John C. Cox

A Theory of the Term Structure of Interest Rates

AbstractThis paper uses an intertemporal general equilibrium asset pricing model to study the term structure of interest rates. In this model, anticipations, risk aversion, investment alternatives, and preferences about the timing of consumption all play a role in determining bond prices. Many of the factors traditionally mentioned as influencing the term structure are thus included in a way which is fully consistent with maximizing behavior and rational expectations. The model leads to specific formulas for bond prices which are well suited for empirical testing.

Option pricing: A simplified approach

This paper presents a simple discrete-time model for valuing options. The fundamental economic principles of option pricing by arbitrage methods are particularly clear in this setting. Its development requires only elementary mathematics, yet it contains as a special limiting case the celebrated Black-Scholes model, which has previously been derived only by much more difficult methods. The basic model readily lends itself to generalization in many ways. Moreover, by its very construction, it gives rise to a simple and efficient numerical procedure for valuing options for which premature exercise may be optimal.

The valuation of options for alternative stochastic processes

Abstract This paper examines the structure of option valuation problems and develops a new technique for their solution. It also introduces several jump and diffusion processes which have not been used in previous models. The technique is applied to these processes to find explicit option valuation formulas, and solutions to some previously unsolved problems involving the pricing of securities with payouts and potential bankruptcy.

An Intertemporal General Equilibrium Model of Asset Prices

This paper develops a continuous time general equilibrium model of a simple but complete economy and uses it to examine the behavior of asset prices. In this model, asset prices and their stochastic properties are determined endogenously. One principal result is a partial differential equation which asset prices must satisfy. The solution of this equation gives the equilibrium price of any asset in terms of the underlying real variables in the economy. IN THIS PAPER, we develop a general equilibrium asset pricing model for use in applied research. An important feature of the model is its integration of real and financial markets. Among other things, the model endogenously determines the stochastic process followed by the equilibrium price of any financial asset and shows how this process depends on the underlying real variables. The model is fully consistent with rational expectations and maximizing behavior on the part of all agents. Our framework is general enough to include many of the fundamental forces affecting asset markets, yet it is tractable enough to be specialized easily to produce specific testable results. Furthermore, the model can be extended in a number of straightforward ways. Consequently, it is well suited to a wide variety of applications. For example, in a companion paper, Cox, Ingersoll, and Ross [7], we use the model to develop a theory of the term structure of interest rates. Many studies have been concerned with various aspects of asset pricing under uncertainty. The most relevant to our work are the important papers on intertemporal asset pricing by Merton [19] and Lucas [16]. Working in a continuous time framework, Merton derives a relationship among the equilibrium expected rates of return on assets. He shows that when investment opportunities are changing randomly over time this relationship will include effects which have no analogue in a static one period model. Lucas considers an economy with homogeneous individuals and a single consumption good which is produced by a number of processes. The random output of these processes is exogenously determined and perishable. Assets are defined as claims to all or a part of the output of a process, and the equilibrium determines the asset prices. Our theory draws on some elements of both of these papers. Like Merton, we formulate our model in continuous time and make full use of the analytical tractability that this affords. The economic structure of our model is somewhat similar to that of Lucas. However, we include both endogenous production and

OPTIMAL CONSUMPTION AND PORTFOLIO POLICIES WHEN ASSET PRICES FOLLOW A DIFFUSION PROCESS

Abstract We consider a consumption-portfolio problem in continuous time under uncertainty. A martingale technique is employed to characterize optimal consumption-portfolio policies when there exist nonnegativity constraints on consumption and on final wealth. We also provide a way to compute and verify optimal policies. Our verification theorem for optimal policies involves a linear partial differential equation, unlike the nonlinear partial differential equation of dynamic programming. The relationship between our approach and dynamic programming is discussed. We demonstrate our technique by explicitly computing optimal policies in a series of examples. In particular, we solve the optimal consumption-portfolio problem for hyperbolic absolute risk aversion utility functions when the asset prices follow a geometric Brownian motion. The optimal policies in this case are no longer linear when nonnegativity constraints on consumption and on final wealth are included. By these examples, one can see that our approach is much easier than the dynamic programming approach.

The relation between forward prices and futures prices

Abstract This paper consolidates the results of some recent work on the relation between forward prices and futures prices. It develops a number of propositions characterizing the two prices. These propositions contain several testable implications about the difference between forward and futures prices. Many of the propositions show that equilibrium forward and futures prices are equal to the values of particular assets, even though they are not in themselves asset prices. The paper then illustrates these results in the context of two valuation models and discusses the effects of taxes and other institutional factors.

A variational problem arising in financial economics

Abstract We provide sufficient conditions for a dynamic consumption–portfolio problem in continuous time to have a solution. When the price processes satisfy a regularity condition, all utility functions that are continuous, increasing, concave, and are dominated by a strictly concave power function admit a solution.

On dynamic investment strategies

Abstract This paper presents a new approach for analyzing dynamic investment strategies. Previous studies have obtained explicit results by restricting utility functions to a few specific forms; not surprisingly, the resultant dynamic strategies have exhibited a very limited range of behavior. In contrast, we examine what might be called the inverse problem: given any specific dynamic strategy, can we characterize the results of following it through time? More precisely, can we determine whether it is self-financing, yields path-independent returns, and is consistent with optimal behavior for some expected utility maximizing investor? We provide necessary and sufficient conditions for a dynamic strategy to satisfy each of these properties.

A continuous-time portfolio turnpike theorem

Abstract We prove a continuous-time portfolio turnpike theorem. The proof uses the theory of martin-gales and is more intuitively appealing than the usual discrete-time mode of proof using dynamic programming. When the interest rate is strictly positive, the present value of any contingent claim having payoffs bounded from above can be made arbitrarily small when the investment horizon increases. Thus an investor concentrates his wealth in buying contingent claims that have payoffs unbounded from above at the very beginning of his horizon. As a consequence, it is the asymptotic property of his utility function as wealth goes to infinity that determines his optimal investment strategy at the very beginning of his horizon.

The Black—Scholes Formula

An introduction to mathematical finance would not be complete without an exposition of its most famous result: the Black—Scholes formula for the price of European call and put options.