significant, as reflected by its -statistic.8 On the other hand, if the scatter plot is curved as in Figure 1.3b the correlation will be low, even though there is obviously a strong relationship between the variables. Fitting a line to the data in Figure 1.3b would give a beta estimate that has a large standard error and correlation will not necessarily be very significant.

Correlation is a limited measure of dependency. Very often correlation estimates in financial markets lack robustness9 so it is not surprising that alternative methods for capturing co-dependency have been considered.10 The concept of a copula goes back to Schweizer and Sklar (1958). A copula is a function of several variables: in fact it is a multivariate uniform distribution function. If , . ., u„ are values of n univariate distribution functions, so each Uj e [0, 1], then a copula is a function ( . . ., u„) -» [0, 1].

Copulas are used to combine marginal distributions into multivariate distributions. They are unique: for any given multivariate distribution (with continuous marginal distributions) there is a unique copula that represents it. They are also invariant under strictly increasing transformations of the marginal distributions. Copulas have long been recognized as a powerful tool for modelling dependence between random variables. A useful general reference text on copulas is Nelsen (1999).

Here are some simple examples of copulas:

(i) ( ,, . . ., u„) - . . . u„

(ii) ( . . ., u„) = min(wb . . ., u„)

(iii) ( ,, . . ., u„) = max(J2U ui ~ (" ~ l), °)

Copula (i) corresponds to the case that the random variables are independent: the joint density will be the product of the marginal densities. Copula (ii) corresponds to counter-monotonic dependency, which is similar to negative correlation. Copula (iii) corresponds to co-monotonic dependency, which is similar to positive correlation.11

8Note that p is significant when p is significant; in fact a simple /-test (§A.2.2) for the significance of correlation is [rV(r-2)/V(i-r2)]~;r„2.

9In Chapter 11 we shall see that correlation estimates will not be very robust if the two series are not jointly covariance-stationary.

10In Chapter 12 we ask whether one should be using return data at all to measure the co-dependency of financial returns. There it is argued that return data have all the memory taken out of them before the analysis even begins. So return data can only be used to pick up short-term associations between returns series. To investigate the possibility of any long-run associations it is necessary to use a long-memory model, such as a cointegration analysis on the price series.

1 ]Two random variables Xx and X2 are counter-monotonic if there is another random variable X such that Xx is a decreasing transformation of X and X2 is an increasing transformation of X. If they are both increasing (or decreasing) transformations of X then Xx and X2 are called co-monotonic. (Note that the transformations do not have to be strictly increasing or decreasing.)

(b) J

Figure 1.3 Highly dependent returns and (a) high correlation; (b) low correlation.

In the last few years copulas have been used as a powerful tool in financial risk management (Embrechts et al., 1999a). They have important applications to the aggregation of individual loss distributions into an overall loss distribution, particularly when correlation is difficult to assess, as it is, for example, in operational risk measurement.

1.2 Volatility and Correlation in Financial Markets

The only significant risks are the irreducible risks: those that cannot be reduced by hedging or diversification. Thus the concerns of a portfolio manager focus not on the total volatility of a portfolio, but on the volatility that is collinear with the market. This volatility is represented by the portfolio beta; it represents the irreducible part of the total volatility of the portfolio. In a capital asset pricing model framework (§8.1.1) a high beta can be attributed

to a high positive correlation with the index and a high relative volatility for the portfolio - this was shown in (1.5) above. Of course, if the relative volatility of the portfolio is very high it can have a large irreducible risk even when the correlation with the market is low (as long as it is positive).

Similarly, what matters for pricing an option is only the volatility of underlying price movements and not the trend in prices. Whatever the trend in an asset price, an option position can be hedged by the proper position on the underlying asset. All market participants would agree about the same fair price of an option if the volatility of underlying price movements could be forecast accurately, but usually it cannot. Then, if sensitivity to volatility changes cannot be hedged away, different traders will have different market views that give rise to a large bid/offer spread.

Thus the estimation and forecasting of volatility and correlation is at the heart of financial risk modelling:

>- Traders writing options need to forecast the volatility of the price process

over the lifetime of the option. >- The risk management of their positions, which is based on optimal hedging,

also requires volatility and correlation forecasts, but mainly over the short

term.

*- Implied volatility and correlation are necessary to compute the appropriate

hedge ratios for their positions. >- Statistical volatility and correlation forecasts for all possible risk factors in

the markets are necessary to net positions and to calculate a total market

risk capital requirement for the entire firm. *- To validate the pricing and hedging models that are used in the front office,

the middle office risk management and control functions will require

independent assessments of all the implied and statistical volatilities and

correlations.

An option price depends on a whole volatility surface so it is a little simplistic to talk about the implied volatility for a time horizon

Implied volatility is the volatility forecast over the life of an option that equates an observed market price with the model price of an option. An option price depends on the choice of model and, after, on a whole volatility surface (§2.2.3) so it is a little simplistic to talk about the implied volatility for a time horizon. However, the Black-Scholes model price of a simple call (or put) option will depend on the implied volatility. This implied volatility is the volatility of the geometric Brownian motion process that is assumed to govern price variations from now until the option matures, that will equate the model price with the market price (Chapter 2). In that sense it is more accurate to refer to the Black-Scholes implied volatility, or for short, the Black-Scholes volatility.

Statistical volatility depends on the choice of statistical model that is applied to historical asset returns data. The statistical model is usually a time series model, such as a moving average or generalized autoregressive conditional heteroscedasticity (GARCH) process (these will be discussed extensively in