Nonlinear diffusion equations in mathematical finance: A study of transaction costs and uncertain volatility

We consider the standard models of contingent claim valuation, Black-Scholes and Cox-Ross-Rubinstein, and relax the following assumptions: (i) no transaction costs for buying or selling the underlying asset; (ii) complete markets with constant volatility.; First, we study a continuous-time pricing model with proportional transaction costs. We show that the associated equations are Black-Scholes-type non-linear diffusion equations with mixed volatilities which depend on the convexity of the value function. As transaction costs grow relative to the size of the time-interval between transactions, the equations become ill-posed. The valuation problem must then be interpreted as the solution to a degenerate diffusion equation in the sense of an obstacle problem.; We also study discrete models of pricing and hedging with transaction costs. We describe the asymptotic analysis of the discrete-time optimal pricing model with transaction costs of Bensaid, Lesne, Pages and Scheinkman. The equations describing the asymptotic behaviour of this scheme are similar to the ones derived in continuous-time modeling. Therefore, it is possible to relate the continuous-time hedging strategies to an optimality principle, without recourse to a specific utility or preference function.; The second topic is uncertain volatility and incomplete markets. We present a contingent claim pricing/hedging model with an uncertain volatility for the underlying asset. The only assumption about volatility is that it is a process bounded between two known volatilities {dollar}sigmasb{lcub}min{rcub}{dollar} and {dollar}sigmasb{lcub}max{rcub}.{dollar} Under this assumption, we show that the lowest price for a riskless hedging strategy is given by the solution to a non-linear diffusion equation of the same type as that of the transaction cost problem.; Finally, we present an explicit finite difference scheme for solving the non-linear PDEs that arise from these problems. It is easy to implement and can be easily adapted to solve American-type contingent claims and to add term structures.