387 386 MARTINGALE MODELS FOR THE SHORT RATE NOTES Proposition 24.9 (Bond options) Using notation as above we have, both for the Hull-White and the Vasicek models, the following bond option formula: fitted exactly to today's observed bond prices with O as c(t, T, K, S) = p(t, S)N(d) - p(t, T) . K . N(d - op), = (1) 0 aT (0, t) + 02t, (24.49) where where f* denotes the observed forward rates. (The observed bond price curve is d = - [ p(t, S) 1 108 ( p(t, T ) K ) Op + 20 p, assumed to be smooth.) Hint: Use the affine term structure, and fit forward rates rather than bond (24.50) prices (this is logically equivalent). op = =(1 - e-a(5-1)}. = (1 - e- 20 ( I-+ )]. Exercise 24.6 Use the result of the previous exercise in order to derive the (24.51 bond price formula in Proposition 24.4. 24.5 Exercises Exercise 24.7 It is often considered reasonable to demand that a forward rate curve always has an horizontal asymptote, i.e. that limT-co f(t, T) exists for all Exercise 24.1 Consider the Vasicek model, where we always assume that a > 0. t. (The limit will obviously depend upon t and r(t)). The object of this exercise (a) Solve the Vasicek SDE explicitly, and determine the distribution of r(t). is to show that the Ho-Lee model is not consistent with such a demand Hint: The distribution is Gaussian (why?), so it is enough to compute (a) Compute the explicit formula for the forward rate curve f(t, T) for the the expected value and the variance. Ho-Lee model (fitted to the initial term structure). (b) As t - oo, the distribution of r(t) tends to a limiting distribution. Show (b) Now assume that the initial term structure indeed has a horizontal that this is the Gaussian distribution Nb/a, o/v2a]. Thus we see that, asymptote, i.e. that limT- f* (0, T) exists. Show that this property is in the limit, r will indeed oscillate around its mean reversion level b/a. not respected by the Ho-Lee model, by fixing an arbitrary time t, and (c) Now assume that r(0) is a stochastic variable, independent of the Wiener showing that f(t, T) will be asymptotically linear in T. process W, and by definition having the Gaussian distribution obtained Exercise 24.8 The object of this exercise is to indicate why the CIR model is in (b). Show that this implies that r(t) has the limit distribution in (b), connected to squares of linear diffusions. Let Y be given as the solution to the for all values of t. Thus we have found the stationary distribution for the Vasicek model. following SDE. (d) Check that the density function of the limit distribution solves the time dy = (2aY + 02) dt + 20 VYaw, Y(0)=yo. invariant Fokker-Planck equation, i.e. the Fokker-Planck equation with the at-term equal to zero. Define the process Z by Z(t) = VY(t). It turns out that Z satisfies a sto- Exercise 24.2 Show directly that the Vasicek model has an affine term struc- chastic differential equation. Which? ture without using the methodology of Proposition 24.2. Instead use the charac- terization of p(t, T) as an expected value, insert the solution of the SDE for r, 24.6 Notes and look at the structure obtained. Basic papers on short rate models are Vasicek (1977), Hull and White (1990), Exercise 24.3 Try to carry out the program outlined above for the Dothan Ho and Lee (1986), Cox et al. (1985), Dothan (1978), and Black et al. (1990). model and convince yourself that you will only get a mess. For extensions and notes on the affine term structure theory, see Duffie and Kan Exercise 24.4 Show that for the Dothan model you have EQ [B(t)] = 0o. (1996). An extensive analysis of the linear quadratic structure of the CIR model can be found in Magshoodi (1996). The bond option formula for the Vasicek Exercise 24.5 Consider the Ho-Lee model model was first derived by Jamshidian (1989). For examples of two-factor models see Brennan and Schwartz (1979) and Longstaff and Schwartz (1992). dr = O(t) dt + odw(t). Assume that the observed bond prices at t = 0 are given by {p* (0, T); TZ 0). Assume furthermore that the constant o is given. Show that this model can be