Derivatives Options on Bonds and Interest Rates 11... · 2012-04-26 · – Short 1 t 2-year zero-coupon (Example: Short 1 2-year zc) • In order to have zero-duration: 1 1 2 2 t

Derivatives Options on Bonds and Interest Rates

Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles

26 April 2012 Derivatives 10 Options on bonds and IR |2

Introduction

•  A difficult but important topic: •  Black-Scholes collapses:

1. Volatility of underlying asset constant 2. Interest rate constant

•  For bonds: –  1. Volatility decreases with time –  2. Uncertainty due to changes in interest rates –  3. Source of uncertainty: term structure of interest rates

•  3 approaches: 1. Stick of Black-Scholes 2. Model term structure : interest rate models 3. Start from current term structure: arbitrage-free models


Interest rate model

•  The source of risk for all bonds is the same: the evolution of interest rates. Why not start from a model of the stochastic evolution of the term structure?

•  Excellent idea •  ……. difficult to implement •  Need to model the evolution of the whole term structure! •  But change in interest of various maturities are highly correlated. •  This suggest that their evolution is driven by a small number of underlying

factors.


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Euro area spot rates (AAA)

Derivatives 10 Options on bonds and IR |5

26 April 2012

Changes in interest rates - summary statistics


Daily data

Euro area (changing composition) - Government bond, nominal, all issuers whose rating is triple A - Svensson model - continuous compounding - yield error minimisation - Yield curve spot rate, 1-year maturity - Euro, provided by ECB

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Modelling the term structure evolution

•  Modelling the evolution of the term structure is complex. •  Assuming that all IR might change by the same amount

(parallel shift) would suppose there exist arbitrage opportunities.

•  To see this, suppose that the term structure is flat. •  The value of a (T-t)-year zero-coupon is:

•  where r is the interest rate (the same of all zero-coupons)

)(100),( tTrT etrP −−×=

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Zero duration portfolio

•  Suppose r=3% •  Create a portfolio with zero-duration:

–  Long n t1 -year zero-coupon bond (Example: Long n 10-year zc) –  Short 1 t2-year zero-coupon (Example: Short 1 2-year zc)

•  In order to have zero-duration: 11

22

PtPtn =

Interest rate 3%Bonds 1 2 Portfolio with zero-‐durationMaturity 10 2 # units ValuePrice 74.08 94.18 Bond 1 0.25 18.84Duration 10 2 Bond 2 -‐1.00 -‐94.18Convexity 100 4 -‐75.34

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Impact of convexity

Conclusion: No arbitrage implies evolution of the term structure more complex than parallel shift

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No-arbitrage in continuous time The no-arbitrage condition can be demonstrated in a continuous time setting

021 =+−= CashPnPV

dzrdtradr )()( σ+=Stochastic process for short rate:

Pricing equation for zero-coupon: )(),( tTretrP −−=

Long-short portfolio:

Zero duration: 11

22

)()(0PtTPtTn

rV

−−

=→=∂∂

0²)(21

2122 =−= dtTTPTdV σBy Ito:

This portfolio has no risk and no investment. It should earn zero dV = 0

This equation holds only if T1 = T2. Conclusion: the bond valuation model implies that arbitrage is possible.

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dtTTPT

dtPtTPtTPtTPtT

dtrCashPtTPtTnrPnrPdV

²)(21

²))²(21²)²(

)()(

21(

²)²(21²)²(

21

2122

221111

22

221121

σ

σσ

σσ

−=

−−−−−

=

⎥⎦

⎤⎢⎣

⎡ +−−−+−=

dzrVdt

rVa

rV

tVdV σσ

∂∂

+∂∂

+∂∂

+∂∂

= ²)²²

21(

11

22

)()(0PtTPtTn

rV

−−

=→=∂∂As:

Details of calculation

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Monte Carlo experiment # period/year 250dt 0.004Drift a 1.00%sigma 3.00%Short rate r 3%

Maturity Yield Price Quantity ValueBond 1 10.00 3.00% 74.08 0.25 18.84Bond 2 2.00 3.00% 94.18 -‐1.00 -‐94.18Cash 75.34Total V 0.00

dr -‐0.17% =a*dt+NORMSINV(RAND())*sigma*SQRT(dt)r+dr 2.83%

Maturity Yield Price Quantity Value DeltaBond 1 t1-‐dt 10.00 2.83% 75.38 0.25 19.16 0.33Bond 2 t2-‐dt 2.00 2.83% 94.51 -‐1.00 -‐94.51 -‐0.34Cash 75.35 0.01Total V+dV 0.0028 0.0028

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26 April 2012

Binomial no-arbitrage model


Suppose that we observe the following term structure:

Expected binomial evolution of the 6-month rate:

Underlying assumptions : Expected 6-month rate = 2% Standard deviation (per annum) = 1% Ru = 2% + 1% * sqrt(0.5) Rd=2% - 1% * sqrt(0.5) More on this later

26 April 2012

Option valuation

•  How to value a 6-month call option on a 1-year ZC with a strike price K=99?

•  Build a binomial tree for the bond price:


Prices calculated using the 6-month rate prevailing at time t = 0.5 using true proba (riskless bond)

This is the market price. It is not calculated!

