Calculo

Journal of Financial Economics 5 (1977) 177-188. 0 North-Holland

Publishing Company

AN EQUILIBRIUM

CHARACTERIZATION STRUCTURE Oldrich VASICEK*

OF THE TERM

Wells Fargo Bank and University of California, Berkeley, CA, U.S.A. Received August 1976, revised version received August 1977 The paper derives a general form of the term structure of interest rates. The following assumptionsare made: (A.l) The instantaneous (spot) interest rate follows a diffusion process; (A.2) the price of a discount bond depends only on the spot rate over its term; and (A.3) the market is efficient. Under these assumptions, it is shown by means of an arbitrage argument that the expected rate of return on any bond in excess of the spot rate is proportional to its standard deviation. This property isthen used to derive a partial differential equation for bond prices. The solution to that equation is given in the form of a stochastic integral representation. An interpretation of the bond pricing formula is provided. The model is illustrated on a specific case.

1. Introduction

Although considerable attention has been paid to equilibrium conditions in capital markets and the pricing ofcapital assets, few results are directly applicable to description of the interest rate structure. The most notable exceptions are the works of Roll (1970, 1971), Merton (1973, 1974), and Long (1974). This paper gives an explicit characterization of the term structure of interest rates in an efficient market. The development of the model is based on an arbitrage argument similar to that of Black andScholes (1973) for option pricing. The model is formulated in continuous time, although some implications for discrete interest rate series are also noted. 2. Notation and assumptions Consider a market in which investors buy and issue default free claims on a specified sum of money to be delivered at a given future date. Such claims will be called (discount) bonds. Let P(t, s) denote the price attime t of a discount bond maturing at time s, t 5 s, with unit maturity value,
P(s,s) = 1.
*The author wishes to thank P. Boyle, M. Garman, M. Jensen, and the referees, R. Roll and S. Schaefer for their helpful comments and suggestions.
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0. Vasicek, Equilibrium and term structure

The yield to maturity R(t, T) is the internal rate of return at time t on a bond with maturity date s= t+T,
R(t,

T) = -;

logP(t,

t+

T),

T> 0.

(1)

The rates R(t, 5”) considered as a function of T will be referred to as the term structure at time t. The forward rate F(t, s) will be defined by the equation
R(t, T) = ;

s
t+T

F(t, r)dr.

f

(2)

In the form explicit for the forward rate, this equation can be written as F(t, s) = ; [(s- t)R(t, s-t)].

(3)

Theforward rate can be interpreted as the marginal rate of return from committing a bond investment for an additional instant. Define now the spot rate as the instantaneous borrowing and lending rate, r(t) = R(t, 0) = lim R(t, T).
T-+0

(4)

A loan of amount W at the spot rate will thus increase in value by the increment d W = Wr(t)dt. (5)

This equation holds with certainty. At any time t, thecurrent value r(t) of the spot rate is the instantaneous rate of increase of the loan value. The subsequent values of the spot rate, however, are not necessarily certain. In fact, it will be assumed that r(t) is a stochastic process, subject to two requirements: First, r(t) is a continuous function of time, that is, it does not change value by an instantaneous jump. Second, it is assumed thatr(t) follows a Markov process. Under this assumption, the future development of the spot rate given its present value is independent of the past development that has led to the present level. The following assumption is thus made : (A.l) The spot rate follows a continuous Markov process.

The Markov property implies that the spot rate process is characterized by a single state variable, namely...