On an Extension Multifractional SABR Model for Pricing Variance and Volatility Swaps

This paper presents a robust methodology for the valuation of options on variance swaps and volatility swaps. While existing literature has often focused on the pricing of the swaps themselves under stochastic volatility models, the valuation of options on these second-order derivative products remains a challenge, particularly within a framework that captures empirical market properties such as volatility clustering and long memory. Our approach addresses this problem by relying on the precise calculation of the optimal exercise prices (or strike rates) for variance and volatility swaps. To achieve this, we introduce and utilize a multifractional extension of the classic SABR model. This enhanced version of the model incorporates a deterministic and time-varying Hurst function, h(t), which allows for the modeling of a richer and more flexible volatility dynamics, capable of replicating the varying persistence phenomena observed in real markets. The proposed methodology is detailed comprehensively. We derive the theoretical formulas for the swap prices, which form the basis for calculating the optimal exercise prices. The valuation of options (call and put) on these swaps is then performed by employing advanced numerical techniques, adapted to the complexity of the multifractional process.