A Simplified Method for Hedging Jump Diffusions

Abstract

Geometric Brownian Motion (GBM) and has been widely used in the Black Scholes option-pricing framework to model the return of assets. However, many empirical investigations show that market returns have higher peaks and fatter tails than GBM. Contrary to the Black Scholes model, an option-pricing model which contains jumps reflects the evolution of stock prices more accurately. Therefore, hedging a model under jump diffusion would be desirable. This thesis develops a simplified method for hedging jump diffusions.

In order to hedge the jump risk, other instruments besides the underlying asset must be used in the hedging procedure. We start with a the Partial Integro Differential Equation (PIDE) that models contingent claims with jumps and consider a dynamic hedging strategy that uses a hedging portfolio with the underlying asset and liquidly traded options. We introduce a simple hedging method, where, at each rebalance time, we minimize the instantaneous jump risk by finding proper weights for the underlying asset and instruments.

We use a simulation method to test our approach using a Truncated SVD method to solve the linear system of equations resulting from our minimization procedure. Our results indicate that the proposed dynamic hedging strategy provides sufficient protection against diffusion and jump risk.

The method also provides a firm theoretical basis for a method which is used in practice.