Option pricing and hedging under liquidity costs

The purpose of this thesis is to study the option pricing and hedging in an illiquid market. In order to decide the optimal strategy, we choose the maximisation of expected utility of terminal wealth as the identification tool. We design an efficient algorithm via the dynamic programming principle to compute the value function for European options, and calculate the optimal strategy numerically in the binomial market. Based on the numerical solution, we prove that the hedging strategy and the option prices would be distinctly identified by market parameters in the illiquid market. The study of option pricing as the function of initial number of shares allows us to observe a new phenomenon: curves of option price in the illiquid markets are intersected by the horizontal replicating price without transaction cost. And those intersections are very close to each other. That phenomenon implies that the price for selling call option can be lower than the replicating price under some conditions. We further observe the smile effect in the implied volatility and explain that the deeply smile of implied volatility in short-expiration options can be caused by the illiquidity effect in the market. Finally, we compare the implied volatilities which are given by the convex liquidity cost and the proportional transaction cost and prove that the convexity of liquidity cost can amplify the effect of transaction cost. We compare implied volatilities from the model to the real market quotes (S&P 500 index) and analyse how the market parameters affect on the implied volatility in the illiquid market. This comparison reveals an estimation of the level of liquidity in the real market.