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Dynamic Hedging Under Jump Diffusion with Transaction Costs

Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1 and Morgan Stanley, New York, New York 10036
David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
School of Accounting and Finance, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
shannon.kennedy{at}morganstanley.com
paforsyt{at}uwaterloo.ca
kvetzal{at}uwaterloo.ca

If the price of an asset follows a jump diffusion process, the market is in general incomplete. In this case, hedging a contingent claim written on the asset is not a trivial matter, and other instruments besides the underlying must be used to hedge in order to provide adequate protection against jump risk. We devise a dynamic hedging strategy that uses a hedge portfolio consisting of the underlying asset and liquidly traded options, where transaction costs are assumed present due to a relative bid-ask spread. At each rebalance time, the hedge weights are chosen to simultaneously (i) eliminate the instantaneous diffusion risk by imposing delta neutrality, and (ii) minimize an objective that is a linear combination of a jump risk and transaction cost penalty function. Because reducing the jump risk is a competing goal vis-à-vis controlling for transaction cost, the respective components in the objective must be appropriately weighted. Hedging simulations of this procedure are carried out, and our results indicate that the proposed dynamic hedging strategy provides sufficient protection against the diffusion and jump risk while not incurring large transaction costs.

Subject classifications: finance; asset pricing.
History: Received April 2006; revision received January 2008; accepted May 2008.