7Projected and reprojected conditional volatilities, Six−month Treasury Bill, 03/53 − 12/99

time

projectedreprojected

53 58 63 68 73 78 83 88 93 98−40

−20

0

20

40

60

80Relative difference between projected and reprojected conditional volatility, Six−month Treasury Bill, 03/53 − 12/99

time

percent

Figure 15: Conditional projected and reprojected second moments and their difference; ”1

Gausssian factor - 4 Hermite polynomials” model, estimated using 11s1s0s1s1400000 SNP

score.Yuriy Kitsul 48

Conclusions

• It is possible to have a flexible diffusion framework with ”no-arbitrage” andclosed-form bond prices.

• EMM-MCMC is very well suited to handle our framework, which is highlynon-linear, with prior restrictions on non-linear functionals of parameteres

• A one Gaussian factor model with a sufficient, but not overly excessivenumber of semi-nonparametric terms can not be rejected by the time-seriesof 6-month to maturity yields data.

• Further work: fit the model to the joint dynamics of several yields of differentmaturities.

Yuriy Kitsul 49

Extensions

• Incorporate macroeconomic variables.

• Apply to a multi-country case.

• Time-varying functional forms.

• Derivatives.

50

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2004-11-29