Unpublished memorandum, 1979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

2

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

4

Now differentiate ...

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