Interest Rate Dynamics, Derivatives Pricing, and Risk Management

157 Pages·1996·4.08 MB·English

by  Lin Chen (auth.)

Lecture Notes in Economies
and Mathematical Systems 435
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Lin ehen
Interest Rate Dynamies,
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ehen, Lin:
Interest rate dynamics, derivatives pricing, and risk
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(Lecture notes in economics and mathematical systems ; 435)
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Preface
There are two types of tenn structure models in the literature: the equilibrium
models and the no-arbitrage models. And there are, correspondingly, two types of
interest rate derivatives pricing fonnulas based on each type of model of the tenn
structure.
The no-arbitrage models are characterized by the work of Ho and Lee (1986),
Heath, Jarrow, and Morton (1992), Hull and White (1990 and 1993), and Black,
Dennan and Toy (1990). Ho and Lee (1986) invent the no-arbitrage approach to the
tenn structure modeling in the sense that the model tenn structure can fit the initial
(observed) tenn structure of interest rates. There are a number of disadvantages
with their model. First, the model describes the whole volatility structure by a sin
gle parameter, implying a number of unrealistic features. Furthennore, the model
does not incorporate mean reversion. Black-Dennan-Toy (1990) develop a model
along tbe lines of Ho and Lee. They eliminate some of the problems of Ho and Lee
(1986) but create a new one: for a certain specification of the volatility function,
the short rate can be mean-fteeting rather than mean-reverting. Heath, Jarrow and
Morton (1992) (HJM) construct a family of continuous models of the term struc
ture consistent with the initial tenn structure data. Unfortunately, the interest rate
models that result from the HJM approach is usually non-Markov (the distribution
of interest rates in the next period depends not only on the current rate but also on
the rates in the earlier periods). There are only a small number of known forward
rate volatility functions that give rise to Markov models. Moreover, except for a few
simple cases, it is difficult to obtain closed fonn solutions for the values of bond
and interest rate derivatives. The approach of Hull and White (1990 and 1993) to
the no-arbitrage model is to extend the equilibrium models by letting parameters
be time-varying. These generalized tenn structure models have more ftexibility to
fit a given yield curve and the tenn structure of volatility. However, except for the
extended Vasicek model, their approach provides no closed fonn solution and has
to rely on numerical methods.
Generally speaking, the main problem with the no-arbitrage approach to the
tenn structure models is that at any date a function for the tenn structure of interest
rates needs to be estimated and there is no guarantee that the estimated function
will be consistent with the previously estimated function. Not surprisingly, despite
the no-arbitrage approach' s ability to fit the initial tenn structure, its empirical per
fonnance is quite disappointing. For example, Flesaker (1993) tests HJM's model
using a Generalized Method of Moments with three years of daily data for Eurodol
lar futures and futures options and shows that HJM' s approach is incompatible with
the data for most sub-periods. Furthennore, Backus, Foresi and Zin (1994) discover
vi Preface
that the pricing of derivatives based on no-arbitrage tenn structure models can actu
ally lead to systematic arbitrage opportunities as a result of its mispricing of some
assets. They demonstrate that the Black-Dennan-Toy model is likely to overprice
call options on long bonds when interest rates exhibit mean reversion and that this
mispricing can be exploited even when no other traders offer the mispriced assets.
They conclude that the time-dependent parameters cannot substitute for sound fun
damentals.
The equilibrium models of the tenn structure of interest rates, such as Varsicek
(1977) and Cox-lngersoIl-Ross (1985) (CIR), have the important advantage that all
interest rate derivatives are valued on a common basis. Nevertheless, these model
tenn structures do not correctly price actual bonds. One primary reason is that there
are too few model parameters to be adjusted. The other problem with their mod
els is that they have not incorporated sufficient empirical realism. Their models
either allow negative interest rates (Varsicek (1977», assume a constant volatility
of interest rates (Varsicek (1977», or assume perfect correlation between volatil
ity and the short rate (CIR 1985). Although one-factor models offer tractability,
there is compelling reason to believe that a single state variable, such as the short
rate, is not sufficient to capture reasonably weIl the direction of future yield curve
changes. Stambaugh (1988) and Pearson and Sun (1993), among others, have docu
mented more econometric evidence in favor of multi-factor models of interest rates.
Brennan and Schwartz (1979) and Schaefer and Schwartz (1984) have developed
two-factor tenn structure models by taking both the short rate and the long rate as
