Numerical pricing of exchange option with stock liquidity under Bayesian statistical method

In the paper, we consider the pricing problem of exchange option where the two underlying stocks are correlated and imperfectly liquid. Firstly, we obtain an explicit pricing formula for the exchange option in the incomplete market by Esscher measure transformation. Then, a Bayesian statistical method is proposed to estimate the unknown parameters, since the parameter calibration of exchange option pricing model has been tricky with the market data on options unavailable. This method considers the prior information and the correlation of market data on the two underlying stocks based on the conditional posterior distributions. Meanwhile, the posterior inference on the exchange option price is performed combining a Markov chain Monte Carlo algorithm. Finally, an empirical analysis is conducted to investigate the sensitivity of exchange option to stock liquidity. Empirical results indicate that the exchange option price with liquidity-adjustment differs from the corresponding option price under Black-Scholes model. The effect of stock liquidity on exchange option price is significant. Moreover, the proposed method provides reference for the parameter estimation of more complicated models with liquidity.