Optimal Execution algorithms for high frequency trading

Faculty: Shashi Jain (Mgmt Studies) and Srikanth Iyer (Math)

Summary:

Institutional traders (pension funds, hedge funds etc) often like to buy or sell large amount of shares while minimizing the adverse movements that result as consequence of their own trades. This is typically done by slicing the parent order into smaller child orders and executing each of these child orders over a period of time. There is a trade-off involved, if the trader executes the order too quickly they will sink through the less protable side of the order book. Rapid buy or sell orders can send a signal that a large institutional order has to be fullfilled to the market, which results in reshuing of the order book in a manner that hurts the trader. A slow execution on the other hand is risky due to uncertainties of the future prices at which the trade would be executed. The trader must therefore formulate a model (trading strategy) to decide on how to optimally liquidate (or acquire) the assets.

Some of the seminal papers on this topic are Bertsimas and Lo [1], Almgren and Chriss [2], and Forsyth et al. [3]. A comprehensive review on algorithmic and high frequency trading in general and optimal execution in particular can be found in Cartea et al. [4]. The typical challenge in formulation of an optimal execution algorithm lies with the choice of risk measure that has to be minimized over the execution horizon. While single period mean-variance optimization is straight forward, multi-period mean variance optimization leads to strategies that are time-inconsistent. We would like to follow the approach of Forsyth et al. [3] who propose the use the quadratic variation as a risk measure to obtain time consistent policies for dynamic strategies. A challenge however is the choice of numerical methods that are suitable for execution algorithms for single asset problems are quite challenging to extend for multi-asset cases.

In this research we would like to work on numerical methods for optimal execution algorithms where the executioner wants to optimally liquidate a basket of correlated assets with an objective to minimize the execution costs and associated inventory risk (in form of quadratic variation). Given the high-dimensional nature of the problem involved, Monte Carlo based methods seem like an attractive option, as numerical integration using MC methods is not signicantly aected by the dimensionality of the problem. A contribution we hope to achieve is development of Monte Carlo schemes that can eciently solve general stochastic optimal control problems that are related to portfolio allocation and execution.

Given the quantum of data that is available, for example the micro second level market order and limit order book data for each asset in the portfolio, we are also interested in applying reinforcement learning to learn from the past data the optimal actions that would maximize the reward objective. Eventually we would like to compare the performance of model free (or more data driven) optimal execution with that of traditional model based execution algorithms. The ideal candidate should have interest in random process, stochastic differential equations, numerical methods and nance. The mathematical focus of the research will be on solution of the Hamilton Jacobi Bellman (HJB) equations and would involve understanding of optimal control problems, Markov decision process, dynamic programming and reinforcement learning.

References

[1] D. Bertsimas, A. W. Lo, Optimal control of execution costs, Journal of Financial Markets 1 (1) (1998) 1{50.

[2] R. Almgren, N. Chriss, Optimal execution of portfolio transactions, Journal of Risk 3 (2001) 5{40.

[3] P. A. Forsyth, J. S. Kennedy, S. Tse, H. Windcli, Optimal trade execution: a mean quadratic variation approach, Journal of Economic Dynamics and
Control 36 (12) (2012) 1971{1991.

[4] A. Cartea, S. Jaimungal, J. Penalva, Algorithmic and high-frequency trading, Cambridge University Press, 2015.