Option Pricing in an Incomplete Market

The purpose of this paper is to illustrate the pricing of options in an incomplete market using the new consistent uplifted martingale measure methodology introduced by Grigorian and Jarrow [2024, Filtration Reduction and Incomplete Markets, Frontiers of Mathematical Finance, 3(1), 78–105; 2023, Filtration Reduction and Completeness in Brownian Motion Models. Working Paper, Cornell University; 2024, Filtration Reduction and Completeness in Jump-Diffusion Models. Working Paper, Cornell University]. We apply it to an incomplete market where a stock has stochastic volatility. Two valuation formulas are generated, depending upon whether the trader is more concerned about volatility or price risk in the construction of a partial replicating portfolio for the option’s payoff.