## Optimal Portfolio Execution in Finite-Depth Limit Order Books

(with Arash Fahim) -- working paper

This project presents an algorithmic approach for the problem of the optimal execution under constraint on the depth of the limit order book at each transaction time. Given the future liquidity of the limit order book, we provide an algorithm for the optimal execution of a large position of the portfolio. The suggested algorithm is based on the resilience model proposed by Obizhaeva and Wang (2013) with a more practical restriction on the supply/demand given by finite limit order book market. For the simplest case where the order book depth stays at a fixed level at all times, we identify the active constraints in a small number of iterations and state the optimal strategy in a closed form. When the depth of order book depth is monotone in time, we apply the KKT (Karush-Kuhn-Tucker) conditions to narrow down the execution strategies to a set of size O(N^2) and then perform a dichotomy-based algorithm to pin down the optimal strategy within O(N logN) time. As for the cases that cannot be classi ed into the two categories above, we propose an algorithm which has time complexity of O(N^2) to find a candidate optimal strategy and numerically confirm that the candidate we found is a KKTpoint of the optimal execution problem.

## Pricing American Options Under GARCH Model <full text>

-- Project Report for MAP 5615 Monte Carlo Methods in Financial Mathematics

This project gives a review of the least-squares Monte Carlo (LSM) American styled option valuation method as well as the generalized autoregressive conditional heteroskedastic (GARCH) option pricing model. There are two main kinds of LSM pricing methods: one is proposed by Longstaff and Schwartz (2001), and the other is proposed by Tsitsiklis and Van Roy (2001). A summary and comparison of these two methods is presented in this work. Numerical results show that Longstaff and Schwartz’s approach not only produces results comparable to those calculated by the finite difference approach but also converges very quickly, while Tsitsiklis and Van Roy’s method often generates highly-biased results, especially for out-of-the-money contracts. Finally, I apply the LSM method to price American options under the GARCH model, and get a comparable result to those presented in early studies.

## Improving Pricing Accuracy for Various Numerical Methods with the General Control Variate Method <full text>

(with C-Y Chiu and T-S Dai ) -- The 22nd Annual Conference on. Pacific Basin Finance, Economics, Accounting, and Management.

The control variate method is a popular variance reduction technique used in Monte Carlo methods, which are frequently used to price complex derivatives. This technique is used in much financial literature to exploit information about the errors in estimates of known quantities -- which are usually values of the derivatives that can be analytically priced -- to reduce the error for estimating an unknown quantity, which is usually the price of a complex derivative of interest. This paper generalizes the core idea of the control variate method so that it can be applied to reduce the pricing errors incurred in many numerical pricing methods, such as the tree method, the characteristic-function-based pricing method, and the convolution-based pricing method. For example, a numerical method for pricing a complex derivative, say an Asian option, may need to calculate the density function of the average price of the underlying asset in the convolution-based pricing method. However, this function can not be analytically solved and must be numerically approximated. Thus we find another analytical function that is close to the function of interest and exploit information about the errors in estimates of the analytical function to reduce the error for estimating the function of interest. Numerical results shows that our approach can effectively increase the pricing efficiency of many numerical pricing methods.