PRICING ILLIQUID OPTIONS WITH N + 1 LIQUID PROXIES USING MIXED DYNAMIC-STATIC HEDGING

We study the problem of the optimal pricing and hedging of a European option written on an illiquid asset Z using a set of proxies: a liquid asset S, and N liquid European options Pi, each written on a liquid asset Yi, i = 1, N. We assume that the S-hedge is dynamic while the multi-name Y-hedge is static. Using the indifference pricing approach with an exponential utility, we derive an HJB equation for the value function, and build an efficient numerical algorithm. The latter is based on several changes of variables, a splitting scheme, and a set of fast Gauss Transforms (FGT), and turns out to be more efficient in terms of complexity and lower local space error than a finite-difference method. While in this paper we apply our framework to an incomplete market version of the credit-equity Merton model, the same approach can be used for other asset classes (equities, commodities, foreign exchange, and so on), for example, for pricing and hedging options with illiquid strikes or illiquid exotic options.