Hedging European and Barrier options using stochastic optimization

Abstract

We hedge European and Barrier options in a discrete time and discrete space setting by using stochastic optimization to minimize the mean downside hedge error under transaction costs. Scenario trees are generated using a method which ensures the absence of arbitrage and which matches the mean and variance of the underlying asset price in the sampled scenarios to those of a given distribution. The stochastic optimization based strategy is benchmarked to the method of delta hedging for the case where the underlying asset price follows a discretized geometric Brownian motion and implemented for the case where the underlying asset price is driven by a discretized Variance Gamma process.

Notes

μ=0.0028 and σ=0.0189 per week.

μ=0.0001 and σ=0.0146 per week.

The Barrier option was written on 1000 GBP and had a barrier equal to 98% of the strike price.

The return on cash was assumed to be constant at 5% per annum.

A branching factor of 20 was used.

To keep the first two moments of the VG processes similar to those of the GBMs of the previous section we chose (σ, ν, θ) equal to (0.0028, 0.0189, 1) for the underlying of the European option and (0.00017, 0.0146, 1) for the underlying of the Barrier option.