Dynamic Hedging of Vanilla and Exotic Options with Transaction Costs

Abstract

The classical finance problem of dynamically hedging a short option in a discrete time environment with transaction costs has generally been approached through either a sub-optimal analytical solution with an instantaneous horizon or through the formulation of a long term horizon dynamic program whose solution is often computationally out of reach. We propose a new methodology to solve the dynamic hedging problem that combines the long term horizon of dynamic programming with computational feasibility, a major feature of the analytic methods. Our methodology has two key performance attributes: first, the ability to significantly reduce expected absolute hedging error on vanilla options where analytic solutions exist and second, the ability to be applied to exotics without analytic solutions where our methodology outperforms heuristic methods. We compare the results of our methodology, which utilizes tools from control theory and stochastic programming, to existing analytic delta hedging methodologies from Black and Scholes (1973) and Leland (1985) when we are shorting vanilla options. Simulation reveals our methodology can produce significantly lower expected absolute hedging error, in both a statistical and economic sense, than these analytic methods. When hedging exotic options where no analytic solution exists we can also apply our methodology. We dynamically hedge a 10 asset basket call option and show that our methodology significantly outperforms the expected absolute hedging error of heuristic hedging methodologies. We perform our tests on simulated underlying assets as well as on empirical S&P 500 data showing our methodology's superiority in each case. We believe this is an exciting new dynamic hedging methodology given this strong performance and applicability to both vanilla and exotic options.