| It is well-known that the liquidity has a huge impact on the financial market and illiquidtiy will cause transaction loss, or so-called liquidity cost. In this paper, we have studied the optimal self-finance trading strategy to hedge the investor’s risk in the position of an European option, within the appearance of liquidity cost.When the investor wants to trade in a small time period (for example, high frequency trading), the chance is that he will have to accept some loss in the prices in order to achieve his deal. This loss could be regarded as the result of the market’s illiquidity. In this paper, we have analyzed the impact of high frequency trading on the price distribution of order book. We have proposed certain form of the liquidity loss function l. We shall assume that the underlying asset follows the log-normal distribution and the trading strategy is a semi-martingale process. We suggest that the investor could construct his optimal portfolio through optimizing following target:minimize the sum of cumulated liquidity loss function and the variance of tracking-error. Here, we have introduced the weighted cumulated liquidity loss function Gt=α∫TO Sβtldt. We have discussed two cases of parameter β. In fact, under certain assumptions, this optimization problem is related to the stochastic linear quadratic (LQ) Control problem.Firstly, we have discussed the case that the trading strategy is continuously differ-entiable with respect to time. Furthermore, we have discussed the case that the trading strategy can be represented as a semi-martingale. According to stochastic control theo-ry, the solution to the optimization problem is referred to the existences of the solutions to two types of backward stochastic differential equations(BSDEs):backward stochastic Riccati differential equation and linear backward stochastic differential equation.We have fully exploited the localization method and quasi-linear method to prove the existence and uniqueness of the solutions to the BSDEs. |
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