15 Markov Decision Processes in Finance and Dynamic Options

In this paper a discrete-time Markovian model for a nancial market is chosen. The fundamental theorem of asset pricing relates the existence of a martingale measure to the no-arbitrage condition. It is explained how to prove the theorem by stochastic dynamic programming via portfolio optimization. The approach singles out certain martingale measures with additional interesting properties. Furthermore, it is shown how to use dynamic programming to study the smallest initial wealth x that allows for super-hedging a contingent claim by some dynamic portfolio. There, a joint property of the set of policies in a Markov decision model and the set of martingale measures is exploited. The approach extends to dynamic options which are introduced here and are generalizations of American options. 15.