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The Contingent-Claims Arms Race Tuning the parameters for fixed-income option models is as important as choosing the model itself. Roland Lochoff (Reprinted with permission from the Journal of Portfolio Management ) ROLAND LOCHOFF is director of fixed income services at BARRA in Berkeley (CA 94704). Although the cold war is over, “rocket” scientists continue to extend and elaborate fixed-income contingent-claims models for evaluating embedded options. The race centers on whose model is bigger and better. In general, this means that the model is more complex — although not necessarily more effective. The models may have different philosophical starting points, but many of the improvements involve adding a function or an extra equation where there was previously a fixed assumption. Increased flexibility is good, but estimation of the parameters that feed the model remains the most important decision the modeler will make. A FRAMEWORK FOR VIEWING CONTINGENT-CLAIMS MODELS The modeler of the future course of interest rates to value interest rate options must make several choices and abide by certain financial truths to make the model work. First, no- arbitrage opportunities must be assumed. Second, schematically, the models must choose two of three choices as inputs, with the third determined as an output: • The term structure of rates. • The term structure of volatility. • The expected change in forward rates. These ingredients are the legs of the “assumption triangle” shown in Exhibit 1. Any two legs will determine the third. There are several variants on each of these legs, which can be increasingly refined. For example, the term structure of volatility could have one or more factors; it could be path- dependent, or independent, and involve several additional equations that may refer to the assumptions in the other legs of the assumption triangle. EXHIBIT 1 THE ASSUMPTION TRIANGLE Once the legs of the triangle are chosen, the technique to estimate actual values numerically must be chosen. Methods include partial differential equations, interest rate trees, finite difference or grid methods, Monte Carlo approaches, or path integrals. The method chosen is not related to the choice of model. As an example, a Cox, Ingersoll, and Ross [1985] model may be implemented through a tree or through differential equations. WHERE DO WELL-KNOWN MODELS FIT IN THIS FRAMEWORK? There are three possible ways of combining the triangle. In practice, only two of the three combinations have been used: • Term structure models and equilibrium models, which choose the change in forward rates and the term structure of volatility to imply the allowable forms of the term structure of rates. • Relative valuation models, which choose the initial term structure of rates and the term structure of volatility, and do not need to specify expected changes in forward rates. The first approaches to analyzing fixed-income options were the term structure models developed by Vasicek [1977], Dothan [1978], Richard [1978], Brennan and Schwartz [1979], and Schaefer and Schwartz [1984]. These models all invoke a no-arbitrage condition, following Black-Scholes [1973], but they are not relative valuation models. They describe the behavior of the entire term structure, and price bonds as well as options. Because they are not relative valuation models, they all require risk premiums and mean reversion parameters as input. A second approach, related to the term structure models, is the equilibrium model developed by Cox, Ingersoll, and Ross in 1985. Their model is effectively another term structure-type model, but they derive it explicitly from an equilibrium model of the economy. This equilibrium model is more firmly grounded in economic theory, but, in practice, it shares the strengths and weaknesses of the other term structures. Relative valuation models represented by Ho and Lee [1986] and Heath, Jarrow, and Morton [1992] price options relative to a given term structure. They give no insight into the term structure itself, and price options only relative to market prices of the underlying instrument. The advantage of relative valuation is precisely that they do not require estimating expected returns — a difficult task. None of the models contradicts each other. So if, for example, we input the Cox, Ingersoll, and Ross (CIR) term structure and volatility into Heath, Jarrow, and Morton (HJM), the HJM option price should equal the CIR price. After all, there can only be one arbitrage-free price. All these fixed-income option modeling approaches share some basic assumptions. They all assume that market prices are arbitrage-free. They all make assumptions about term structure volatility. CALIBRATING