A New Analytical Approach for Pricing Variance and Volatility Swaps: Incorporating Liquidity and Self‐Exciting Jumps

ABSTRACT Our investigation focuses on the valuation of variance and volatility swaps by incorporating both clustered jump behaviors and liquidity‐related risk factors. The proposed modeling framework characterizes stock price movements using a jump‐diffusion process, where Hawkes processes govern the jump component in the absence of liquidity effects. Stochastic liquidity is further integrated through a discount factor that modifies the underlying asset value. To preserve model generality, we introduce a complete correlation framework across all Brownian motions involved. A change of measure is applied to express the dynamics under a risk‐neutral pricing measure. Subsequently, the forward characteristic function associated with the log‐price is expressed analytically, enabling the derivation of closed‐form swap valuation formulae. Using daily data from the Brent crude oil futures continuous contract, we calibrate relevant model parameters, providing empirical evidence supporting jump clustering. Numerical implementation of the formulae further offers insights into their sensitivity to variations in model parameters.