Cointegration can have a substantial influence on spread option prices when volatilities are stochastic

has the continuous limit of a system driven by four correlated Brownian motions, one for each asset (in which the ECM is incorporated) and one for each stochastic volatility. The stochastic processes for the asset returns have an additional disequilibrium term to ensure that deviations from equilibrium are corrected, so stationary spreads are imposed. These processes are the continuous time equivalent of an ECM. Their Monte Carlo results show that cointegration can have a substantial influence on spread option prices when volatilities are stochastic. But when volatilities are constant the model simplifies to one of simple bivariate Brownian motion and the standard Black-Scholes results are recovered.

12.4.5 Term Structures

No financial systems have higher cointegration than term structures. In §12.2 some term structures of interest rates and futures prices were used to illustrate how cointegration should be analysed in a multivariate system. These methods lend themselves to any term structures of integrated variables, and there is a large academic literature in this area. Cointegration and correlation go together in the yield curve, and we often find strongest cointegration at the short end where correlations are highest. See Bradley and Lumpkin (1992), Hall et al. (1992), Alexander and Johnson (1992, 1994), Davidson et al. (1994), De Gennaro et al. (1994), Lee (1994), Boothe and Tse (1995) and Brenner et al. (1996).

12.4.6 Market Integration

Market indices in different countries should be cointegrated if purchasing power parity holds. There is some evidence of cointegration in international bond markets and in international equity markets, but arbitrage possibilities seem quite limited (Karfakis and Moschos, 1990; Kasa, 1992; Smith et al, 1993; Corhay et al, 1993; Clare et al, 1995). However, in recent years the US market does appear to be somewhat of a leader in international equity and bond markets (Alexander, 1994; Masih, 1997). More cointegration has been found between international equity markets than between international bond markets, in fact in some studies international bond markets have shown no evidence of cointegration: see Andrade et al. (1991).

In equity markets, the analysis in §12.3 shows that the Dutch, German and French equity indices have been highly cointegrated since 1996, with most of the causality on a daily basis coming from the German market. Some time ago Taylor and Tonks (1989) used the Engle-Granger method to demonstrate a high level of cointegration between UK, Dutch, German and Japanese stock markets between 1979 and 1986, with Granger causality running from the UK to these other markets, but not vice versa. But during a similar period Andrade et al. (1991) found no evidence of cointegration between equities in the UK, US, Germany and Japan based on the Johansen method. However, both these papers use common currency indices, which were greatly influenced by

movements in exchange rates. Cointegration between equity markets should be examined using local currency indices. Alexander and Thillainathan (1995) examine Asian-Pacific equity markets and find evidence of cointegration, but only when indices are expressed in local currency terms. GARCH volatility models of the dollar exchange rates show that FX volatility is extremely variable in the Asian-Pacific region, and would swamp any equity effects if index prices were converted to dollar amounts.

Cointegration between equity markets should I examined using local currency indices

Since an equity index is by definition a weighted sum of the constituents there should be some sufficiently large basket that is cointegrated with the index, assuming the index weights do not change too much over time (Cerchi and Havenner, 1988; Pindyck and Rothenberg, 1993). This is, in fact, the theoretical basis for cointegration index tracking models (§12.5). The sector indices within a given country should also be cointegrated when industrial sectors maintain relatively stable proportions in the economy. By the same token, a basket of equity indices in the Morgan Stanley Capital International (MSCI) world index, or the Morgan Stanley European, Asian and Far Eastern (EAFE) index, should be cointegrated with the aggregate index.

12.5 Applications of Cointegration to Investment Analysis

The cointegration methodology is a powerful tool for long-term investment analysis. If the allocations in a portfolio are designed so that the portfolio tracks an index, then the portfolio should be cointegrated with the index. The portfolio and the index can deviate in the short term, but in the long term they should be tied together. A number of asset management firms are now basing allocations on cointegration analysis.

This section describes the cointegration-based hedge fund management technology that has been developed by myself, Dr Ian Giblin and Wayne Weddington III at Pennoyer Capital Management, New York. The results presented here represent several years of model development and validation before the fund started trading in May 2000. Since then the long-short S&P 100 equity fund has returned 17.07% between May and December 2000 and between January and March 2001 (the time of writing) it returned - 0.54%. This should be compared with the returns of -12.05% and -25.94% on the S&P 100 during the same two periods. More details of the fund are available from www.pennoyer. net.

We have already seen that when portfolios are constructed on the basis of returns analysis, frequent rebalancing will be necessary to keep the portfolio in line with the index (§7.2.5). The power of cointegration analysis is that optimal portfolios may be constructed on the basis of common long-run trends between asset prices, and they will not require so much rebalancing. Commonly it is possible to find cointegrating baskets with relatively few stocks.

If the allocations in a portfolio are designed so that the portfolio tracks an index, then the portfolio should be cointegrated with the index

When portfolios are based on mean-variance analysis there is nothing to ensure that tracking errors are mean-reverting

Any investment strategy that guarantees stationary tracking errors must be based on cointegration

12.5.1 Selection and Allocation

The basic problem is still one of stock selection and asset allocation, and as such bears much relation to the mean-variance analysis described in Chapter 7. But rather than seek portfolio weights to minimize the variance of the portfolio for a given level of return, the criteria that are used in cointegration analysis are to maximize the stationarity (and minimize the variance) of the tracking error. This criterion contrasts with the efficient frontier criterion: in the Markovitz framework portfolios are based on mean-variance analysis and there is nothing to ensure that tracking errors are mean-reverting. Although the portfolios will be efficient, the tracking errors may be random walks, so the replicating portfolio could drift arbitrarily far from the benchmark unless it is frequently rebalanced.

Using cointegration it is possible to devise allocations that have mean-reverting tracking errors. Indeed, any investment strategy that guarantees stationary tracking errors must be based on cointegration. When tracking errors are stationary the portfolio will be tied to the index: it cannot drift too far from the index because the tracking error is mean-reverting.

Benchmarking or index tracking models that are based on cointegration normally employ a linear regression of log prices. The dependent variable is the log index price or some other benchmark, such as LIBOR, that is used to evaluate the performance of the portfolio. In the case of tracking an index plus alpha per cent per annum, the dependent variable will be the log of the index plus a small increment that amounts to a% over the year. The explanatory variables are the log prices of the assets in the tracking portfolio, and the residuals are the tracking errors.

There are two parts to the problem: first select the assets, and then optimize the portfolio weights. The asset selection process is perhaps the hardest but most important part, and can be approached in a number of ways. Selection methods include a brute force approach (such as when the number of assets is fixed and then linear models are fitted for all possible portfolios with this number of assets), methods that are tailored to investors preferences over various types of stocks, and proprietary technical analysis.

The optimal allocation process uses least squares regression analysis: allocations are made according to a cointegrating regression to ensure that the fitted portfolio will be cointegrated with the benchmark and the tracking error will be stationary. Suppose a benchmark with log price index is to be tracked with a number of assets with log prices xu . . ., x„. The Engle-Granger cointegration method is to regress on a constant and and then to test the residuals

for stationarity. The coefficients ab . . ., a„ in the Engle-Granger regression

, - + + . . . + a„x„ + e,

(12.7)