Interest Rate Modelling

275 Pages·2004·4.077 MB·English

by  Simona Svoboda (auth.)

Interest rate modelling
SimonaSvoboda
©SimonaSvoboda2004
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Contents
Introduction ix
Part I: Interest Rate Models 1
Chapter 1. The Vasicek Model 3
1.1. Preliminaries 3
1.2. Term structure equation 5
1.3. Risk-neutral valuation 6
1.4. Note on empirical estimation of the market risk premium 8
1.5. Specific model 9
1.6. Conclusion 15
Appendix 16
Chapter 2. The Cox, Ingersoll and Ross Model 19
2.1. General equilibrium in a simple economy 19
2.2. Equilibrium risk-free rate of interest 23
2.3. Equilibrium expected return on any contingent claim 24
2.4. Value of any contingent claim in equilibrium 25
2.5. A more specialised economy 26
2.6. Term structure model 27
2.7. Distribution of the interest rate 28
2.8. Bond pricing formula 36
2.9. Properties of the bond price under the CIR model 40
2.10. Extending the model to allow time-dependent drift 42
2.11. Comparison of the Vasicek and CIR methods of derivation 44
2.12. More complicated model specifications 46
2.13. Conclusion 46
Chapter 3. The Brennan and Schwartz Model 49
3.1. The generic model 49
3.2. Specific models 54
3.3. Conclusion 57
iii
iv CONTENTS
Chapter 4. Longstaff and Schwartz: A Two-Factor Equilibrium Model 59
4.1. General framework 59
4.2. Equilibrium term structure 66
4.3. Option pricing 72
4.4. Conclusion 75
Chapter 5. Langetieg’s Multi-Factor Equilibrium Framework 77
5.1. Underlying assumptions 77
5.2. Choice of generating process 77
5.3. Multivariate elastic random walk 78
5.4. The bond pricing model 81
5.5. Conclusion 93
Chapter 6. The Ball and Torous Model 95
6.1. Holding period returns 95
6.2. Brownian bridge process 96
6.3. Option valuation 100
6.4. Conclusion 102
Chapter 7. The Hull and White Model 103
7.1. General model formulation 103
7.2. Extension of the Vasicek model 104
7.3. PricingcontingentclaimswithintheextendedVasicekframework 108
7.4. The extended Cox-Ingersoll-Ross model 113
7.5. Fitting model parameters to market data 116
7.6. Conclusion 119
Chapter 8. The Black, Derman and Toy One-Factor Interest Rate
Model 121
8.1. Model characteristics 121
8.2. Pricing contingent claims 122
8.3. Calibrating the lattice to an observed term structure 123
8.4. Continuous time equivalent 125
8.5. A fundamental flaw 128
8.6. Conclusion 133
Chapter 9. The Black and Karasinski Model 135
9.1. The lognormality assumption 135
9.2. Specification of the binomial tree 136
9.3. Matching the lognormal distribution 138
9.4. Conclusion 138
Chapter 10. The Ho and Lee Model 141
10.1. Assumptions 141
10.2. Binomial lattice specification 142
CONTENTS v
10.3. Arbitrage-free interest rate evolution 143
10.4. Relationship to Vasicek and CIR models 152
10.5. Pricing contingent claims 158
10.6. Conclusion 160
Chapter 11. The Heath, Jarrow and Morton Model 161
11.1. Initial specifications 162
11.2. Specifications of the various processes 163
11.3. Arbitrage-free framework 173
11.4. Eliminating the market prices of risk 192
11.5. The problem with forward rates 196
11.6. Unifying framework for contingent claim valuation 196
11.7. Ho and Lee model within the HJM framework 199
11.8. Comparison of equilibrium and arbitrage pricing 201
11.9. Markovian HJM model 206
11.10. Conclusion 211
Chapter 12. Brace, Gatarek and Musiela Model 213
12.1. Introduction 213
12.2. Initial framework 214
12.3. Model of the forward LIBOR rate 215
12.4. Forward risk-neutral measure 218
12.5. Forward LIBOR rate with respect to the forward measure 220
12.6. Derivative pricing 222
12.7. Calibration to market volatilities 225
12.8. Conclusion 226
Part II: Calibration 227
Chapter 13. Calibrating the Hull–White extended Vasicek approach 229
13.1. Using information from the observed term structure of interest
rates and volatilities 229
13.2. Call option on a coupon paying bond 230
13.3. Using market data 232
13.4. Cubic spline interpolation 233
13.5. Constant mean reversion and volatility parameters 234
13.6. Flat volatility term structure 236
13.7. Calibration methodology 237
Appendix 240
vi CONTENTS
Chapter 14. Calibrating the Black, Derman and Toy discrete time
model 247
14.1. Initial Term Structure 247
14.2. Calibrating to interest rate term structure only 248
14.3. Forward Induction: making use of state prices 248