THE MODEL To get a sense of which inputs are important, let’s explore how several models react to changes in inputs. We start with extremely simple models and proceed to the most complex. In all cases it becomes clear that volatility assumptions matter much more than the quality of the model. We will start with the simplest model we can imagine. In this model, interest rates change so that a non-callable bond is priced anywhere from $95 to $105 with uniform probability. Suppose we also have a bond that is callable at par in this interest rate environment. Exhibit 2 shows that the expected value of the option in this framework is $1.36. EXHIBIT 2 Simple Call Model Price of Price of Contingent Expected Non-Callable Callable Value Probability Value 95 95 0 0.091 0.00 96 96 0 0.091 0.00 97 97 0 0.091 0.00 98 98 0 0.091 0.00 99 99 0 0.091 0.00 100 100 0 0.091 0.00 101 100 1 0.091 0.09 102 100 2 0.091 0.18 103 100 3 0.091 0.27 104 100 4 0.091 0.36 105 100 5 0.091 0.45 Option Value (Undiscounted) 1.36 Standard Deviation 3.16 Now we double the volatility. Exhibit 3 illustrates the mechanics. It should come as no surprise that the expected value of the option has doubled. EXHIBIT 3 Increased Volatility Price of Non- Price of Contingent Expected Callable Callable Value Probability Value 90 90 0 0.091 0.00 92 92 0 0.091 0.00 94 94 0 0.091 0.00 96 96 0 0.091 0.00 98 98 0 0.091 0.00 100 100 0 0.091 0.00 102 100 2 0.091 0.18 104 100 4 0.091 0.36 106 100 6 0.091 0.55 108 100 8 0.091 0.73 110 100 10 0.091 0.91 Option Value (Undiscounted) 2.73 Standard Deviation 6.32 We now create another model that is quite different, but equally simple. In this model, interest rates change in such a way that the price of the non-callable bond increases linearly, but with a probability that also increases linearly until par, at which point it decreases linearly. Again we presume a call on the bond at par. Exhibit 4 illustrates the mechanics of this model. EXHIBIT 4 Another Simple Model Price of Non- Price of Contingent Expected Callable Callable Value Probability Value 92 92 0.00 0.00 0.00 93 93 0.00 0.01 0.00 94 94 0.00 0.03 0.00 95 95 0.00 0.05 0.00 96 96 0.00 0.06 0.00 97 97 0.00 0.08 0.00 98 98 0.00 0.10 0.00 99 99 0.00 0.11 0.00 100 100 0.00 0.13 0.00 101 100 1.00 0.11 0.11 102 100 2.00 0.10 0.19 103 100 3.00 0.08 0.24 104 100 4.00 0.06 0.25 105 100 5.00 0.05 0.23 106 100 6.00 0.03 0.17 107 100 7.00 0.01 0.09 108 100 8.00 0.00 0.00 Option Value (Undiscounted) 1.28 Standard Deviation 3.16 Exhibit 5 contrasts the price distributions of the two simple models. Notice that when the standard deviations of the two simple models are equal (Exhibits 2 and 4), the expected value of the option varies by only 8 cents ($1.36 versus $1.28). EXHIBIT 5 PRICE DISTRIBUTIONS FOR TWO SIMPLE MODELS Exhibit 6 s hows the differences in the option values calculated by the two models for different strike prices. Clearly, the level of “at-the-moneyness” makes little difference to the valuations. EXHIBIT 6 OPTION VALUES FOR DIFFERENT CALL PRICES To reiterate, the conclusions are that, in comparing two simple models with different price distributions but identical volatilities: • The at-the-money option prices are very close. • Even for in-the-money or out-of-the-money options, the model prices are very close. Now we have to address the hard question. These conclusions hold with simple models, but do they apply to more realistic models (or to the real market?). What we will do is compare the simple model (we’ll use simple model 1) with the CIR-based model in the BARRA B2 system. We will use three Treasury bonds with different coupons with four years to maturity that are callable at par in two years. To illustrate, we assume a flat 9% yield curve and choose coupons of 8%, 9%, and 10% to represent, respectively, out-of-the-money, at-the- money, and in-the-money embedded options. Exhibit 7 illustrates that naive model gives answers within 4 cents of the sophisticated model, which is our first indication that the conclusions above apply to complicated models. EXHIBIT 7 Comparing Option Values Bond 1 Bond 2 Bond 3 BARRA’s Model 0.57 1.34 2.37 Simple Model 0.61 1.36 2.41 It is not our intention to claim that the BARRA B2 model is only as good as an extremely naive model, but to show that volatility inputs are critical. Exhibit 8 shows results derived using the state-of-the-art HJM model valuing embedded options under scenarios of different volatility. A startling observation emerges: the value is linear with volatility. This result appears to hold for any of the models mentioned at the outset. EXHIBIT 8 HJM MODEL VALUING EMBEDDED OPTIONS UNDER SCENARIOS OF DIFFERENT VOLATILITY Since volatility is critical, it is clear that specifying volatility