Regime-Switching Fischer–Margrabe Options Pricing with Liquidity Risk and Stochastic Volatility

This article presents a model for pricing an exchange option considering stochastic volatility and liquidity risk. The impact of liquidity risk on an asset price is considered by utilizing a liquidity discount process that is influenced by both market and asset-specific liquidity. Girsanov’s theorem is applied to transform from the real-world probability measure to equivalent probability measures, such as the risk-neutral probability measure. The Feynman–Kac theorem is applied to transform the exchange option pricing formula into the vanilla option pricing formula. The analytical expression is derived through the characteristic function approach. The accuracy of the proposed formula is validated through comparisons with Monte Carlo simulation, where the relative error remains below 0.93% across different values of S(0) and τ. Furthermore, numerical experiments highlight that incorporating liquidity risk leads to higher option prices. As the maturity increases from 0.1 to 2.0, the percentage gap between the option prices increases from 1.65% to 20.2%. Finally, sensitivity analysis is conducted to examine the influence of various parameters and to demonstrate the impact of stochastic volatility and liquidity in exchange option valuation.