A Study On Stochastic Liquidity, Trading Strategy And Pricing For The Assets

The liquidity is everything to the market, the outbreak of a number of major financial events during the last more than a decade brings more attention to liquidity and liquidity risk. And model them by mathematical finance. Introduction of liquidity risk needs to amend a lot of the classic financial theory, such as the asset allocation, pricing and the risk management. Unlike other literature associated with the research for the liquidity risk, the article studies the trading strategy, pricing and hedging in the background of the randomness in the liquidity, especially, discusses some new topics triggered by the stochastic liquidity. This article is divided as the following chapters to investigate them respectively:First, the paper is focus on the introduction to the background, and moreover, concludes some main features of the liquidity risk from the actual phenomenon, and to provide a practical basis for the modeling of the entire thesis.Unlike the credit risk or the market risk, the study on the liquidity risk is far less advance, and as we are known, the methods and conclusions associated with the former tow research have been standardized, however, there does not exist a consistent definition to the liquidity risk, and even does not have the qualitative concept. Therefore, this chapter sums up the definitions and the measurement methods refer to the current literature. And subsequently, we build the comprehensive stochastic liquidity model based on the order-driven trading system, which contains the four common elements:immediacy, tightness, depth, resilience. This is not only more realistic modeling to the liquidity risk, but also, to some extent, develops the microstructure theory. In addition, the model is the core foundation of the entire paper.The introduction of stochastic liquidity makes the optimization problem becomes more complex, which does not like the traditional model that can get the explicit solution of the optimal trading strategy. Firstly, we obtain the quasi-variational inequalities through minimizing the expected the execution cost, and then makes use of the linearity of the optimal value function to reduce its dimension, and while uses the barrier ratio structure to solve the optimal trading strategy and the necessary and sufficient conditions to avoid the price manipulation. Second, we also solve the minimization problem which is the mean-variance formation, unfortunately, the optimal value function have no linearity, thus we resort to the general stochastic optimal control method to prove that the optimal value function is the only (discontinue) viscosity solution to the quasi-variational inequalities.Opposed to the profound paper written by Qetin et al.(2004), we focus on considering the two elements of the liquidity, namely, the market depth and immediacy, so that the cost from the liquidity risk can not be vanished even in the continuous bounded variation strategy. Since we simultaneously assume the stochastic liquidity, the stochastic volatility and some constrains on the trading strategy, the traditional completely hedge fails, therefore, we adopt the sup-replication pricing and hedge associated with those extension. Under some extreme conditions, the conclusion of the extended model reduces to the B-S formula. In this sense, this model can be seen as an extension of the traditional model.Finally, the paper integrates the stochastic liquidity model with the MMDH model, and then use the random-sum central limit theorem to obtain the asymptotic distribution of the assets yield, it is concluded that its conditional variance is function of the liquidity process and information arrival process. Based on theoretical analysis, this chapter chooses high frequency data of actively trading stocks for the calibration, traded in the Shanghai and Shenzhen stock exchanges in china. The calibrated results are consistent with the theory findings. Moreover, the chapter uses the efficient method of moments to identify the adequacy of the models. The formal statistics show the former better and t ratio of scores show that the macro fundamental factor results the persistency for the distribution of the volatility and the market microstructure factor leads to the fat tail.