On the performance of delta hedging strategies in exponential Lévy models

Abstract

We consider the performance of non-optimal hedging strategies in exponential Lévy models. Given that both the payoff of the contingent claim and the hedging strategy admit suitable integral representations, we use the Laplace transform approach of Hubalek et al. [Ann. Appl. Probab., 2006, 16(2), 853–885] to derive semi-explicit formulas for the resulting mean-squared hedging error in terms of the cumulant generating function of the underlying Lévy process. In two numerical examples, we apply these results to compare the efficiency of the Black–Scholes hedge and the model delta with the mean–variance optimal hedge in a normal inverse Gaussian and a diffusion-extended CGMY Lévy model.

Keywords:

• Laplace transform approach
• Mean–variance hedging
• Delta hedging
• Lévy processes
• Model misspecification

Classification:

• G1
• G11
• G13

Acknowledgments

We thank the three anonymous referees for their constructive comments that led, in particular, to the numerical example of section 5.2. Moreover, the third author gratefully acknowledges partial financial support from Sachbeihilfe KA 1682/2-1 of the Deutsche Forschungsgemeinschaft. The fourth author gratefully acknowledges financial support from the National Centre of Competence in Research ‘Financial Valuation and Risk Management’ (NCCR FINRISK), Project D1 (Mathematical Methods in Financial Risk Management). The NCCR FINRISK is a research instrument of the Swiss National Science Foundation.

Notes

Note that we assume here that this quantity is given exogenously. Typically, it will be chosen to match some arbitrage-free price the investor receives for selling the option. However, it can also include other initial holdings or liabilities.