Inflation and Foreign Exchange Rates Under Production and Monetary Uncertainty

Abstract

Modern contingent pricing theory (CPT) dates its genesis from the pioneering work of Arrow [1] and Debreu [9] in the context of complete markets. Beja [2, 3] demonstrated the application of contingent pricing concepts to incomplete markets. The approach has been applied to the valuation of options (Cox and Ross [7]; Rubinstein [30]) and a variety of other financial instruments (e.g., Ross [28])- Tne fundamental insight of CPT is that in arbitrage-free markets complex securities may always be viewed as additive combinations of simple “state-claims†having positive value which, in effect, pay off one unit if and only if a given state is attained at a given date. Concurrently, the continuoustime viewpoint pioneered by Black and Scholes [4] and Merton [22] has grown in significance. The basic simplification of the continuous-time approach is that relevant valuation quantities may all be expressed in terms of the first two moments, i.e., mean and variance, of the state variable distributions employed. When CPT adopts a continuous-time format, it has been shown (Garman [13]) that a basic differential equation holds for all securities; that differential equation involves, of course, the state-claim values, the distributional parameters of state variable evolution, and the prices and dividends of securities. Alternatively, somewhat stronger assumptions which lead to the existence of a rational consensus investor allow thedifferential equation to be expressed in terms of marginal utilities (Cox, Ingersoll, and Ross [8]). This paper applies the techniques of continuous-time CPT to the foreign exchange market. Since we wish to substantively treat inflationary and productive sources of risk in two countries, four state variables are necessarily involved. In a sense, therefore, this is an ambitious attempt since the mostcomplex continuous-time models to date (e.g.. Brennan and Schwartz [5]), have substantively treated only two state variables. Such complexity is simplified through the use of some compact notation, but not by the use of ad hoc modeling. Indeed, it should be emphasized that the present treatment is a full-equilibrium approach, and that while the compact quality of the notation might be made to incorporate a significant amount of possible additional structure, nothing here is inconsistent with a complete equilibrium.