260 Mathematics for Finance
we need a model of possible movements of bond prices for each maturity. The
bond prices with different maturities have to be consistent with each other. As
we have seen above, the specification of
a) a model of possible short rates,b) a model of possible values of a bond with the longest maturity (consistent
with the initial term structure)
determines the structure of possible prices of all bonds maturing earlier. An
alternative approach is to specify
a) a model of possible short rates,b) the probabilities at each state,
and, taking these probabilities as risk-neutral ones, to compute the prices of all
bonds for all maturities. The latter method is conceptually simpler, especially
if we take the same probability in each state. The flexibility of short rate mod-
elling allows us to obtain sufficiently many models consistent with the initial
term structure.
If so, the simplest choice of probability 1/2 at each state of the binomial
tree appears to be as good as any, and we can focus on constructing short rate
models so that their parameters are consistent with historical data. A general
short-rate model in discrete time can be described as follows. Lettn=nτ.
Then the following relations specify a tree of the rate movements
r(tn+1)−r(tn)=μ(tn,r(tn))τ+σ(tn,r(tn))ξnwhereξn =±1 with probability 1/2 each, andμ(t, r),σ(t, r) are suitably
chosen functions. In the continuous time limit (in the spirit of Section 3.3.2)
these relations lead to a stochastic differential equation of the form
dr(t)=μ(t, r(t))dt+σ(t, r(t))dw(t).There are many ways in which the functionsμandσcan be specified, but none
of them is universally accepted. Here are just a few examples:μ(t, r)=b−ar,
σ(t, r)=σ(Vasiˇcek model),μ(t, r)=a(b−r),σ(t, r)=σ
√
r(Cox–Ingersoll–
Ross model), orμ(t, r)=θ(t)r, σ(t, r)=σr(Black–Derman–Toy model).
Given the short-rates, the next step is to compute the bond prices. These
will depend on the functionsμandσ. Two problems may be encountered:
- The model is too crude, for example these functions are just constants. Then
we may not be able to adjust them so that the resulting bond prices agree
with the initial term structure. - The model is too complicated, for example we take absolutely general func-
tionsμ, σ.Fitting the initial term structure imposes some constraints on
the parameters, but many are left free and the result is too general to be of
any practical use.