Jumping hedges on the strength of the Mellin transform

With more looming uncertainties in our present financial climate and environment, models with jump–diffusion more than ever are necessary. They are suited to reproduce the large and sudden fluctuations in the level of the underlying variable, and mimic various statistical properties in observed time series. The jump–diffusion modelling setup, however, brings complexity to the valuation and hedging of derivative securities. This paper delves into the subject of hedging along with the illustration of hedging's intimate interplay with pricing. We harness the power of the Mellin transform and its convolution property to establish hedging sensitivities that capture the many dimensions of risk in option positions. Our methodology allows for a wide class of generalised payoffs (i.e. all piecewise linear functions). Each hedging parameter is shown to contain the impact of jumps and an explicit metric to quantify the jump risk is defined. The systematic and efficient calculations of higher-order sensitivities are demonstrated and their implications, from a practical perspective, are also considered. Inclusion of some numerical examples illustrates the ease of implementation of our results.