The valuation of American options with the stochastic liquidity risk and jump risk

Both liquidity risk and jump risk are factors that cannot be ignored in the financial markets. In this paper, we investigate the pricing issue of American options within the framework of underlying asset prices encompassing the aforementioned two risks. First, based the empirical statistical analysis, we propose a novel model to capture the stochastic liquidity risk and jump risk of the underlying asset, and then convert the model into a model under a risk-neutral measure, and derive the characteristic function by solving the governing PDEs. Second, the analytical approximation formulas for the prices of the American options are obtained by using the COS method and Richardson extrapolation scheme. Finally, we conduct detailed numerical experiments. The numerical results compared with the Monte Carlo simulation demonstrate the accuracy and efficiency of our pricing formulas. We perform sensitivity analysis to elucidate the influence of liquidity risk and jump risk on option prices, examining variations in American option prices across various parameters of the model. Additionally, we further substantiated the rationale for incorporating liquidity risk and jump risk into the model by examining the price behavior of option prices under different models.