#### Year

2012

#### Degree Name

Doctor of Philosophy

#### Department

School of Mathematics and Applied Statistics

#### Recommended Citation

Puspaningrum, Heni, Pairs trading using cointegration approach, Doctor of Philosophy thesis, School of Mathematics and Applied Statistics, University of Wollongong, 2012. https://ro.uow.edu.au/theses/3450

#### Abstract

Pairs trading strategy works by taking the arbitrage opportunity of temporary anomalies between prices of related assets which have long-run equilibrium. When such an event occurs, one asset will be overvalued relative to the other asset. We can then invest in a two-assets portfolio (a pair) where the overvalued asset is sold (short position) and the undervalued asset is bought (long position). The trade is closed out by taking the opposite positions of these assets after the asset prices have settled back into their long-run relationship. The profit is captured from this short-term discrepancies in the two asset prices. Since the profit does not depend on the movement of the market, pairs trading can be said as a market-neutral investment strategy.

There are four main approaches used to implement pairs trading: distance approach, combine forecasts approach, stochastic approach and cointegration approach. This thesis focuses on cointegration approach. It uses cointegration analysis in long/short investment strategy involving more than two assets. Cointegration incorporates mean reversion into pairs trading framework which is the single most important statistical relationship required for success. If the value of the portfolio is known to fluctuate around its equilibrium value then any deviations from this value can be traded against. Especially in this thesis, a pairs trading strategy is developed based on the cointegration coefficients weighted (CCW) rule. The CCW rule works by trading the number of unit in two assets based on their cointegration coefficients to achieve a guaranteed minimum profit per trade. The minimum profit per trade corresponds to the pre-set boundaries upper-bound *U* and lower-bound *L* chosen to open trades. The optimal pre-set boundary value is determined by maximising the minimum total profit (MTP) over a specified trading horizon. The MTP is a function of the minimum profit per trade and the number of trades during the trading horizon. This thesis provides the estimated number of trades. The number of trades is also influenced by the distance of the pre-set boundaries from the long-run cointegration equilibrium. The higher the pre-set boundaries for opening trades, the higher the minimum profit per trade but the trade numbers will be lower. The opposite applies for lowering the boundary values. The number of trades over a specified trading horizon is estimated jointly by the average trade duration and the average inter-trade interval. For any pre-set boundaries, both of those values are estimated by making an analogy to the mean first-passage times for a stationary process.

Trading duration and inter-trade interval are derived using a Markov chain approach for a white noise and an AR(1) processes. However, to apply the approach to higher order AR(p) models, p > 1, is difficult. Thus, an integral equation approach is used to evaluate trading duration and inter-trade interval. A numerical algorithm is also developed to calculate the optimal pre-set upper-bound, denoted Uo, that would maximize the minimum total profit (MTP). The pairs trading strategy is applied to three empirical data examples. The pairs trading simulations show that the strategy works quite well for the data used.

Considering cointegration error ϵ_{t} may not follow a linear stationary AR(p) process but a nonlinear stationary process such as a nonlinear stationary exponential smooth transition autoregressive (ESTAR) model, this thesis extends Kapetanios’s and Venetis’s tests by considering a unit root test for a k-ESTAR(p) model with a different approach to Venetis. By using the approach in this thesis, the singularity problem can be avoided without adding the collinear regressors into the error term. For some cases, simulation results show that our approach is better than Venetis test, Kapetanios test and the Augmented Dickey-Fuller (ADF) test.

We also show that the mean first-passage time based on an integral equation approach can also be applied to the evaluation of trading duration and inter-trade interval for an 1-ESTAR(1) model and an 1-ESTAR(2) model. With some adjustments, a numerical algorithm for an AR(2) model can be used for an ESTAR model to calculate the optimal pre-set upper-bound. An application to pairs trading assuming ϵ_{t} is a nonlinear stationary ESTAR model is examined using simulated data.

This thesis also discusses cointegration and pairs trading between future prices and spot index prices of the S&P 500. There are three major issues discussed in Chapter 8. The first issue is to identify an appropriate model for the S&P 500 basis. The S&P 500 basis is defined as the difference between the log of future prices and the log of spot index prices. Following the same analysis procedures and using currently available data period, we reached different conclusion to Monoyios. Instead of concluding the basis follows an ESTAR model, we concluded that it follows a LSTAR model. Furthermore, even though we can conclude that there is possibility nonlinearity in the basis, there is no significant difference between a nonlinear LSTAR model and a linear autoregressive model in fitting the data. We also have a concern in the way the basis is constructed. By pairing up the spot price with the future contract with the nearest maturity, it may produce artificial jumps at the time of maturity. The longer the time to maturity, the higher the difference between the log future price and the log spot price as described by a cost-carry model. Therefore, the second issue of the chapter examined the cointegration of ft and st with a time trend for each future contract. Cointegration analysis with a time trend based on the Engle-Granger approach and the Johansen approach are used. Only 19 out of 44 future contracts conclude that they are cointegrated with a time trend using both the Engle-Granger approach and the Johansen approach. Perhaps, high volatility during financial crisis in the data period affects the cointegration test and also limited number of observations. Based on the Engle-Granger approach and then running nonlinearity tests for the regression residuals, we can conclude that there is no strong evidence of nonlinearity in the relationship between future and spot prices of S&P 500 index. It gives further evidence on the basis for a linearity rather than nonlinearity. The third issue is to examine the application of pairs trading between the future and spot index prices. The stationary characteristics of an AR(p) model are used to develop pairs trading strategy between future contract and corresponding spot index. Pairs trading simulations between future contract and CFD (Contract For Difference) the S&P 500 index data during 1998 show very good results resulting a total return more than 75% during the 3 months trading periods.