The Martingale Options Price Valuation for European Puts Using Stochastic Differential Equation Models

Authors: H. C. Chinwenyi, H. D. Ibrahim, F. A. Ahmed

Abstract:

In modern financial mathematics, valuing derivatives such as options is often a tedious task. This is simply because their fair and correct prices in the future are often probabilistic. This paper examines three different Stochastic Differential Equation (SDE) models in finance; the Constant Elasticity of Variance (CEV) model, the Balck-Karasinski model, and the Heston model. The various Martingales option price valuation formulas for these three models were obtained using the replicating portfolio method. Also, the numerical solution of the derived Martingales options price valuation equations for the SDEs models was carried out using the Monte Carlo method which was implemented using MATLAB. Furthermore, results from the numerical examples using published data from the Nigeria Stock Exchange (NSE), all share index data show the effect of increase in the underlying asset value (stock price) on the value of the European Put Option for these models. From the results obtained, we see that an increase in the stock price yields a decrease in the value of the European put option price. Hence, this guides the option holder in making a quality decision by not exercising his right on the option.

Keywords: Equivalent Martingale Measure, European Put Option, Girsanov Theorem, Martingales, Monte Carlo method, option price valuation, option price valuation formula.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.3669271

References:

[1] Statistics Department –International Monetary Fund (1998). Eleventh Meeting of the IMF Committee on Balance of Payments Statistics on Financial Derivatives. Washington, D.C., October 21-23, 1998.
[2] Heath, D. & Schweizer, M. (2000). Martingales versus PDEs in Finance: An Equivalence Result with xamples. Journal of Applied Probability 37, 947-957 Equivalence Result with Examples. A Journal of Applied Probability, 37, 947-957
[3] Privault, N. (2016). Notes on Stochastic Finance. Chapter 6 on Martingale Approach to Pricing and Hedging, December 20, 2016.
[4] Haugh, M. (2010). Introduction to Stochastic Calculus. Financial Engineering: Continuous-Time Models.
[5] Rouah, F. D. (2017). “Four Derivations of the Black ScholesPDE”. http://www.frouah.com/finance%20notes/Black%20Schooles%20PDE.pdf
[6] Cox, J. C. (1975). The Constant Elasticity of Variance Option Pricing Model. Journal of Portfolio Management, Special Issue December, 1996, 15-17.
[7] Black, F. & Karasinski, P. (1991). Bond and Option Pricing when Short Rates are Lognormal. Financial Analysts Journal, July-August 1991, 52-59.
[8] Wu, Z. (2012). Pricing American Option using Monte Carlo Method. A thesis submitted for the degree of Master of Science in Mathematics and Computational Finance, St Catherine’s College, University of Oxford.
[9] Lu, B. (2012). Monte Carlo Simulations and Option Pricing. Undergraduate Mathematics Department, Pennsylvania State University, 2012.
[10] http://www.nse.com.ng/market_data-site/trading-statistics-site/other-market-information/weekly- report