Often, an interest rate model is not enough to determine the value of a fixed-income security or interest rate derivative. To value mortgage- backed securities or collateralized mortgage obligations, one also needs a prepayment model. To value bonds or interest rate derivatives with signif- icant credit risk, one needs a model of default and recovery. To value interest-sensitive annuities and insurance liabilities, one needs models of lapse and other policyholder behaviors. In all of these behavioral models, the levels of certain interest rates are important explanatory variates, meaning that, for example, the prepayment speeds in a CMO valuation system are driven primarily by the interest rate scenarios.
Common practice has been to estimate parameters for prepayment, default, and lapse models using regression on historical data about inter- est rates and other variables. Then, in the valuation process, the analyst uses the interest rates from a set of risk neutral scenarios to derive esti- mates for the rates of prepayment, default, or lapse along those scenar- ios. This borrower behavior information is combined with the interest rates to produce cash flows and, ultimately, prices. Unfortunately, this practice leads to highly misleading results.
The primary problem here is that the regressions have been estimated using historical data, reflecting the real probability distributions of bor- rower behavior, and then used with scenarios from a risk neutral model,
2-Fitton/McNatt Page 35 Thursday, August 29, 2002 10:02 AM
with an artificial probability distribution. The risk neutral model is not a process for the short rate; rather, it is a process for the risk adjusted short rate. Since the real world is risk averse, the risk adjusted short rate usu- ally has an expected value much higher than the market’s forecast of the short rate; the extra premium for interest rate risk permits one to value optionable default-free bonds by reference to the forward rate curve.
The same procedure can be applied to corporate bonds. Corporate bonds are exposed to default risk in addition to interest rate risk. One may construct a behavioral model of failure to pay based on historical data about default rates and recovery, perhaps using bond ratings as explanatory variates in addition to interest rates. One can then attempt to compute the present value of a corporate bond by finding the expected value of the discounted cash flows from the two models in combination: a risk neutral model of the Treasury curve, and a realistic model of default behavior as a function of interest rates and other vari- ables. Because the cash flows of the bond, adjusted for default, will be less than the cash flows for a default-free bond, the model will price the corporate bond at a positive spread over the Treasury curve.
This spread will almost certainly be substantially too low in compari- son to the corporate’s market price. The reason for this is that, just as investors demand a return premium for interest rate risk, they demand an additional return for default risk. The application of an econometrically estimated model of default to pricing has ignored the default risk pre- mium encapsulated in the prices of corporate bonds. Market practice has evolved a simple solution to this; one adjusts the default model to fit (sta- tistically, in the equilibrium case; exactly, in the arbitrage-free case) the current prices of active corporates in the appropriate rating class. By using the market prices of active corporates to imbed the default risk pre- mium in the model, the analyst is really applying the principle of risk neu- tral valuation to the default rate. The combined model of risk adjusted interest rates and risk adjusted default rates now discounts using the cor- porate bond spot rate curve instead of the Treasury spot curve.
The same technique of risk neutralizing a model by embedding information about risk premia derived from current market prices can be applied to prepayment models as well. The results of a prepayment model can be risk adjusted by examining the prices of active mortgage- backed securities. Unfortunately, one can only guess at the appropriate expected return premium for insurance policy lapse risk or mortality risk. Nevertheless, these quantities should be used to “risk neutralize”
these models of behavior to the extent practical. The integrity of risk neutral valuation depends on risk adjusting all variables modeled; oth- erwise, model prices will be consistently overstated.
The Four Faces of an Interest Rate Model 37
A final note can be made in this regard about option adjusted spread (OAS). OAS can be understood in this context as a crude method to risk adjust the pricing system to reflect all risk factors not explicitly mod- eled.
Realistic and Arbitrage-Free
A realistic, arbitrage-free model starts by exactly matching the term structure of interest rates implied by a set of market prices on an initial date, then evolves that curve into the future according to the realistic probability measure. This form of a model is useful for producing sce- narios for evaluation of hedges or portfolio strategies, where it is impor- tant that the initial curve in each scenario exactly matches current market prices. The difficulty with such an approach lies in the estima- tion; realistic, arbitrage-free models are affected by confounding, where it is impossible to discriminate between model misspecification error and the term premia. Since the model parameters have been set to match market prices exactly, without regard to historical behavior, too few degrees of freedom remain to estimate both the term premia and an error term. Unless the model perfectly describes the true term structure process (that is, the time dependent parameters make the residual pric- ing error zero at all past and future dates, not just on the date of estima- tion), the term premia cannot be determined. The result is that realistic, arbitrage-free models are not of practical use.
Realistic and Equilibrium
Since the arbitrage-free form of a realistic model is not available, the equilibrium form must be used for stress testing, Value at Risk (VAR) calculations, reserve and asset adequacy testing, and other uses of realis- tic scenarios.
