Projected SOR, Penalty and Policy Iteration for Real-Options Variational Inequalities

An option to invest whose payoff depends on two random factors is valued by a variational inequality; after a monotone discretisation it is a linear complementarity problem with an M-matrix, which has one solution. This report compares projected SOR, the penalty method, policy iteration and, for an option that expires, Ikonen–Toivanen splitting, on three problems: the McDonald–Siegel option, checked against its closed form on the full two-factor grid; an option on a mean-reverting price, checked by simulating the rule its boundary defines; and an option with a finite life. All methods reach the same discrete solution, to $2.1 \times 10^{-9}$, and the boundary converges to the closed form $X^{\ast} = 2.8560$ at second order, to $2.4 \times 10^{-4}$ on $512 \times 512$ cells. With a sparse direct solver on one thread, projected SOR was the fastest on the perpetual problems at the larger grids, 2.6 s against 26 s and 27 s at $512 \times 512$. Policy iteration from the payoff moved the boundary about one cell per iteration, so its count grew with the grid, from 4 to 23. The penalty error fell as one over the penalty, which must be set by a rule. The nine-point stencil, never an M-matrix, converged and was the most accurate at strong correlation. For the expiring option, splitting with one factorisation matched the exact step solvers at a fraction of their time. The recommendations hold for a direct solver, one thread and problems of this size.