determined based on the data in the immediate past. The process could be
repeated at regular time intervals and the spread levels determined on a con-
tinuous basis.
Sharpe Ratio Calculations
We will desist from calculating the Sharpe ratio for a single pair because the
portfolio of pairs traded determines in a great way the eventual Sharpe ratio.
Time-Based Stops
As noted in Chapter 6, the mere passage of time adds to the risk of holding
an unconverged spread. This is because of the residual common factor ex-
posure resulting in the phenomenon of mean drift. With the passage of time,
the SNR ratio deteriorates. We therefore need to calibrate the SNR ratio at
which we would end up unwinding the position regardless of convergence.
In some cases it may be worthwhile looking at unwinding at the mean
value or zero spread instead of waiting for the spread to swing to the oppo-
site direction.
SUMMARY
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 non-
parametric 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.FURTHER READING MATERIAL
Hidden Markov Models
MacDonald, Iain L., W. Zuchinni and W. Zucchii. Hidden Markov and other Mod-
els for Discrete Valued Time Series. (Boca Raton, Florida: CRC Press, 1997).
Ergodic Theory
Walters, Peter. An Introduction to Ergodic Theory. (New York: Springer Verlag,
2000).
136 STATISTICAL ARBITRAGE PAIRS