factors. Longstaff and Schwartz (1992) have recently developed another two-factor
model of interest rates. The choice of the model' s two factors, the short rate and the
volatility of the short rate, allow interest rate derivatives prices to reftect the current
levels of the short rate and volatility. However, as pointed out by some authors, the
joint process of the two factors was chosen for its analytical tractability rather than
its empirical realism. Also. it is argued that the interest rate \'olatility in their model
is not really stochastic, as it is related to the short rate in a way similar to how the
volatility is related to the short rate in CIR' s model.
It is generally believed that equilibrium derivatives pricing fonnulas will be use
ful for practical purposes only until substantial improvements have been made. This
paper attempts to make such an improvement by bringing more empirical realism
to the equilibrium models of derivatives valuation while maintaining the model' s
tractability. Although our model, along with other equilibrium models, has the merit
that all interest rate derivatives are valued within a unified and consistent framework,
it has the disadvantage that it may not fit perfectly a given initial tenn structure.
However, this problem is less severe in our model; there are ten adjustable parame
ters which can be estimated to give the correct (i.e. market-priced) valuations for a
Preface vii
handful of actively traded bonds by calibrating the model. Then the model can price
other less actively traded bonds relative to these benchmark bonds. The appealing
attributes of the model presented are not only its ability to fit data better but also
its ability to explicitly relate the dynamics of the short mean and volatility to the
movements of the term structure and the values of interest rate derivatives. It is this
very property that makes the model useful in dealing with the practical day-to-day
problems of pricing derivatives and managing risks of fixed income securities. An
other major desirable feature of our method is that. with the Green' s function. our
approach has turned the valuation of derivative securities into a numerical integra
tion. This feature has certain advantages over the currently available methods of
derivatives valuation which normally deal with the solving of the valuation partial
differential equation directly. Given the general issues of sensitivity and stability of
numerical solutions to partial differential equation. our approach of valuing deriva
tives by evaluating integrals instead of solving partial differential equation is more
reliable and efficient.
Acknowledgments
This book is based on my dissertation submitted to Harvard University. I would
like to thank the following individuals whose helps had made possible my graduate
studies at Harvard and Stanford: Li Guo-G!1ang, Li Yu-Chang, Chen Yi-Sheng, John
Y. Campbell, Jarnes Coleman, Sanjiv R. Das, Joseph Kalt, Glenn Loury, Robert C.
Merton, F. M. Scherer, Jarnes H. Stock, Daniel W. Stroock, Nancy B. Tuma, and
Louisa van Baalen. I would also like to thank Christian Gilles, Gary Anderson,
Mark Fisher, Parnela Gerbino, Noah Williarns and many other supportive colleagues
at the Federal Reserve Board for hel ps.
LinChen
Washington, D.C. USA
November 1,1995
Contents
1 A Three-Factor Model of the Term Structure of Interest Rates 1
1.1 Introduction . 1
1.2 TheModel .. 4
1.3 Benchmark Case 7
1.4 Green' s Function 10
1.5 Derivatives Pricing 14
1.6 The Term Structure of Interest Rates 21
1.7 Expected Future Short Rate. 29
1.8 Forward Rates . . . . . . . . 35
2 Pricing Interest Rate Derivatives 39
2.1 Introduction . . . . . . . 39
2.2 Bond Options . . . . . . 40
2.3 Caps, Floors, and Collars 43
2.4 Futures Price and Forward Price 45
2.5 Swaps ......... . 50
2.6 Quality Delivery Options 52
2.7 Futures Options .. 55
2.8 American Options . 58
3 Pricing Exotic Options 63
3.1 Introduction.... 63
3.2 Green' s Function in the Presence of Boundaries 63
3.3 Derivatives with Payoffs at Random Times. 65
3.4 Barrier Options . . 66
3.5 Lookback Options. 67
3.6 Yield Options . . . 68
4 Fitting to a Given Term Structure 73
4.1 Introduction.......... 73
4.2 Merging to the Heath-Jarrow-Morton Framework 73
4.3 Whole-Yield Model . . . . . . . . . . . . . . . . 75
xii Contents
5 A Discrete-Time Version of the Model 77
5.1 Introduction ........... . 77
5.2 Construction ofthe Four-Dimensional Lattice 78
5.3 Applications ................. . 89
6 Estimation of the Model 95
6.1 Introduction . . . . . 95
6.2 KaIman Filter . . . . 95
6.3 Maximum Likelihood . 99
6.4 Method ofMoments 101
6.5 Simulated Moments . . 102
7 Managing Interest Rate Risk 105
7.1 Introduction . . . . . . . . . . . . . 105
7.2 Generalized Duration and Convexity 107
7.3 Hedging Ratios . . . . . . . 110
7.4 Hedging: General Approach 112
7.5 Hedging Yield Curve Risk 113
8 Extensions of the Model 119
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 119
8.2 Extension I: Jumping Mean and Diffusing Volatility . 120
8.3 Extension 11: Jumping Mean and Jumping Volatility . 122
9 Concluding Remarks 125
A Proof of Lemma 1 127
B Proof of Proposition 2 131
C Proof of Lemma 2 135
D Proof of Proposition 8 137
EIntegral Equation for Derivative Prices 141
Bibliography 143