The Non-Uniqueness of Partial Differential Equations Options Price Valuation Formula for Heston Stochastic Volatility Model

Authors: H. D. Ibrahim, H. C. Chinwenyi, T. Danjuma

Abstract:

An option is defined as a financial contract that provides the holder the right but not the obligation to buy or sell a specified quantity of an underlying asset in the future at a fixed price (called a strike price) on or before the expiration date of the option. This paper examined two approaches for derivation of Partial Differential Equation (PDE) options price valuation formula for the Heston stochastic volatility model. We obtained various PDE option price valuation formulas using the riskless portfolio method and the application of Feynman-Kac theorem respectively. From the results obtained, we see that the two derived PDEs for Heston model are distinct and non-unique. This establishes the fact of incompleteness in the model for option price valuation.

Keywords: Option price valuation, Partial Differential Equations, Black-Scholes PDEs, Ito process.

References:

[1] Heath, D. & Schweizer, M. (2000). Martingales versus PDEs in Finance: An Equivalence Result with Examples. A Journal of Applied Probability, 37, 947-957.
[2] Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. The Journal of Political Economy, 81(3), 637-654.
[3] Lorig, M. & Sircar, R. (2016). Stochastic Volatility: Modeling and Asymptotic Approaches to Option Pricing and Portfolio Selection. Financial Signal Processing and machine Learning, 135-161.
[4] Haugh, M. (2010). Introduction to Stochastic Calculus. Financial Engineering: Continuous-Time Models.
[5] Kluge, T. (2002). Pricing Derivatives in Stochastic Volatility Models using the Finite Difference Method. Diploma thesis, Technische UniversitÄat Chemnitz FakultÄat fÄur Mathematik. 5-36.
[6] Heston, S.L. (1993). A closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Review of Financial Studies, 6(2): 327-343.