The Mathematics of Financial Modelingand Investment Management

19-EquityPort Page 576 Friday, March 12, 2004 12:40 PM

576 The Mathematics of Financial Modeling and Investment Management

ri = αi + βirM + ε

where the noise term exhibits only local correlation and tends to zero

over large portfolios. Market return is stochastic and therefore uncer-

tain. Suppose, however, that there are many returns with similar betas

but with different alphas. The no-arbitrage condition forbids this situa-

tion for an infinite economy but leaves open the possibility that a finite

number of such situations exist.

For each beta, or more likely for each beta band as betas will not be

strictly equal, invest in a long portfolio with the positive alphas and a

short portfolio with the negative alphas. Repeat the operation for each

band of beta. The resulting portfolio will implement a simple statistical

arbitrage strategy. It will be nearly market-neutral, with profit depending

only on the spreads between alphas and not on the direction of the market.

There are several caveats. First, the appropriate distribution of betas

and alphas must exist. This is an empirical question that cannot be solved

a priori. Second, there are residual risks, as the noise term will be reduced

but not completely eliminated and betas will not be strictly equal. Third,

the factor model might be misspecified and therefore unstable.

Contrarian strategies where managers go short on overpriced stocks

and long on underpriced stocks are also possible. Long-short strategies

of this type started in the 1980s with so-called pair trading reportedly

initiated by a trading group working at Morgan Stanley. Under the

direction of Nunzio Tartaglia, this group’s strategy consisted in forming

pairs of stocks that had a small distance measured by the relative vari-

ance. Setting appropriate thresholds, underpriced stocks are bought and

overpriced stocks sold.

The ideas underlying contrarian strategies are ultimately formalized

by the concepts of cointegration and error correction. When applied to

price, processes error correction represents changes in returns when prices

diverge from some common trend. Many efforts at building true statisti-

cal arbitrage techniques therefore make use of cointegration techniques.

In terms of cointegration, one implements statistical arbitrage by search-

ing for cointegrating relationships. Each cointegrating relationship repre-

sents a stationary, mean-reverting portfolio. Being autocorrelated, these

portfolios are more predictable than other portfolios or individual stocks.

As most implementations are proprietary, the different approaches

are only partially described. The key problem is to find true cointegrated

portfolios. In practice, there are several approaches; these include:

■ Searching for cointegrated pairs of stocks. This can be performed with

standard cointegration tests and techniques. However results are very