The main objective of this thesis is to study the microstructure of the exchange. I address some problems related to the optimal execution and the high frequency order book data.

I address the optimal execution as a multiple exercise option and we find an optimal trading strategy as a function of the number of lots to buy or to sell, and the time left before a given deadline. The multiple exercise approach can be used with any type of model for the price process. I model the dynamics of an order book directly, which allows me to capture the possible illiquidity of the market after an execution of a large trade.

I also present a study of the notion of the price and the inference about the order book. The state of the exchange is characterized by an order book - number of lots that are asked or offered for each given price. Modeling the entire order book is too complex since there are too many variables to match, and the resulting model may not be realistic. My approach is to model the state of the exchange using only a small number of parameters, where one parameter represents a price of the asset, and another parameter represents liquidity of the market. This is a compromise between modeling the entire order book, and modeling only one dimensional process that represents the price.