Pairs Trading: Quantitative Methods and Analysis (Ganapathy Vidyamurthy, 2004)

Introduction

• The elephant and 6 blind men.
• Predicting the market is like the 6 blind men describing what are they touching.
• The CAPM model helps separate out portfolio returns into a market component and a residual component. 
• Portfolios with a zero market component are called market neutral portfolios. 
• Market neutral strategies involve the trading of market neutral portfolios, and the returns generated by such strategies are uncorrelated with the market. 
• Pairs trading is a genre of market neutral strategies in which a portfolio has only two assets.

Time series

• A time series is constructed by periodically drawing samples from probability distributions that vary with time. 
• The white noise process is the most elementary form of time series and is generated by drawing samples from a fixed distribution at every time instance. 
• Auto-regressive moving average (ARMA).
• ARMA time series are generated using fixed linear combinations of white noise realizations. 
• Time series forecasting for ARMA processes involves deciphering the linear combination and the white noise sequence used to generate the given data and using it to predict the future values. 
• A random walk process is the time series where the current value is a simple sum of all the white noise realizations up to the present time. 
• A random walk is a nonstationary time series. Nonstationary time series are usually transformed to stationary time series using differencing. 
• The logarithm of the stock price series is usually modeled as a random walk.

Factor Models

• Factor models are models that are used to explain the risk return characteristics of assets and come in many flavors. 
• Even though the details may vary, factor models are firmly based on the principles of arbitrage pricing theory (APT). 
• A factor model is considered fully specified by the factor exposures, the factor covariance matrix, and the specific variance matrix. 
• A factor model may be used as a framework to estimate many commonplace parameters that may be needed in the course of the investment process. 
• Examples of such computations include the estimation of risk on a portfolio, the evaluation of portfolio beta, and computing the contents of a tracking basket. 
• The factor covariance matrix is a crucial piece of information that the factor model provides. However, it must be used with care.

Kalman Filtering

• Control theory and the rate of reaction.
• The Kalman filter is an optimal state estimation process applied to a dynamic system that involves random perturbations. 
• Inherent in any discussion on the Kalman filter are the notions of state and observation.
• Kalman filtering may be summarized as a three-step process comprising prediction, observation, and correction, or reconciliation of the prediction with the observation. 
• The simplest case of the Kalman filter reduces to finding the average of n numbers. 
• The recursive least squares method is also a special case of the Kalman filter that may be applied to filtering random walks. 
• When the state and observation variances are the same, that is, the signal-to-noise ratio is unity, then the estimation of the Kalman states for a random walk boils down to a weighted average of the observations, with the weights formed by ratios of Fibonacci numbers. 
• The degree of smoothness to be achieved in a random walk can be controlled by varying the sampling rate of the random walk sequence.

Statistical Arbitrage Pairs

• Statistical pairs trading is a relative value arbitrage on two securities and is based on the premise that there is a long-run equilibrium between the prices of the stocks composing the pair. 
• The degree of deviation from the long-run equilibrium is called the spread and represents the extent of mutual mispricing. 
• Any deviation from the long-run equilibrium is compensated for in subsequent movements of the time series. 
• Pairs trading involves trading on the oscillations about the equilibrium value. 
• The econometric paradigm of cointegration and error correction is central to the analysis of the pairs-trading strategy.

Pairs Selection in Equity Markets

• The candidate list of potentially cointegrated stock pairs can be compiled by the process of identifying similar stocks. 
• The notion of similarity is formalized using a distance measure between two stocks. 
• The distance measure is based on an APT model possibly with fundamental risk factors. 
• The candidate list of pairs is determined by choosing pairs with distance values within a certain threshold. 
• The distance measure is the absolute value of the common factor correlation between the two stocks. 
• If the common factor correlation is +1 or –1 and the integration of the specific return series of the stocks involved are stationary, then conditions for cointegration are satisfied. 
• It may be possible to trade pairs of stocks even though they deviate from ideal conditions of cointegration. 
• The signal-to-noise ratio as defined is a measure of the deviation from the ideal condition of cointegration.

Testing For Tradability

• Tradability testing is a two-step process consisting of evaluating the linear relationship and measuring the degree of mean reversion of the residual. 
• The linear relationship between the log-price series of the two stocks is characterized by the cointegration coefficient and the stock premium. 
• They may be estimated in a multifactor framework or by ordinary least squares regression. 
• The spread series can be calculated by applying the linear relationship. 
• The degree of mean reversion of a series is quantifiable in terms of the zero-crossing frequency. 
• The zero-crossing frequency can be directly estimated using the bootstrap procedure. 
• The reciprocal of the zero-crossing frequency is indicative of the trading period, and a pair may be deemed tradable if we are satisfied with the range of trade periods or zero crossing frequencies generated by the bootstrap.

Trading Design

• When trading the spread, it is desirable to trade at threshold levels that yield the maximum profits. 
• A large threshold value trades infrequently for a large profit, and a small threshold value trades frequently for a small profit. 
• The optimal value for the threshold is between the extremes. 
• Finding the optimal value for the threshold is easier done using nonparametric methods rather than parametric methods. 
• The profit function to be maximized can be constructed from sample data using a two-step process of ensuring monotonicity of the crossover distribution followed by Tikhonov-Miller regularization.
• The abcissa for the maximum value of the profit function is the desired threshold value.

Risk Arbitrage Mechanics

• Risk arbitrage relates to trading around corporate events, especially mergers and acquisitions. 
• The practice of risk arbitrage has a long history and is a widely practiced arbitrage technique.
• The mechanics involve putting on a spread position when the deal is announced and unwinding it on deal completion. 
• The spread is a key market variable that the arbitrageur relates to. 
• Its value is roughly equal to the dollar profit per share of target and also indicative of the inherent risk of deal break. 
• We also discuss a typical trading strategy.

Trade Execution

• The arbitrageur normally executes the paired transaction through a broker. 
• Verifying the executions by pairing them is an approach fraught with inconsistencies. 
• It makes sense for the arbitrageur to insist on a firm average spread or better. 
• It is possible to put on a spread position during the pricing period and be perfectly hedged. 
• In such situations, however, the exact position size of the target stock is gradually revealed. 
• Putting on a spread position typically involves a short sale and must be executed in accordance with the uptick rule.

How market is being cleared

• Dinic’s algorithm, max flow
• Lazy allocation algorithm

The Market Implied Merger Probability

• The risk neutral probability of merger is the probability implied by the observed spread between the bidder and target firms. 
• It is similar to the implied volatility parameter for options. 
• The evolution of the risk neutral probability of merger can be related to the spread dynamics. 
• It is useful in the design of more appropriate value at risk measures.

Spread Inversion

• The spread observed in the marketplace between two companies involved in a merger is likely to be distorted due to other market effects like the bid-ask spread effects and market maker inventory adjustments. 
• The Kalman filtering approach is a suitable smoothing technique for estimating the actual spread levels. 
• The filtered spread could be used for the risk-neutral probability and also assist in timing executions.