A Discrete Time Equivalent Martingale Measure

An equivalent martingale measure selection strategy for discrete time, continuous state, asset price evolution models is proposed. The minimal martingale law is shown to generally fail to produce a probability law in this context. The proposed strategy, termed the Girsanov principle, performs a multiplicative decomposition of asset price movements into a predictable and martingale component with the measure change identifying the discounted asset price process to the martingale component. It is shown that the proposed measure change is relevant for economies in which investors adopt hedging strategies that minimize the variance of a risk adjusted cost of hedging that uses risk adjusted asset prices in calculating hedging returns. Risk adjusted prices deflate asset prices by the asset’s excess return. The explicit form of the change of measure density leads to tractable econometric strategies for testing the validity of the Girsanov principle. A number of interesting applications of the Girsanov principle are also developed. A DISCRETE TIME EQUIVALENT MARTINGALE MEASURE The theory of asset pricing has established the equivalence of the absence of arbitrage opportunities in markets to the existence of an equivalent probability measure under which asset prices, discounted by the accumulation in a money market 1 account, are martingales. This measure is termed an equivalent martingale measure and arbitrage free contingent claims are priced by evaluating the expectations of payoffs, discounted by the money market account, under this equivalent martingale measure. Additionally, the literature on continuous time models and discrete time finite state models contains many examples of contexts in which the equivalent martingale measure is unique and many applied contingent claim valuation models exploit this feature for explicit valuation procedures. For discrete time continuous state processes however, the martingale measure is typically not unique and we need to develop a finer understanding of equivalent martingale measure selection strategies in this context. The context of discrete time continuous state processes is important for applications as the econometric and time series literature is particularly rich in models of this type and the development of asset pricing methods for such processes helps bridge the gap between our statistical knowledge of asset prices and the 2 valuation of contingent claims. For continuous time continuous state processes, ----------------------------------------------------------------------------------------------------------------------------------------------------------------1For infinite dimensional state spaces the notion of no arbitrage needs to be refined to that of absence of free lunches. For further details the reader is referred to Stricker (1990), Back and Pliska (1991), Delbaen (1992), Lakner (1993), Schachermayer (1994). 2 Considerable advances are also currently being made in bridging this gap by