A generalized European option pricing model with risk management

Risk control systems in financial markets with numerous innovative financial products are characterized by infrequent and significant fluctuations (e.g., financial crises and minor disturbances occurring anytime and anywhere). Given that the traditional Black–Scholes (BS) model is difficult to adapt to ever-changing financial markets, to better describe real financial markets, this paper presents a generalized European option-pricing model with stochastic volatility and stochastic interest rates and pure jumps under Levy processes, which are stochastic processes with both stationary and independent increments. We use the Levy–Ito formula and measurement tools to transform logarithmic stock prices into conditions under risk neutral measures, and the characteristic functions of logarithmic stock prices are obtained using the decomposed characteristic function form. We in turn obtain a characteristic function solution based on the Fourier and inverse Fourier transform. Finally, we conduct a Monte Carlo (MC) simulation which highlight the adaptability, accuracy and efficiency of the FFT algorithm. The proposed model is superior from an economic and mathematics perspective, as it not only inherits advantages of the BS model but also better depicts the leptokurtosis and jump phenomenon in option markets. •A generalized model depicting the leptokurtosis and jump phenomenon in optionmarkets is presented.•Good complex stochastic partial differential equation results are obtained.•An adaptable, highly accurate and efficient Fourier transform algorithm is identified.•The theoretical basis of option pricing under risk management is extended.