Variance and volatility swap valuation with nonlinear liquidity-feedback stochastic volatility and regime switching

This paper values discretely sampled variance swaps and absolute-return-based volatility swaps under regime-switching stochastic volatility and stochastic liquidity. Extending an existing stochastic-liquidity model, a quadratic liquidity-stress term shifts the variance-factor mean-reversion target. An exponential-quadratic forward transform yields pricing formulas through coupled Riccati equations and a finite-state Markov backward equation. Explicit finite-horizon sufficient conditions establish martingale and transform admissibility. Recomputed numerical experiments assess simulation agreement, integration accuracy and nested liquidity channels. Under the uncalibrated baseline parameterization, delayed variance feedback contributes more than the direct return-liquidity channel, while their cross-channel interaction is small. These conclusions concern the stated illustrative parameter settings and the specified absolute-return contract, whose payoff differs from the square root of realized variance.