Option Pricing In Illiquid Market

Asset pricing issue under liquidity risk has become the most popular research focus in the field of finance, and the key is to analyze the effect of the market illiquidity on the stock return. The main risk faced by the security trade comes from illiquidity, which makes market more uncertain and illiquid stock hold high risk premium to attract the investor. As the new security market develops, the research about liquidity risk is more and more important and lots of scholars explore the impact of illiquidity to the asset pricing.Our paper analyzes how to replicate the payoff of European option under the shock of underlying asset caused by trading when the market is illiquid. Firstly, we setup generalized Black–Scholes model inclusive the initial trading cost, which is the first attempt and different from the current literature. And then we introduce the price shock function applied by Huberman & Stanzl(2001)which is linear to the trade amount and obtain a generalized Black–Scholes pricing PDE . Furthermore, it is found that the precision replication is always cheaper than super replication. Additionally, we provide the numerical results of European call option and discover that the agent should buy more stock or borrow more to replicate the payoff of a call because of the price shock during the trading, and to the contrary, he must short and lend more to replicate a put. Different from the classical Black-Scholes, price shock results in stochastic volatility and an out-of-money option has lower implied volatility than an in-the-money option, which is a strong discovery to the empirical analysis about volatility smile. Finally, we explore the impact of the dividend to the price of the option and find that the dividend leads to a lower price for a call and high price for a put.