# Optimization Methods for Algorithmic Trading

A group of researchers at Department of Mathematics and Informatics, Faculty of Science University of Novi Sad, in close cooperation with a London based hedge fund, is developing mathematical models for Algorithmic Trading.

Algorithmic Trading, also known as Algorithmic Execution, is the automated process of trading exogenous orders in electronic (stock) exchanges. There are many aspects to algorithmic trading that make it attractive. Most real world quantitative trading problems involve working with high levels of uncertainty – hence requiring complex mathematical models.

The key structure we are concerned with in Algorithmic trading is market microstructure, although different traders might assume different properties as the most important characteristics of market microstructure. Among the generally accepted are the price process, visible and hidden liquidity pattern, impact measure, bid/ask spread, depth and detail of the order book, tick size etc. The mathematical models of market microstructure are based on High Frequency Data. Main properties of high frequency financial data (HFFD) are irregular temporal spacing, discreteness, diurnal patterns and temporal dependence. Furthermore multiple transactions occur within a second with different transaction prices and transaction volumes. These properties make HFFD more difficult to analyze and built reliable forecasting models. On the other hand HFFD are an extremely rich source of information that has been widely used in recent years for understanding market dynamics and building models.

One important topic in our research is Optimal Trading Trajectory for Atomic Orders. Atomic order is the basic component of any algorithm for execution and it is determined by short time execution windows (measured in minutes) and quantities of up to 15% of the average traded volume within the considered time window. The principal idea is to define an optimization procedure that yields an optimal trading trajectory applicable in live trading for a large universe of instruments. The optimal trajectory we define consists of both types of orders, market and limit, and takes advantage of any temporary price or liquidity improvements available at a particular trading venue. Thus it provides a systematic way of employing both passive and aggressive trading strategies in order to minimize risk and maximize gain. The key ingredient of the optimiz

ation model we employ is Fill Probability Function that estimates the probability of fill for passive orders. This function is modeled from HFFD and allows one to combine passive and aggressive orders in an optimal way. Any optimization procedure meant to be deployed in a real trading environment must be computationally affordable and applicable in real time for potentially large portfolios of securities. Thus modelling simplifications need to be carefully designed. Another intriguing question is the multi-market environment, which provides additional opportunities but also adds complexity to the market microstructure. The complex multi-market environment yields a bilevel nonlinear optimization problem.

Another direction in this research project we pursue actively is the question of an objective and fair Benchmark for Trading Algorithms. The selected benchmark is used as a measure of trader’s performance. It is undoubtably difficult to define one standard measure for all trade executions given that the objectives can be very different given that orders come with many constraints. There are many types of benchmarks, some are established before trading like Arrival Price, others are established during the trading process like VWAP, others quantify delays, or measure the performance with respect to the closing price etc. The most important performance measures are VWAP (Volume Weighted Average Price) and IS (Implementation Shortfall or Arrival Price). Both measures are widely used in practice and represent the standard in financial industry. A number of algorithms is developed in order to minimize the slippage to VWAP and IS, but of the problems with measuring slippage, whether it be VWAP or ES, they either distort the slippage measure or do not represent the true nature of slippage. In the case of VWAP, by its own definition this measure distorts slippage with increasing order size due to the market impact caused by ones own orders. Although a good algorithm, the slippage measure is fundamentally flawed. In the case of the other equally dominant algorithm, IS, although it is unbiased in terms of measuring slippage caused by price drift and market impact, it does not reflect the true slippage due to its reference to a static reference price. In other words, it does not capture the absolute slippage. The measure we are currently developing takes a posteriori view of market conditions and its main characteristic is that it is completely objective. Roughly speaking, a posterior approach allows us to determine what would have been the optimal order placement if we knew the complete market information during the trading window. Thus we define the performance measure as the difference between that optimal trading position and the trading position that is actually executed. This difference is calculated taking into account all process and traded quantities within the considered time window. With this measure we are capturing the impact caused by our own trading as a cost that affects all trades, including our own and thus avoid the main problem with VWAP in the case of large trades.

Contact person: Natasa Krejic, natasak@uns.ac.rs