Unpublished memorandum, 1979
Let the price at time u of a discount bond maturing at time v be described by the stochastic differential equation
where is a Wiener process. As shown in Vasicek (1977) (Chapter 6 of this volume), the mean and volatility σ(u, v) of the instantaneous rate of return are related by
where is the spot rate and is the market price of risk. Eq. (1) can be written as
Integrate Eq. (3) over u from t to . We get
Now differentiate ...