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

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:

jump-diffusion model , stochastic volatility model , Black-Scholes model