Approximate hedging with constant proportional transaction costs in financial markets with jumps

We study a problem of option replication under constant proportional transaction costs in models where stochastic volatility and jumps are combined to capture the market's important features. Assuming some mild condition on the jump size distribution, we show that transaction costs can be approximately compensated by applying the Leland adjusting volatility principle and the asymptotic property of the hedging error due to discrete readjustments. In particular, the jump risk can be approximately eliminated, and the results established in continuous diffusion models are recovered. The study also confirms that, for the case of constant trading cost rate, the approximate results established by Kabanov and Safarian [Finance Stoch., 1 (1997), pp. 239–250] and by Pergamenschikov [Ann. Appl.Probab., 13 (2003), pp. 1099–1118] are still valid in jump-diffusion models with deterministic volatility.