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Market price of interest rate risk


= ln( 99.0198.02

) / 0.5

= 2.00%!1.50%= 99.01! e"1.50%!0.50 " 98.02

Shape of term structure determined by 2 forces: -  Expected future spot rate -  Risk aversion (required risk premium)

26 April 2012

Different possible views of current term structure


Small IR increase High risk premium

High IR increase Smal risk premium


Review: forward on zero-coupons

•  Borrowing forward ↔ Selling forward a zero-coupon

•  Long FRA:

0 T T* τ

+M

-M(1+Rτ)

)1

)(ττ

rRrM

+

−


Options on zero-coupons

•  Consider a 6-month call option on a 12-month zero-coupon with face value 100

•  Current spot price of zero-coupon = 98.020 •  Exercise price of call option = 99 •  Payoff at maturity: Max(0, ST – 99) •  The spot price of zero-coupon at the maturity of the option depend on the

6-month interest rate prevailing at that date. •  ST = 100 / (1 + rT 0.50) •  Exercise option if:

•  ST > 99 •  rT < 2.02%


Payoff of a call option on a zero-coupon

•  The exercise rate of the call option is R = 2.02% •  With a little bit of algebra, the payoff of the option can be written as:

•  Interpretation: the payoff of an interest rate put option •  The owner of an IR put option:

•  Receives the difference (if positive) between a fixed rate and a variable rate

•  Calculated on a notional amount •  For an fixed length of time •  At the beginning of the IR period

Max(0, 99(2.02%! rT )0.501+ rT 0.50

)


European options on interest rates

•  Options on zero-coupons •  Face value: M(1+Rτ) •  Exercise price K A call option •  Payoff:

Max(0, ST – K) A put option •  Payoff:

Max(0, K – ST )

•  Option on interest rate

•  Exercise rate R A put option •  Payoff:

Max[0, M (R-rT)τ / (1+rTτ)]

A call option •  Payoff:

Max[0, M (rT -R)τ / (1+rTτ)]

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Example


26 April 2012

Back to 1-period binomial pricing


Based on the previous number, we could calculate the usual binomial parameters (u,d,p)

We then use the 1-period valuation formula.

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We have a problem!


Solution vary with expected change in interest rate (dr) -  Risk premium >0 or <0 -  RN proba >1 or <0!!!

Reason: IR lattice not risk neutral Solution: adjust dr

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Example


E(dr) such that Risk premium = 0

98.020 = (0.50!98.409+ 0.50!99.107)! e"1.50%!0.50

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Multiperiod tree


We can generalize the previous approach to several periods. The idea is to build a binomial tree for the interest rate that fits the initial term structure (zero-coupon prices calculated with the model are equal to market prices). Two popular models in which RN proba = ½:

- Ho-Lee model - Simple Black, Derman and Toy (BDT) model

26 April 2012

Ho-Lee model


ri+1, j = ri, j +!i !"t +" ! "t

ri+1, j+1 = ri, j +!i !"t #" ! "t

The thetas are parameters used to fit the term structure.

Continuous-time model in a risk neutral world (see Hull Chap 30):

dr =!(t)dt +"dz



Cap

•  A cap is a collection of call options on interest rates (caplets). •  The cash flow for each caplet at time t is:

Max[0, M (rt – R) τ] •  M is the principal amount of the cap •  R is the cap rate •  rt is the reference variable interest rate •  τ is the tenor of the cap (the time period between payments)

•  Used for hedging purpose by companies borrowing at variable rate •  If rate rt < R : CF from borrowing = – M rt τ •  If rate rt > R: CF from borrowing = – M rT τ + M (rt – R) τ = – M R τ


Floor

•  A floor is a collection of put options on interest rates (floorlets). •  The cash flow for each floorlet at time t is:

Max[0, M (R –rt) τ] •  M is the principal amount of the cap •  R is the cap rate •  rt is the reference variable interest rate •  τ is the tenor of the cap (the time period between payments)

•  Used for hedging purpose by companies investing at variable rate •  If rate rt > R : CF from investing = M rt τ •  If rate rT < R: CF from investing = M rT τ + M (R – rt ) τ = M R τ

•  Collar: long cap + short floor




26 April 2012

Swaption


A 6-month swaption on a 1-year swap Option maturity: 6 month Swap maturity: 1 year (at option maturity) Swap rate: 1.25% per period (1 period = 6 month)

Remember: Swap = Floating rate note - Fix rate note Swaption = put option on a coupon bond

Bond maturity: 1.5 year Coupon: 2.5% (semi-annual) Option maturity: 6-month Strike price = 100


Valuation formula

•  The value of any bond or derivative in this model is obtained by discounting the expected future value (in a risk neutral world). The discount rate is the current short rate.

trijiijii

ij jiecouponfpfp

f Δ++++ +−+

=,

11,1,1 )1(

i is the number of period j is the number of “downs” Δt is the time step


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Black Derman Toy


zi+1, j = zi, j +!i !"t +" ! "t

zi+1, j+1 = zi, j +!i !"t #" ! "t

ri, j = ezi, j

Lognormal model of short rate evolution

26 April 2012

BDT illustration



Documents

Derivatives Options on Bonds and Interest Rates 11... · 2012-04-26 · – Short 1 t 2-year zero-coupon (Example: Short 1 2-year zc) • In order to have zero-duration: 1 1 2 2 t