Some analysts express concern that, because the predicted initial curve under the equilibrium model does not perfectly match observed market prices, then the results of scenario testing will be invalid. How- ever, the use of an equilibrium form does not require that the predic- tions be used instead of the current market prices as the first point in a scenario. The scenarios can contain the observed curve at the initial date and the conditional predictions at future dates. This does not introduce inconsistency, because the equilibrium model is a statistical model of term structure behavior; by taking this approach we explicitly recognize that its predictions will deviate from observed values by some error. In contrast, the use of an arbitrage-free, realistic model implicitly assumes that the model used for the term structure process is absolutely correct.
2-Fitton/McNatt Page 37 Thursday, August 29, 2002 10:02 AM
Summary of the Four Faces
Exhibit 2.1 summarizes the uses of the four faces of an interest rate model.
Exhibit 2.2 shows the mathematical form of a commonly used interest rate model, disseminated by Black and Karasinski,4 under each of the modeling approaches and probability measures. In each equation, u is the natural logarithm of the short rate.
In the above models, σ is the instantaneous volatility of the short rate process,κ is the rate of mean reversion, θ is the mean level to which the nat- ural logarithm of the short rate is reverting, and λ represents the term pre- mium demanded by the market for holding bonds of longer maturity. The value of the state variable u at the time of estimation is represented by u0.
The realistic model forms can be distinguished from the risk neutral forms by the presence of the term premium function λ. The difference between the arbitrage-free forms and the equilibrium forms can be discerned in that the parameters of the arbitrage-free forms are functions of time.
EXHIBIT 2.1 When to Use Each of the Model Types
Model Classification Risk Neutral Realistic
Arbitrage-free • Current pricing, where input data (market prices) are reliable
• Unusable, since term pre- mium cannot be reliably estimated
Equilibrium • Current pricing, where inputs (market prices) are unreliable or unavailable
• Horizon pricing
• Stress testing
• Reserve and asset ade- quacy testing
EXHIBIT 2.2 Four Forms of the Black-Karasinski Model Model
Classification
Risk
Neutral Realistic
Arbitrage- free
du = κ(t) (θ(t)−u)dt + σ(t)dz du = κ(t) (θ(t)− λ(u,t)−u)dt + σ(t)dz
•u0 and θ(t) matched to bond prices
•κ(t) and σ(t) matched to cap or option prices
•u0 and θ(t) matched to bond prices
•κ(t) and σ(t) matched to cap or option prices
•λ(u,t) cannot be reliably estimated Equilibrium du = κ(θ−u)dt + σdz du = κ(θ - λ(u) - u)dt + σdz
•u0 statistically fit to bond prices
•u0 statistically fit to bond prices
•κ,θ,σ historically esti- mated
•κ,θ,σ,λ(u) historically estimated
4Fischer Black and Piotr Karasinski, “Bond and Option Pricing when Short Rates are Lognormal,” Financial Analysts Journal (July–August 1991), pp. 52–59.
CHAPTER 3
39
A Review of No Arbitrage Interest Rate Models
Gerald W. Buetow, Jr., Ph.D., CFA President BFRC Services, LLC Frank J. Fabozzi, Ph.D., CFA Adjunct Professor of Finance School of Management Yale University James Sochacki, Ph.D.
Associate Professor of Applied Mathematics Department of Mathematics and Statistics James Madison University
nterest rates are commonly modeled using stochastic differential equa- tions (SDEs). One-factor models use an SDE to represent the short rate and two-factor models use an SDE for both the short rate and the long rate. The SDEs used to model interest rates must capture some of the market properties of interest rates such as mean reversion and/or a vola- tility that depends on the level of interest rates. There are two distinct approaches used to implement the SDEs into a term structure model:
equilibrium and no arbitrage. Each can be used to value bonds and interest rate contingent claims. Both approaches start with the same SDEs but apply the SDE under a different framework to price securities.
I
3-Buetow/Sochacki Page 39 Thursday, August 29, 2002 10:01 AM
Equilibrium models such as those developed by Vasicek ,1 Cox, Ingersoll, and Ross,2 Longstaff,3 Longstaff and Schwartz,4 and Brennan and Schwartz5 all start with an SDE model and develop pricing mecha- nisms for bonds under an equilibrium framework. The actual implemen- tation may vary depending on the model. Vasicek and CIR develop analytic pricing expressions while Backus, Foresi, and Telmer6 present econometric and recursive approaches to implement the equilibrium models. Brennan and Schwartz use a finite difference scheme that approximates a partial differential equation.
No arbitrage models such as Black and Karasinski ,7 Black, Derman, and Toy,8 Ho and Lee,9 Heath, Jarrow, and Morton,10 and Hull and White11 begin with the same or similar SDE models as the equilibrium approach but use market prices to generate an interest rate lattice. The lattice represents the short rate in such a way as to ensure there is a no arbitrage relationship between the market and the model. The numerical
1O. Vasicek, “An Equilibrium Characterization of the Term Structure,” Journal of Financial Economics (1977), pp. 177–188.
