Hedging Options under Transaction Costs and Stochastic Volatility

In this paper, we consider the problem of hedging a contingent claim on a stock under transaction-costs and stochastic volatility. Extensive research during the last two decades has clearly demonstrated that the volatility of most stocks is not constant over time. Writers of over-the-counter stock options should take account of the effects of stochastic volatility while pricing and hedging contracts, as the volatility of the underlying is the crucial factor in estimating the price of options. Pricing methods for options under stochastic volatility processes are widely available, but practical methods for hedging under stochastic volatility are rare. The simple delta-vega hedging scheme adds option contracts to the portfolio in order to neutralize the volatility exposure during a short interval of time. This method requires frequent rebalancing of the portfolio, which could be costly due to the bid-ask spread on traded option contracts. Static hedging aims at replication of the final payoff with a fixed portfolio of traded options. The static hedging approach fails however when the traded claims do not match the maturity and the moneyness of the over-the-counter products. In this paper we use a stochastic optimization approach to construct short term delta-vega hedges that take account of future rebalancing and transaction costs. The size of the stochastic optimization model grows exponentially with the number of trading dates considered. We show that the decomposition method PDCGM combined with the interior point solver HOPDM allows for an efficient implementation of the stochastic optimization model in a parallel computing environment. This integration of high performance computing and state-of-the-art decomposition methods provides the means for solving the stochastic volatility hedging model with multiple portfolio rebalancing dates.