Final_1.pdf

Rearranging the terms a little bit, we have

(5.7)

Therefore, the return on the portfolio is the increment to the spread value in
the time period i. We have successfully associated a portfolio with a sta-
tionary time series. The one thing that remains is providing an interpretation
forg, the cointegration coefficient. We will discuss this in later chapters.

A Trading Strategy

We now construct a simple trading strategy. The idea is to trade on the os-
cillations about the equilibrium value for the spread. We could put on a
trade upon deviation from the equilibrium value and unwind the trade when
equilibrium is restored. However, note that the equilibrium value is also the
mean value of the time series. Therefore, given that the spread swings
equally in both directions about the equilibrium value, we could potentially
unwind the trade when the spread deviates in the other direction. This
would reduce the trading frequency on average by a factor of two. Given
that stocks have a bid-ask spread, we would incur a trading slippage every
time a trade is executed. Reducing the trading frequency reduces the effect
of this slippage.
Let us therefore consider the strategy where the trades are put on and
unwound on a deviation of ∆on either direction from the long-run equilib-
riumm. We buy the portfolio (long Aand short B) when the time series is ∆
below the mean and sell the portfolio (sell Aand buy B) when the time se-
ries is ∆above the mean in itime steps.

(5.8)

The profit on the trade is the incremental change in the spread, 2∆.

Example

The Data
Consider two stocks AandBthat are cointegrated with the following data:

Cointegration Ratio = 1.5

Delta used for trade signal = 0.045

Bid price of Aat time t= $19.50

Ask price of Bat time t= $7.46

log log

log log

pp

pp

t

A

t

B

ti

A

ti

B

()− ()=−

()++− ()=+

γμ

γμ

∆

∆

[]log(pppptiA++)−γγlog(tiB ) −−[]log(tA) log( )tB =−spreadti+spreadt

82 STATISTICAL ARBITRAGE PAIRS