No-Arbitrage Valuation of Contingent Claims in Discrete Time

Abstract

Following Harrison and Kreps (1979) and Harrison and Pliska (1981), the valuation of contingent claims in continuous-time and discrete-time finite state space settings is generally based on the no-arbitrage principle, and the use of an equivalent martingale measure. In contrast, for some of the most popular discrete time processes used in finance, such as GARCH processes, the existing literature exclusively uses equilibrium arguments based on the specification of a pricing kernel or a representative agent. We demonstrate that contingent claims can be valued in a conditionally lognormal, discrete time, infinite statespace setup using only the no-arbitrage principle and an equivalent martingale measure which we characterize. Our valuation framework allows for conditionally heteroskedastic stock returns (e.g. GARCH) and generalizes the processes for stock returns investigated in Duan (1995) and Heston and Nandi (2000) by allowing for time-varying interest rates and a time-varying price of risk.