precisely becomes the key input to any contingent- claims model. The models that allow more choices for volatility are probably going to perform better over a full range of interest rate environments than those with fewer. When we say volatility, we typically think of a single number, perhaps the volatility of short rates, or the implied volatility of T-bond futures. Yet as one obvious contradiction, we know that short rates are almost always more volatile than long. In fact it makes more sense to think in terms of a term structure of volatility. Various models allow specification of the term structure of volatility in different ways. The BARRA B2 model (a variant of the Cox, Ingersoll, and Ross model) uses a mean reversion parameter linked to the volatility assumption. The HJM model allows explicit specification of this function. HJM models may have two or more ways of specifying this volatility. EXHIBIT 9 Two-Factor Model for Callable Government Bonds — Option Values Coupon 6.0 7.0 8.0 9.0 10.0 25-Year Maturity Callable at Par in 20 Years 0.43 0.82 1.25 1.80 2.41 15-Year Maturity Callable at Par in 10 Years 0.82 1.56 2.47 3.50 4.75 10-Year Maturity Callable at Par in 5 Years 0.57 1.41 2.55 4.16 6.32 EXHIBIT 10 “Tuned” One-Factor Model versus Two-Factor Model Differences in Option Values Coupon 6.0 7.0 8.0 9.0 10.0 25-Year Maturity -0.11 -0.09 0.01 0.03 0.07 Callable at Par in 20 Years 15-Year Maturity -0.02 0.11 0.12 -0.02 -0.09 Callable at Par in 10 Years 10-Year Maturity 0.00 0.05 0.00 -0.09 -0.01 Callable at Par in 5 Years Clearly, this provides a more realistic range of outcomes in that there are more degrees of freedom in the process. Nevertheless, it also confronts the modeler with choices that may not be obvious and presents the risk of straying into the realm of misplaced rigor. Exhibits 9 and 10 compare the performance of a “well- tuned” two-factor model and a well-tuned one-factor model. For this range of contingent claims, the extra complexity of the two-factor model (with carefully chosen parameters) gains us little. As you can see in this example, the contingent claims are long-dated options in relatively long bonds. The one-factor model is tuned so that it gives good performance for bonds like these. For a long bond with a short call, this single-factor model may not have the flexibility to stretch to all ranges of outcomes with the same precision as a two-factor model. And, for example, for an option on the spread between short and long rates, the parameters chosen for the long option example may be completely inappropriate. We have tuned the B2 model so that it performs best in areas where options can have the most material effect on the value and risk of the assets it covers. Portfolios that contain a high proportion of derivative instruments, where the value of the contingent claim can be a high proportion of the value of the total underlying, will probably need a two- (or more) factor model to get reasonable values. We hope we have made the case that while option modeling technology has become very advanced and useful, it is also surrounded by a body of cant that can confound the unsuspecting practitioner. To conclude: • Volatility is the most important model parameter. • Even bad models can be tuned to give good results for most simple options. • Good models are good because they work over a wider range of options. REFERENCES Black, F., and M. Scholes. “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy, Vol. 81 (May-June 1973), pp. 637-654. Brennan, M.J., and E.S. Schwartz. “A Continuous Time Approach to the Pricing of Bonds.” Journal of Banking and Finance, Vol. 3 (July 1979), pp. 133-155. Cox, John C., Jonathan E. Ingersoll, Jr., and S.A. Ross. “A Theory of the Term Structure of Interest Rates.” Econometrica, Vol. 53, No. 2 (March 1985). Dothan, U. “On the Term Structure of Interest Rates.” Journal of Financial Economics, Vol. 6 (March 1978), pp. 59-70. Heath, D., R. Jarrow, and A. Morton. “Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claim Valuation.” Econometrica. Vol. 60 (1992), pp. 77-105. Ho, T.S.Y., and S.B. Lee. “Term Structure Movements and Pricing Interest Rate Contingent Claims.” Journal of Finance, Vol. XLI, No. 5 (December 1986). Richard, S. “An Arbitrage Model of the Term Structure of Interest Rates.” Journal of Financial Economics, Vol. 6 (1978), pp. 33-57. Schaefer, S., and E.S. Schwartz. “A Two-Factor Model of the Term Structure: An Approximate Analytical Solution.” Journal of Financial and Quantitative Analysis, Vol. 19, No. 4 (December 1984). Vasicek, O. “An Equilibrium Characterization of the Term Structure.” Journal of Financial Economics, Vol. 5 (November 1977), pp. 177-188.

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