We propose the use of a mean-quadratic-variation criteria to determine an optimal trading strategy 5 in the presence of price impact. We derive the Hamilton Jacobi Bellman (HJB) Partial Differential 6 Equation (PDE) for the optimal strategy, assuming the underlying asset follows Geometric Brownian

A jump diffusion model coupled with a local volatility function has been suggested by Andersen and Andreasen (2000). This model is attractive in that it shows promise in terms of being able to capture observed market cross-sectional implied volatilities, without being unduly complex. By generating a discrete set of American option prices assuming a jump diffusion with known parameters (i.e. in a synthetic market), we investigate two crucial challenges intrinsic to this type of model: calibration of parameters and hedging of jump risk. Our investigation suggests that it can be difficult to estimate the model parameters that govern the jump size distribution. However, the local volatility function is easier to estimate when an appropriate regularization (e.g. splines) is used to avoid over-fitting. In general, even though the estimation problem is ill-posed, it appears that combining jump diffusion with a local volatility function produces a model which can be calibrated with sufficient accuracy to prices of liquid vanilla options. With regard to hedging jump risk, two different hedging strategies are explored: a semi-static approach which uses a portfolio of the underlying and traded short maturity options to hedge a long maturity option, and a dynamic technique which involves frequent trading of options and the underlying. Simulation experiments in the synthetic market suggest that both of these methods can be used to sharply reduce the standard deviation of the hedging portfolio relative profit and loss distribution.

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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. Since 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.