Differential game-theoretic thoughts on option pricing and transaction costs (Q2701834)

A zero-sum differential game is defined, where the two decision-makers are an investor and ``Nature'', respectively. The future price behaviour of a security is given by a stochastic differential equation including the future uncertainty. This uncertainty is taken over by ``Nature'' as a decision-maker who counteracted the investor as much as possible. The Pontryagin maximum principle is applied in order to obtain an expression for a ``fair'' price (``fair'' price for a call option given the optimal investment behaviour of the investor against ''Nature'' playing purely against the investor). NEWLINENEWLINENEWLINEThe same problem is solved again in terms of the HJB (Hamilton-Jacobi-Bellman) approach. A relationship between a restricted version of the Black-Scholes and the Hamilton-Jacobi-Bellman partial differential equations is given. Transaction costs are added to the cost function of the investor. It is shown that how these costs change the behaviour of the investor. NEWLINENEWLINENEWLINEIn the second part of the paper, a nonzero-sum game with three players is considered. Here, the third player is the bank interested in maximising its own profits by choosing the right formula for transaction costs. Therefore a three-person nonzero-sum game, with a special kind of Stackelberg information structure, results. In simple models it is shown that the bank can essentially absorb all the extra profit envisaged by the investor.