A Jump Diffusion Model with Fast Mean-Reverting Stochastic Volatility for Pricing Vulnerable Options

Abstract

The Black–Scholes–Merton option pricing model is a classical approach that assumes that the underlying asset prices follow a normal distribution with constant volatility. However, this assumption is often violated in real-world financial markets, resulting in mispricing and inaccurate hedging strategies for options. Such discrepancies may result into financial losses for investors and other related market inefficiencies. To address this issue, this study proposes a jump diffusion model with fast mean-reverting stochastic volatility to capture the impact of market price jumps on vulnerable options. The performance of the proposed model was compared under three different error distributions: normal, Student-t, and skewed Student-t, and under different market scenarios that consist of bullish, bearish, and neutral markets. In a simulation study, the results show that our model under skewed Student-t distribution performs better in pricing vulnerable options than the rest under different market scenarios. Our proposed model was fitted to S&P 500 Index by maximum likelihood estimation for the mean and volatility processes and Gillespie algorithm for the jump process. The best model was selected based on AIC and BIC. Samples of the simulated values were compared with the S&P 500 values and MSE computed at various sample sizes. Values of MSE at different sample sizes indicate significant decrease to actual MSE values demonstrating that it provides the best fit for modeling vulnerable options.