Jump Diffusion & Stochastic Volatility Models for Option Pricing (Application in Python & MATLAB)

Document Type : Original Article

Authors

1 Master of Finance, Department of Finance and Accounting, Tehran Faculty of Petroleum, Petroleum University of Technology, Tehran, Iran

2 Assistant Professor, Department of Finance and Accounting, Tehran Faculty of Petroleum, Petroleum University of Technology, Tehran, Iran

Abstract

The Black-Scholes model assumes that the price of the underlying asset follows a geometric Brownian motion. This assumption has two implications: first, log-returns over any horizon are normally distributed with constant volatility σ and the second, stock price evolution is continuous, therefore, there is no market gaps. These conditions are commonly violated in practice: empirical returns typically exhibit fatter tails than a normal distribution, volatility is not constant over time, and markets do sometimes gap. The existence of volatility skew will misprice options price. Derived from these flaws, a number of models have proposed. In this paper we will analyze, simulate and compare two most important models which have widespread using: jump diffusion model and stochastic volatility model. Each of the aforementioned models have programmed in MATLAB and Python, then their results have been compared together in order to provide a robust understanding of each of them. Our results show that in comparison to Black-Scholes model these two models yield better performance.

Keywords

References

1)     Golbabai, A., O. Nikan, and T. Nikazad. 2019. Numerical analysis of time fractional Black–Scholes European option pricing model arising in financial market. Computational and Applied Mathematics 38 (4):173.

2)     Li, Z., W.-G. Zhang, Y.-J. Liu, and Y. Zhang. 2019. Pricing discrete barrier options under jump-diffusion model with liquidity risk. International Review of Economics & Finance 59:347-368.

3)     Ma, J., S. Huang, and W. Xu. 2020. An Efficient Convergent Willow Tree Method for American and Exotic Option Pricing under Stochastic Volatility Models. The Journal of Derivatives 27 (3):75-98.

4)     Tian, Y., and H. Zhang. 2020. European option pricing under stochastic volatility jump-diffusion models with transaction cost. Computers & Mathematics with Applications 79 (9):2722-2741.

5)     Das S and Sundaram K, 1999, ‘Of Smiles and Smirks: A Term Structure Perspective’, Journal of Financial and Quantitative Analysis, Vol. 34, Issue 2, Pages 211-239.

6)     Silverio Foresi, and Liuren Wu, 2005, ‘Crash-O-Phobia: A Domestic Fear or A Worldwide Concern?’, Journal of Derivatives, Issue 3, Pages 8-21.

7)     Heston Steven, 1993, ‘A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options’, The Review of Financial Studies, Vol. 6, Issue 2, Pages 327–343.

8)      Hull John and White Alan, 1993, ‘One-Factor Interest-Rate Models and the Valuation of Interest-Rate Derivative Securities’, Journal of Financial and Quantitative Analysis, Vol. 28, Issue 2, Pages 235-254.

9)      Jorion Philippe, 1988, ‘On Jump Processes in the Foreign Exchange and Stock Markets’, The Review of Financial Studies, Vol.1, Issue 4, Pages 427–445.

10)  Merton Robert, 1976, ‘Option pricing when underlying stock returns are discontinuous’, Journal of Financial Economics, Vol.3, Issues 1-2, Pages 125-144

Files

Share

How to cite

Statistics