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

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 which 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.

Guaranteed Minimum Withdrawal Benefit in Variable Annuities

“…(2007), we propose an efficient finite difference scheme following the penalty approximation approach to solve for the fair value of the annuities. The numerical procedure of using the penalty approximation approach represents a nice contribution to the family of numerical methods for solving singular stochastic control problems (Kumar and Muthuraman, 2004;Forsyth and Labahn, 2006). In addition, we design the finite difference scheme that allows for discrete jumps across discrete withdrawal dates for solving the discrete time withdrawal model.…”

Analysis of a penalty method for pricing a guaranteed minimum withdrawal benefit (GMWB)

“…We use a policy iteration (Forsyth and Labahn, 2008) to solve the discretized PDE in (4.30). Let we only have three possible values because ϕ, ψ ∈ {0, 1} and ϕψ = 0.…”

Implications of a regime-switching model on natural gas storage valuation and optimal operation

“…Following the proof in (Barles and Jakobsen, 2007;Forsyth and Labahn, 2007) and using Lemma 4.1, we can obtain the monotonicity of scheme (4.21). Remark 4.6.…”

“…We omit the details here. Readers can refer to (D'Halluin et al, 2005, Theorem 5.5) and Forsyth and Labahn (2007) for complete stability proofs of the semi-Lagrangian fully implicit scheme for American Asian options and the finite difference schemes for controlled HJB equations, respectively.…”