14.4. Pricing contingent claims – Backward Induction 250
14.5. Calibration methodology for a constant volatility parameter 251
Chapter 15. Calibration of the Heath, Jarrow and Morton framework 257
15.1. Volatility Function Specifications 258
15.2. Implied Volatility Specification 261
15.3. Historical Volatility Specification 262
15.4. Principal Component Analysis 262
15.5. Choosing the number of volatility factors 266
15.6. Combining Historical and Implicit Calibration 267
Closing Remarks 269
Bibliography 271
Index 273
Introduction
Growth in the derivatives markets has brought with it an ever-increasing
volume and range of interest rate dependent derivative products. To allow
profitable, efficient trading in these products, accurate and mathematically
sound valuation techniques are required to make pricing, hedging and risk
management of the resulting positions possible.
The value of vanilla European contingent claims such as caps, floors and
swaptions depends only on the level of the yield curve. These types of in-
strumentsarepricedcorrectlyusingthesimplemodeldevelopedbyBlack[5].
This model makes several simplifying assumptions which allow closed-form
valuation formulae to be derived. This class of vanilla contingent claims has
become known as ‘first-generation’ products.
Theseinstrumentsexposeinvestorstotheleveloftheunderlyingyieldcurve
at one point in time. They reflect the investors’ view of the future changes
in the level of the yield curve, not their view of changes in the slope of the
curve. ‘Second-’ and ‘third-generation’ derivatives, such as path-dependent
and barrier options, provide exposure to the relative levels and correlated
movements of various portions of the yield curve. Rather than hedging these
exotic options with the basic underlying instrument, i.e. the bond, the ‘first
generation’ instruments are used. Therefore, the Black model prices of these
‘first generation’ instruments are taken as given. This does not necessarily
imply a belief in the intrinsic correctness of the Black model. Distributional
assumptions which are not included in the Black model, such as mean rever-
sion and skewness, are incorporated by adjusting the implied volatility input.
The more sophisticated models developed allow the pricing of instruments
dependentonthechanginglevelandslopeoftheyieldcurve. Acrucialfactor
is that these models must price the exotic derivatives in a manner that is
consistent with the pricing of vanilla instruments. When assessing the cor-
rectness of any more sophisticated model, its ability to reproduce the Black
pricesofvanillainstrumentsisvital. Itisnotamodel’sa prioriassumptions,
butratherthecorrectnessofitshedgingperformancethatplaysapivotalrole
in its market acceptance.
The calibration of the model is an integral part of its specification, so the
usefulness of a model cannot be assessed without considering the reliability
and robustness of parameter estimation.
vii
viii INTRODUCTION
General framework
The pricing of interest rate contingent claims has two parts. Firstly, a finite
number of pertinent economic fundamentals are used to price all default-free
zero coupon (discount) bonds of varying maturities. This gives rise to an
interest rate term structure, which attempts to explain the relative pricing of
zero coupon bonds of various maturities. Secondly, taking these zero coupon
bond prices as given, all interest rate derivatives may be priced.
As with asset prices, the movement of interest rates is assumed to be de-
termined by a finite number of random shocks, which feed into the model
through stochastic processes. Assuming continuous time and hence also con-
tinuousinterestrates, thesesourcesofrandomnessaremodelledbyBrownian
motions (Wiener processes).
When modelling interest rates we do not have a finite set of assets, but
rather a one-parameter family of assets: the discount bonds, with the matu-
rity date as the parameter. The risk-free rate of interest (short-term interest
rate) is not specified exogenously (as in stock price models), but is the rate
of return on a discount bond with instantaneous maturity. Also, unlike in
assetpricing theory, thefundamental assets– thediscount bonds, may them-
selves be viewed as derivatives. Hence the modelling of the interest rate term
structure may be viewed as tantamount to interest rate derivative pricing.
