Analytical Pricing of Discretely Sampled Volatility Swaps Under the 4/2 Stochastic Volatility Model

This paper develops a unified analytical framework for pricing discretely sampled volatility-average swaps under the 4/2 stochastic volatility model. The model accommodates a broad range of volatility dynamics by combining affine and inverse-affine components in the instantaneous volatility specification, thereby unifying and extending the structural features of the classical Heston and 3/2 stochastic volatility models. Closed-form expressions for the conditional complex moments of the asset price are derived and serve as the fundamental building blocks for obtaining explicit analytical pricing formulas for volatility-average swaps under discrete sampling. The validity of the proposed pricing formulas is rigorously established within the admissible parameter space of the model. Extensive numerical experiments verify the accuracy and computational efficiency of the analytical results when compared with Monte Carlo simulations. The numerical analysis further reveals that discretely sampled volatility swap prices converge to their continuous-time counterparts in a manner that may be monotonic or non-monotonic, depending on the interaction between the volatility and inverse-volatility components of the 4/2 model, thereby emphasizing the importance of sampling effects in volatility derivative valuation. A detailed sensitivity analysis demonstrates how variations in the parameters governing the volatility and inverse-volatility components influence the fair strike prices, underscoring the structural flexibility of the 4/2 stochastic volatility model. Overall, the proposed framework provides an analytically tractable and computationally efficient approach for pricing volatility-linked derivatives under discrete sampling, offering valuable insights for both theoretical research and practical applications in volatility markets.