Valuation of Financial Derivatives Under the 4/2 Stochastic Volatility Model

Abstract

This research focuses on the valuation of European derivatives and real options under a rescaled double mean-reverting 4/2 stochastic volatility, which extends the 4/2 stochastic volatility model originally proposed by Grasselli in 2017 by combining fast and slow mean-reverting terms for asset volatility. Moreover, closed-form approximate pricing formulas are obtained for European derivatives by using an asymptotic analysis approach under the rescaled (double) mean-reverting 4/2 stochastic volatility model(s). The approximate pricing formulas use the Black–Scholes price as the leading term, combined with correction terms from the asymptotic expansion. The accuracy of closed-form approximate pricing formulas for European derivatives is verified through MonteCarlo simulation and market option data. The results show that the option values obtained from pricing formulas can be well fitted by the simulation results under different maturities and strike price conditions (moneyness), and are also well validated by S&P 500 market data. The advantages of the 4/2 stochastic volatility model in reflecting option price sensitivity are demonstrated by adjusting parameters and comparing the results with those from the Heston model and the 3/2 stochastic volatility model.

For real options, closed-form approximate pricing formulas are obtained for the values of real options and investment thresholds by using an asymptotic analysis approach under the double mean-reverting 4/2 stochastic volatility model. In numerical experiments, sensitivity analysis is performed with respect to the model parameters (the "Heston" and "3/2" factors) and other associated variables. Additionally, the accuracy of the real option approximate pricing formula is verified by comparisons with simulated values obtained using the Least Squares Monte Carlo method. The results show that all relative errors are below 0.3%, confirming the reliability of the proposed method.

In conclusion, this research provides closed-form approximate pricing formulas for the values of European derivatives, real options and investment thresholds under the rescaled double mean-reverting stochastic volatility model. The accuracy and robustness of these formulas are verified through numerical methods.