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# Centre of Financial Mathematics Seminar Series

## September 18, 2017 @ 1:30 pm - 2:30 pm

##### Speaker

Professor Yuji Yamada (University of Tsukuba, Japan)

##### Title

Model Predictive Control for Pairs Trading Portfolio:

Incorporation of Transaction Cost and Gross Exposure Constraints

##### Abstract

Model Predictive Control (MPC) is a flexible yet tractable technique in control engineering that recently has gained much attention in finance, particularly for its application to portfolio optimization. In this research, we consider an MPC approach for constructing pairs trading portfolio, where pairs trading is a popular hedge fund investment strategy based on the spread of a pair of stocks, i.e., one stock price minus some multiple of the other, with a predictive mean-reverting property. Although a normal pairs trading leads to a positive profit by taking and clearing the long-short position by setting certain thresholds, the MPC approach provided here intends to form a portfolio consisting of spreads of pairs and control the monetary value invested in each spread.

In the current presentation, we extend our previously developed MPC strategy for optimal pairs trading portfolio in Yamada and Primbs (2012) by incorporating the following two important and practical issues: The first issue is gross exposure (GE), which is the total value of long and short positions invested in risky assets (or stocks) as a proportion of the wealth possessed by a hedge fund. This quantity measures the leverage of a hedge fund, and the fund manager may limit the amount of leverage by imposing an upper bound, i.e., a GE constraint. The second issue is related to transaction costs, where the MPC algorithm in Yamada and Primbs (2012) may require frequent trades of many stocks leading to large transaction costs in practice. Here we assume that the transaction cost is proportional to the change in the amount of money (i.e., the change of absolute values of long or short positions) invested in each stock. We formulate the MPC strategy via a conditional mean-variance problem which we show reduces to a convex quadratic problem, even with gross exposure and proportional transaction cost constraints. Based on numerical experiments using Japanese stock data, we demonstrate that the incorporation of the transaction cost constraint improves the empirical performance of the wealth in terms of Sharpe ratio, which may be improved further by adding the GE constraint