Variance optimal hedging with application to electricity markets

In this paper, we use the mean-variance hedging criterion to value contracts in incomplete markets. Although the problem has been well studied in continuous and even discrete frameworks, very few works that incorporate illiquidity constraints have been undertaken, and no algorithm is available in the literature to solve this problem. We first show that a valuation problem incorporating illiquidity constraints with a mean-variance criterion admits a unique solution. We then develop two least squares Monte Carlo algorithms based on the dynamic programming principle to efficiently value these contracts. In these methods, conditional expectations are classically calculated by regression using a dynamic programming approach discretizing the control. The first algorithm calculates the optimal value function. The second algorithm calculates the optimal cashflow generated along the trajectories. Next, we provide an example of the valuation of a load curve contract coming from an energy market. In such contracts, incompleteness comes from the uncertainty surrounding a customer's load, which cannot be hedged, and very tight illiquidity constraints are present. We compare the strategies given by two algorithms and by a closed formula ignoring constraints. Finally, we show that hedging strategies can be very different. A numerical study of the convergence of the algorithms is also given.