Martingale Measures and Hedging for Discrete-Time Financial Markets

Source abstract

The price of stocks is modelled by a discrete-time, square-integrable, vector-valued process X. No further boundedness condition on X is imposed. Contingent claims H are described by square-integrable random variables. One looks for values v of the initial wealth v that allow for super-hedging H by some portfolio plan. In several cases, the smallest value v is known to coincide with the maximal expectation of H under equivalent martingale measures. Here, within an L 2 -framework, another sufficient condition is provided which can be looked upon as a stronger form of the no-arbitrage condition. The mathematical tool and one of the main contributions is an optional decomposition theorem for a process which is a supermartingale under any equivalent martingale measure. The upper price process for a contingent claim is shown to be a typical example for such a process. Moreover it is shown that in a Markovian model one can restrict attention to Markovian portfolio plans and to Markovian martingale measures.