The theory of interest rate dynamics relies on a degree of abstraction in
that the fundamental assets (the discount bonds) are assumed to be perfect
assets, that is default-free and available in a continuum of maturities.
Approaches to term structure modelling
Themodellingofthetermstructureofinterestratesincontinuoustimelends
itself to various approaches. The most widely used approach has been to
assume the short-term interest rate follows a diffusion process1. Bond prices
arethendeterminedassolutionstoapartialdifferentialequationwhichplaces
restrictions on the relationship of risk premia of bonds of varying maturities.
Unfortunately it is particularly difficult and cumbersome to fit the observed
term structure of interest rates and volatilities within this simple diffusion
model.
Onefactormodels. Oneofthefirstmodelstomakeasignificantimpact
oninterestratemodellingwasbyVasicek[50]. Althoughhispaperisentitled
“An Equilibrium Characterisation of the Term Structure” he does not make
anyassumptionsaboutequilibriumwithinanunderlyingeconomy,nordoeshe
makeuseofanequilibriumargumentinthederivation. Instead,hisderivation
1A diffusion process is a Markovian process for which all realisations or sample func-
tions {Xt,t ∈ [0,∞)} are continuous. A Markovian process has the characteristic that
given the value of Xt, the values of Xs,s > t do not depend on the values of Xu,u < t.
Brownianmotionisadiffusionprocess. [34]
APPROACHES TO TERM STRUCTURE MODELLING ix
relies on an arbitrage argument, much like the one used by Black and Scholes
in the derivation of their option pricing model. Vasicek makes assumptions
about the stochastic evolution of interest rates by exogenously specifying the
process describing the short-term interest rate.
A later approach utilised by Cox, Ingersoll and Ross (CIR) [18] begins
with a rigorous specification of an equilibrium economy which becomes the
foundation for the model specifications. Assumptions are made about the
stochastic evolution of exogenous state variables and about investor prefer-
ences. The form of the short-term interest rate process and hence the prices
of contingent claims are endogenously derived from within the equilibrium
economy. The CIR model is a complete equilibrium model, since bond prices
are derived from exogenous specifications of the economy, that is: production
opportunities, investors’ tastes and beliefs about future states of the world.
Mostmodels,includingtheVasicekmodel, arepartialequilibriumtheories,
since they take as input beliefs about future realisations of the short-term
interest rate (depicted within the functional form of the short-term interest
rateprocess)andmakeassumptionsaboutinvestors’preferences(specifiedby
the market prices of risk). The resulting discount bond yields are based on
these assumptions.
The equilibrium approach has the advantage that the term structure, its
dynamicsandthefunctionalformofthemarketpricesofriskareendogenously
determinedbymeansoftheimposedequilibrium. CIR[18]criticisethepartial
equilibrium approach, since it applies an arbitrage argument to exogenously
specified interest rate dynamics and allows an arbitrary choice of the form of
the market prices of risk which may lead to internal inconsistencies.
The assumption implicit within one-factor models is that all information
about future interest rates is contained in the current instantaneous short-
term interest rate and hence the prices of all default-free bonds may be rep-
resented as functions of this instantaneous rate and time only. Also, within a
one-factorframeworktheinstantaneousreturnsonbondsofallmaturitiesare
perfectly correlated. These characteristics are inconsistent with reality and
motivate the development of multi-factor models.
Multi-factor models. Brennan and Schwartz [10] propose an interest
rate model based on the assumption that the whole term structure can be
expressed as a function of the yields of the longest and shortest maturity
default-free bonds. Longstaff and Schwartz [38] develop a two factor model
of the term structure based on the framework of Cox, Ingersoll and Ross
[18]. The two factors are the short-term interest rate and the instantaneous
varianceofchangesinthisshort-terminterestrate(volatilityoftheshort-term
interest rate). Therefore the prices of contingent claims reflect the current
levels of the interest rate and its volatility. Langetieg [36] develops a general
framework where the short-term interest rate is expressed as the sum of a