Electron. J. Diff. Equ., Vol. 2014 (2014), No. 165, pp. 1-19.

Option pricing with transaction costs and stochastic volatility

Ionut Florescu, Maria C. Mariani, Indranil SenGupta

In a realistic market with transaction costs, the option pricing problem is known to lead to solving nonlinear partial differential equations even in the simplest model. The nonlinear term in these partial differential equations (PDE) reflects the presence of transaction costs. In this article we consider an underlying general stochastic volatility model. In this case the market is incomplete and the option price is not unique. Under a particular market completion assumption where we use a traded proxy for the volatility, we obtain a non-linear PDE whose solution provides the option price in the presence of transaction costs. This PDE is studied and under suitable regularity conditions, we prove the existence of strong solutions of the problem.

Submitted November 20, 2013. Published July 30, 2014.
Math Subject Classifications: 35R09, 91G20, 91G80.
Key Words: Stochastic volatility models; transaction costs models; nonlinear PDEs; financial market.

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Ionut Florescu
Financial Engineering Division and Hanlon Financial Systems Lab
Stevens Institute of Technology
Castle Point on Hudson
Hoboken, NJ 07030, USA
email: ifloresc@stevens.edu
Maria C. Mariani
Department of Mathematical Sciences
University of Texas at El Paso, Bell Hall 124
El Paso, TX 79968-0514, USA
email: mcmariani@utep.edu
Indranil SenGupta
Department of Mathematics
North Dakota State University
NDSU Dept # 2750, Minard Hall 408E12
Fargo, ND 58108-6050, USA
email: indranil.sengupta@ndsu.edu

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