TWO-FACTOR E QUILIBRIUM MODELS A number of researchers have investigated the properties of two-factor equilibrium models. For example, Brennan and Schwartz have developed a model where the process for the short rate reverts to a long rate, which in turn follows a stochastic process. 6 The long rate is chosen as the yield on a perpetual bond that pays $1 per year. Because the yield on this bond is the reciprocal of its price, Ito's lemma can be used to calculate the process followed by the yield from the process followed by the price of the bond. The bond is a traded security. This simplifies the analysis because the expected return on the bond in a risk-neutral world must be the risk-free interest rate. Another two-factor model, proposed by Longstaff and Schwartz, starts with a general equi librium model of the economy and derives a term structure model where there is stochastic volatility. 7 The model proves to be analytically quite tractable. NO-ARBITRAGE MODELS The disadvantage of the equilibrium models presented in the preceding few sections is that they do not automatically fit today's term structure. By choosing the parameters judiciously, they can be made to provide an approximate fit to many of the term structures that are encountered in practice. But the fit is not usually an exact one and, in some cases, there are significant errors. Most traders find this unsatisfactory. Not unreasonably, they argue that they can have very little confidence in the price of a bond option when the model does not price the underlying bond correctly. A 1 % error in the price of the underlying bond may lead to a 25% error in an option price. 6 See M. J. Brennan and E. S. Schwartz, "A Continuous Time Approach to Pricing Bonds," Journal ofBanking and 7 See F. A. Longstaff and E. S. Schwartz, "Interest Rate Volatility and the Term Structure: A Two Factor General A no-arbitrage model is a model designed to be exactly consistent with today's term structure of interest rates. The essential difference between an equilibrium and a no-arbitrage model is therefore as follows. In an equilibrium model, today's term structure of interest rates is an output. In a no-arbitrage model, today's term structure of interest rates is an input. In an equilibrium model, the drift of the short rate (i.e., the coefficient of dt) is not usually a function of time. In a no-arbitrage model, this drift is, in general, dependent on time. This is because the shape of the initial zero curve governs the average path taken by the short rate in the future in a no-arbitrage model. If the zero curve is steeply upward sloping for maturities between t\ and t 2 , r has a positive drift between these times; if it is steeply downward sloping for these maturities, r has a negative drift between these times. It turns out that some equilibrium models can be converted to no-arbitrage models by including a function of time in the drift of the short rate. 8 THE HO-LEE MODEL Ho and Lee proposed the first no-arbitrage model of the term structure in a paper in 1986. 8 They presented the model in the form of a binomial tree of bond prices with two parameters: the short- rate standard deviation and the market price of risk of the short rate. It has since been shown that the continuous-time limit of the model is dr = 9(t)dt + crdz (23.12) where a, the instantaneous standard deviation of the short rate, is constant and Q(t) is a function of time chosen to ensure that the model fits the initial term structure. The variable 6{t) defines the average direction that r moves at time t. This is independent of the level of r. Interestingly, Ho and Lee's parameter that concerns the market price of risk proves to be irrelevant when the model is used to price interest rate derivatives. This is analogous to risk preferences being irrelevant in the pricing of stock options. The variable 6(t) can be calculated analytically. It is e(t) = F t (0,t) + a 2 t (23.13) where the F(0, t) is the instantaneous forward rate for a maturity ( as seen at time zero and the subscript t denotes a partial derivative with respect to t. As an approximation 9(t) equals F,(0, r). This means that the average direction that the short rate will be moving in the future is approximately equal to the slope of the instantaneous forward curve. The Ho-Lee model is illustrated in Figure 23.3. The slope of the forward curve defines the average direction that the short rate is moving at any given time. Superimposed on this slope is the normally distributed random outcome. stock option trading | forex trader | making money |