Existence and Uniqueness of Martingale Solutions to Option Pricing Equations with Noise
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Lithuanian Mathematical Journal
Existence and uniqueness of martingale solutions to option pricing equations with noise∗ Jun Zhao, Ru Zhou, and Peibiao Zhao1 Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China (e-mail: [email protected]; [email protected]; [email protected]) Received September 14, 2019; revised April 2, 2020
Abstract. We introduce a new option pricing equation with noise in a frictional financial market, which is fully different from the classical option pricing equation, and arrive at the existence of martingale solutions of this option pricing equation regardless of incompressibility. Furthermore, we also discuss the uniqueness of martingale solutions. MSC: 35Q30, 91B24 Keywords: martingale solution, existence and uniqueness, incomplete hedging, option pricing equation with noise
1 Introduction At the end of the 1970s, Black and Scholes [4] proposed the classical option pricing equation in a frictionfree complete financial market, which has made a great contribution to the development of option pricing. As is well known, the celebrated Black–Scholes (BS) equation is deduced by constructing a risk-free portfolio Π = ΔS − V such that dΠ = rΠ dt, where r stands for the risk-free rate, V is the contingent claim, and S is the price of risky asset. In particular, the number of units of risky asset S is Δ = ∂V /∂S . This is the well-known delta-hedging principle. The existence of the risk-free hedging principle, roughly speaking, is equivalent to the fact that the market is completely frictionless. In other words, the requirement dΠ = rΠ dt is a very ideal condition, which essentially means that the portfolio Π is risk-free and the contingent claim V can be perfectly achievable. However, such a portfolio Π making it possible to achieve complete hedging is not always accessible in reality due to the incompleteness of financial markets caused by all kinds of frictions. Therefore in frictional financial markets, the incomplete hedging is not only intrinsic, but also brings essential difficulties to the solution of the new option pricing equation in the sense of incomplete hedging. As stated in [23], there are two major approaches that have been developed in searching for solutions of option pricing in incomplete markets. One is picking a specific martingale measure for pricing according to some optimal criterion (see [3,9,11,12,25]), and the other is the optimization method (see [14,22]). Of course, there are some studies on the relationship between risk measurement and hedging [7, 8]. ∗ 1
This work was supported by the National Natural Science Foundation (NNSF) of China (Nos. 11871275, 11371194) and Postgraduate Research & Practice Innovation Program of Jiangsu Province. Corresponding author.
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J. Zhao, R. Zhou, and P. Zhao
As far as we know, the study of the existence and uniqueness of solutions to option pricing equations with incomplete hedging in fricti
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