Higher Dimensional Fair Option Pricing and Hedging Under Hara and Cara Utilities

Abstract

By means of the optimal portfolio approach, and in particular by finding what we call the "fundamental matrix of derivatives pricing and hedging", we have formulated a very compact, yet very general form of a Black-Scholes type pricing PDE, and of Black-Scholes type hedging, for any contract in complete or incomplete, multi-factor, multi-tradable, Itô-SDE-setting, under HARA or CARA utility. Our approach to pricing is the one first introduced by Kallsen, called "neutral derivative pricing", which assumes that the considered derivative is traded actively. Furthermore, we discuss the relationship between HARA and CARA option pricing theories, and provide an example of a model with stochastic interest rates (in an incomplete market) for which between HARA and CARA ("neutral") pricing theories, only HARA theory is possible.

Keywords: neutral derivative pricing, optimal portfolio, stochastic interest rates, incomplete markets, interst rate derivatives, hedging

JEL Classification: G13, G11