Numerical Methods for Nonlinear PDEs in Finance

Many problems in finance can be posed in terms of an optimal stochastic control. Some well-known examples include transaction cost/uncertain volatility models [17, 2, 25], passport options [1, 26], unequal borrowing/lending costs in option pricing [9], risk control in reinsurance [23], optimal withdrawals in variable annuities[13], optimal execution of trades [20, 19], and asset allocation [28, 18]. A recent survey on the theoretical aspects of this topic is given in [24]. These optimal stochastic control problems can be formulated as nonlinear Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs). In general, especially in realistic situations where the controls are constrained (e.g. in the case of asset allocation, we may require that trading must cease upon insolvency, that short positions are not allowed, or that position limits are imposed), there are no analytical solutions to the HJB PDEs. At first glance, it would appear to be a formidable task to develop a numerical method for solving such complex PDEs. In addition, there may be no smooth classical solutions to the HJB equations. In this case, we must seek the viscosity solution [12] of these equations. However, using the powerful theory developed in [7, 5, 3] we can devise a general approach for numerically solving these HJB PDEs. This approach ensures convergence to the viscosity solution. The contributions of this article are as follows: