Unpublished memorandum, 1979

Let the price at time u of a discount bond maturing at time v be described by the stochastic differential equation

1

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

3

Integrate Eq. (3) over u from t to . We get

4

Now differentiate this equation with respect to v. This produces

5

where

6

is the forward rate, and

7

is the volatility of the forward rate . The stochastic differential equation corresponding to the integral form (5) is

8

We note that the drift of forward rates is fully determined by their volatilities and the pricing of risk.

The dynamic of the spot rate is described by

9

where is the volatility of the spot rate. The drift of the spot rate is equal to the slope of the forward rate curve at the origin, less the market price of risk multiplied by the volatility of the spot rate. This is consistent with equations (21) and (22) in Vasicek (1977).

Put in Eq. (5). Then

10

Taking the expectation as of time t yields the equation

11

The liquidity premium (or term premium, as it should be called) is given by

12

The liquidity premium in a term structure of interest rates has two components. The first component is driven by the market price of risk. It is equal to the expected integral over the span of the forward rate of the forward rate volatility multiplied by the market price of risk. There is, however, a second component, which is present even if the market price of risk is zero. This component, equal to the negative of the expected aggregate over the forward rate span of the bond price volatility times the forward rate volatility, arises as a result of the nonlinear relationship between prices and rates.