Professor Samar Mukhopadhyay’s article was accepted by IEEE Transactions on Industrial Informatics.



Abstract
Several empirical studies show that, under multiple risks like stochastic volatility and jump risks, markets exhibit many new properties, such as volatility smile and cluster fueled by the explosion of transaction data. The traditional Black-Scholes model fails to fit these newly-developed characteristics. This paper attempts to capture these newer features, using the valuation of European options as a vehicle. Statistical analysis performed on the data collected from the currency option market clearly shows the existence of mean reversion, jumps, volatility smile, and leptokurtosis and fat tail.

We characterize the dynamics of the underlying asset in this kind of environment by establishing a coupled stochastic differential equation model with triple characteristics of mean reversion, non-affine stochastic volatility and mixed-exponential jumps. However, the traditional no-arbitrage option pricing theory no longer applies for analytical solution of this model. Moreover, the commonly used Monte Carlo simulation to numerically calculate the option prices takes a long time, especially for a huge amount of data. We propose a characteristic function method to derive the closed-form pricing formula. We also present a Fast Fourier Transform (FFT) algorithm-based numerical solution method.

Finally, extensive numerical experiments are conducted to validate both the modeling methodology and the numerical algorithm.Results demonstrate that the model behaves well in capturing the properties observed in the market, and the FFT numerical algorithm is both accurate and efficient in addressing large amount of data.