Hedging with Energy: Simple Error Bounds for Mis-Specified Diffusions

Abstract

In the setting of diffusion models for price evolution, we suggest an easily implementable approximate evaluation formula for measuring the errors in option pricing and hedging implied in volatility mis-specification. The main tool we use in this paper is a fundamental inequality for the L-2 norm of the solution, and the derivatives of the solution, of a partial differential equation (the so called inequality). This result allows us to give bounds to the errors implied in the use of approximate models for option valuation and hedging. When statistical or a-priori information is available on the volatility, the error measure can be minimized w.r.t. the parameters of the approximating model. In this case the usual approximate hedging procedure, where hedging parameters are computed using an approximate model calibrated to observed prices (e.g. implied volatility), is sub optimal. The method suggested in this paper can help in conjugating statistical estimation of the volatility function derived from flexible but computationally cumbersome statistical models, with the use of analytically tractable approximate models calibrated using error estimates derived from energy inequalities. The performance of the new method is compared with the standard implied volatility hedging method in an example where the true model is a lognormal mixture while the approximating model is the standard lognormal model.