2J. Cox, J. Ingersoll, and S. Ross, “A Theory of the Term Structure of Interest Rates,”Econometrica (1985), pp. 385–408.
3F. Longstaff, “A Non-linear General Equilibrium Model of the Term Structure of Interest Rates,” Journal of Financial Economics (1989), 23, pp. 195–224 and “Mul- tiple Equilibria and Term Structure Models,” Journal of Financial Economics (1992), pp. 333–344.
4F. Longstaff and E. Schwartz, “Interest Rate Volatility and the Term Structure: A Two-Factor General Equilibrium Model,” Journal of Finance (1992), pp. 1259–1282.
5M. Brennan and E. Schwartz, “A Continuous Time Approach to the Pricing of Bonds,”Journal of Banking and Finance (1979), pp. 133–155, and, “An Equilibri- um Model of Bond Pricing and a Test of Market Efficiency,” Journal of Financial and Quantitative Analysis (1982), pp. 301–329.
6D. Backus, S. Foresi, and C. Telmer, “Affine Term Structure Models and the For- ward Premium Anomaly,” Journal of Finance (2001), pp. 279–304.
7F. Black and P. Karasinski, “Bond and Option Pricing when Short Rates are Log- normal,”Financial Analyst Journal (July–August 1991), pp. 52–59.
8F. Black, E. Derman, and W. Toy, “A One Factor Model of Interest Rates and Its Application to the Treasury Bond Options,” Financial Analyst Journal (January–
February 1990), pp. 33–39.
9T. Ho and S. Lee, “Term Structure Movements and Pricing Interest Rate Contin- gent Claims,” Journal of Finance (1986), pp. 1011–1029.
10D. Heath, R. Jarrow, and A. Morton, “Bond Pricing and the Term Structure of Interest Rates: A New Methodology,” Econometrica (1992), pp. 77–105.
11J. Hull and A. White, “Pricing Interest Rate Derivative Securities,” Review of Fi- nancial Studies (1990), 3, pp. 573–592, and, “One Factor Interest Rate Models and the Valuation of Interest Rate Derivative Securities,” Journal of Financial and Quan- titative Analysis (1993), pp. 235–254.
A Review of No Arbitrage Interest Rate Models 41
approach used to generate the lattice will depend on the SDE model(s) being used to represent interest rates.
No arbitrage models are the preferred framework to value interest rate derivatives. This is because they minimally ensure that the market prices for bonds are exact. Equilibrium models will not price bonds exactly and this can have tremendous effects on the corresponding con- tingent claims. No arbitrage lattices also allow for a systematic valua- tion approach to almost all interest rate securities.
Three general SDE functional forms are considered in this work. The first is the Hull-White (HW) model. The HW model is a more general version of the Ho and Lee (HL)12 approach except that it allows for mean reversion. Implementing the HW in a binomial framework removes a degree of freedom and in this case the HW model collapses to the HL model if a constant time step is retained. The second model we consider is the Black-Karasinski (BK) model. The BK model is a more general form of the Kalotay, Williams, and Fabozzi (KWF) model.13 The BK model (like the HW model) in the binomial setting does not have enough degrees of freedom to be properly modeled and so the time step must be allowed to vary. The third is the Black, Derman, and Toy model.
We implement the HW and BK trinomial models using the Hull and White approach. Within the trinomial setting the time step remains con- stant and mean reversion can be explicitly incorporated. We discuss the SDEs, the properties of the SDEs, the numerical solutions to the SDEs, and the binomial and trinomial interest rate lattices for these models.
The focus of our presentation is on the end user and developer of interest rate models. We will highlight some significant differences across models. Most of these are due to the different distributions that underlie the models. This is done to emphasize the need to calibrate all models to the market prior to their use. By calibrating the models to the market we reduce the effects of the distributional differences and ensure a higher level of consistency in the metrics produced by the models.
The outline of this chapter is as follows. In the next section we present the SDEs and some of their mathematical properties. We also use the mathematics to highlight properties of the short rate. We then develop the methodology used to implement our approach in both the binomial and trinomial frameworks. A comparison of some numerical results across the different models including some interest rate risk and valuation metrics is then presented.
12T. Ho and S. Lee, “Term Structure Movements and Pricing Interest Rate Contin- gent Claims.”
13A. Kalotay, G. Williams, and F.J. Fabozzi, “A Model for the Valuation of Bonds and Embedded Options,” Financial Analyst Journal (May–June 1993), pp. 35–46.
3-Buetow/Sochacki Page 41 Thursday, August 29, 2002 10:01 AM
42 INTEREST RATE AND TERM STRUCTURE MODELING