Analytical valuation for geometric Asian options in illiquid markets

Classical option pricing theories are usually built on the paradigm of competitive and frictionless markets, while ignoring the impact of market liquidity on asset prices. In this paper, we study the pricing problems of the continuously monitored fixed and floating strike Asian options in imperfectly liquid markets. First, we assume that the dynamics of the underlying asset price follow a liquidity-adjusted Black-Scholes model. And then we derive the partial differential equation for the continuous fixed and floating strike geometric Asian options based on the delta-hedging strategy. Meanwhile, we also present the PDE governing the price of the continuous fixed and floating strike arithmetic Asian options. Second, the analytical pricing formulas for the continuous fixed and floating strike geometric Asian call options are derived by using the PDE method. Besides, we demonstrate the put-call parity relations for the continuous fixed and floating strike geometric Asian options. Finally, numerical experiments are performed to illustrate the accuracy and efficiency of the proposed liquidity-adjusted option pricing model through comparing the analytical solution with Monte Carlo simulation. Furthermore, we investigate the sensitivity of the continuous arithmetic and geometric Asian option prices to the liquidity factors. The numerical results support our idea of introducing market liquidity effect into option pricing framework.