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    Trading Pairs and Principle Component Analysis

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    1. Explain the principle of "Trading Pairs" between two equities through cointegration testing between two equity instruments.

    2. Explain the concept of Principle Component Analysis for Asset Pricing through an example.

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    Cointegration tests between two equities:
    Two securities can be effective in a "Pair Trade" when the correlation between them weakens. This means that when one of the securities moves up in price, the other security moves down and an effective strategy is to short the outperforming stock and go long on the underperforming stock. The premise of such a trading strategy is that the 'spread' between the two securities would eventually converge.
    To determine if a pair of securities can be pair traded effectively, the spread between the prices of two securities should be a 'stationary' time series. A 'stationary' time series can be defined as a stochastic process, whose cumulative distribution function does not change when shifted in time. In otherwise, the time-series shows a repetitive pattern and consequently, the parameters Mean and Variance does not change over time.
    Mathematically, for any stochastic process St
    FS(St, St+1, St+2, ......., Stk) = FS(St+τ, St+1+ τ, St+2+ τ, ......., Stk+ τ)
    Where FS is the cumulative distribution function of St
    This can be established through cointegration tests between the prices of the two securities.
    The 'Engle-Granger two step method' can be used to test for cointegration between the two securities' traded prices.
    Let xt and yt represent the prices of two securities X and Y. The prices (over time periods) of the two securities are cointegrated when a linear relationship between them is 'stationary'.
    y_t- βx_t= ...

    Solution Summary

    This solution gives equations and detailed explanations for the principle of "trading pairs" and the Principle Component Analysis for asset pricing.