Title: What is the right price of a European option in an incomplete market
Abstract: We study option prices in financial markets where the risky asset prices are modelled by jump diffusions. For simplicity we put the risk free asset price equal to 1. Such markets are typically incomplete, and therefore there are in general infinitely many arbitrage-free option prices in these markets. We consider in particular European options with a terminal payoff F at the terminal time T, and propose that the right price of such an option is the initial wealth needed to make it possible to generate by a self-financing portfolio a terminal wealth which is as close as possible to the payoff F in the sense of variance. We show that such an optimal initial wealth \hat{z} with corresponding optimal portfolio \hat{\pi} exist and are unique. We call \hat{z} the minimal variance price of F and denote it by p_{mv}(F). In the classical Black-Scholes market this price coincides with the classical Black-Scholes option price. If the coefficients of the risky asset prices are deterministic, we show that p_{mv}(F)=E_{Q^{*}}[F], for a specific equivalent martingale measure (EMM) Q^{*}. This shows in particular that the minimal variance price is free from arbitrage. The, for the general case we apply a suitable maximum principle of optimal stochastic control to relate the minimal variance price \hat{z}=p_{mv}(F) to the Hamiltonian and its adjoint processes, and we show that, under some conditions, \hat{z}=p_{mv}(F)=E_{Q_0}[F] for any Q_0 in a family \mathbb{M}_0 of EMMs, described by the set of solutions of a system of linear equations. Finally, We illustrate our results by looking at specific examples.
Publication Year: 2020
Publication Date: 2020-12-17
Language: en
Type: preprint
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