Numerical methods for controlled Hamilton-Jacobi-Bellman PDEs in finance

TLDR

Nonlinear option pricing problems are often cast as optimal control problems, yielding Hamilton–Jacobi–Bellman or Hamilton–Jacobi–Bellman–Isaacs equations. The authors employ a piecewise constant control approximation within a general framework, illustrated by numerical examples for option pricing with asymmetric borrowing/lending costs and stock borrowing fees. Monotone discretization schemes converge to the viscosity solution, and for HJB-type equations a Newton-type policy iteration is guaranteed to converge, though it may fail or be expensive for HJBI cases or jump processes.

Abstract

Many nonlinear option pricing problems can be formulated as optimal control problems, leading to Hamilton–Jacobi–Bellman (HJB) or Hamilton– Jacobi–Bellman–Isaacs (HJBI) equations. We show that such formulations are very convenient for developing monotone discretization methods that ensure convergence to the financially relevant solution, which in this case is the viscosity solution. In addition, for the HJB-type equations, we can guarantee convergence of a Newton-type (policy) iteration scheme for the nonlinear discretized algebraic equations. However, in some cases, the Newton-type iteration cannot be guaranteed to converge (for example, the HJBI case), or can be very costly (for example, for jump processes). In this case, we can use a piecewise constant control approximation. While we use a very general approach, we also include numerical examples for the specific interesting case of option pricing with unequal borrowing/lending costs and stock borrowing fees