The 90-day realized volatility jumps up 90 days before a large market movement, so it is even more difficult to predict. Not surprisingly the models in Table 5.3 have much lower R2 for 90-day realized volatility in all three rates. Figure 5.4c shows the combined forecast from the GBP-USD model in the second column of Table 5.3.

The linear regression approach to the construction of combined volatility forecasts allows one to use standard results on confidence intervals for regression predictions (§A.5.1) to construct confidence intervals for volatility forecasts

The linear regression approach to the construction of combined volatility forecasts allows one to use standard results on confidence intervals for regression predictions (§A.5.1) to construct confidence intervals for volatility forecasts. For example, a two-sided confidence interval for the realized volatility will be

(a, - \„ a, + , where rj, is the combined volatility forecast value at time /, and

z.va+RrvXxrR,),

(5.5)

where Za is an appropriate critical value (in this case normal since very many points were used for the regression), s is the estimated standard error of the regression, X is the matrix of in-sample data on the different volatility forecasts and R, - (1, rj„, d2„ . . ., aml) the vector of the individual volatility forecasts at the time when the combined forecast is made.

As an example, return to the realized volatility forecasts shown Figure 5.3. The X matrix for the GBP 30-day realized combined forecast 2 in Figure 5.4a contains data on Is (for the constant), 30-day GARCH volatility, 30-day historic and 30-day implied, and

(XX)"1 =

/ 0.04200 -0.00543 0.00274 V-0.00110

-0.00543 0.00115

-0.0066 0.00001

0.00274 -0.0066

0.00042 -0.00021

-0.001104 0.00001

-0.00021 0.00011/

The vector R, depends on the time period. For example, on 7 July 1995 it took the value (1, 9.36, 9.55, 8.7) and so R,(XX) 1R( = 0.0023. Since the estimated in-sample standard error of regression was s - 2.936, and the 5% critical value of N(0, 1) is 1.645, the value for a 90% confidence interval based on (5.5) for the realized volatility on 7 July 1995 was 4.83. The point prediction of realized volatility on that day was 8.44, so the interval prediction is 8.44 ±4.83. Therefore from this model one can be 90% sure that realized volatility would be between 3.61% and 13.27%. Similarly, a 95% confidence interval is (2.68%, 14.2%). These interval predictions are very imprecise. Similar calculations for the other rates and for 90-day volatilities show that all interval predictions are rather wide. This is because the standard errors of regression are relatively large and the goodness of fit in the models is not particularly good.

5.3 Consequences of Uncertainty in Volatility and Correlation

Volatility is very difficult to predict - and correlation perhaps even more so. This section considers how one should account for the uncertainty in volatility and correlation when valuing portfolios. When portfolios of financial assets are valued, they are generally marked-to-market (MtM). That is, a current market price for each option in the portfolio is used to value the portfolio. However, there may not be any liquid market for some assets in the portfolio, such as OTC options. These must be valued according to a model, that is, they must be marked-to-model. Thus the MtM value of a portfolio often contains marked-to-model values as well as marked-to-market values.

Uncertainty in volatility is captured by a distribution of the volatility estimator; this will result in a distribution of mark-to-model values

There are uncertainties in many model parameters; volatility and correlation are particularly uncertain. When uncertainty in volatility (or correlation) is captured by a distribution of the volatility (or correlation) estimator; this distribution will result in a distribution of mark-to-model values. In this section the MtM value of an options portfolio is regarded as a random variable. The expectation of its distribution will give a point estimate of MtM value. However, instead of the usual MtM value, this expectation will be influenced by the uncertainty in volatilities. The adjustment to the usual value will be greatest for portfolios with many OTM options; on the other hand, the variance of the adjusted MtM value will be greatest for portfolios with many ATM options.

We shall show that volatility and correlation uncertainty give rise to a distribution in MtM values. Distributions of MtM values are nothing new. This is exactly what is calculated in VaR models. However, in VaR models the value distribution arises from variations in the risk factors of a portfolio, such as the yield curves, exchange rates or equity market indices. This section discusses how to approach the problem of generating a value distribution where the only uncertainty is in the covariance matrix forecast.

5.3.1 Adjustment in Mark-to-Model Value of an Option

In the first instance let us consider how the uncertainty in a volatility forecast can affect the value of an option. Suppose that the volatility forecast is expressed in terms of a point prediction and an estimated standard error of this prediction. The point prediction is an estimate of the mean E(a) of the volatility forecast, and the square of the estimated standard error gives an estimate of the variance V(<j) of the volatility forecast. Denote by /(a) the value of the option as a function of volatility, and take a second-order Taylor expansion of /(a) about E(cs):

/(a) */( )) + (df/da)(a - E(a)) + \ (d2f/da2)(a - ))2. (5.6) Thus the expectation of the option value is

E(f(a)) « ) + \ (d2f/da2)V(a), (5.7)

When the uncertainty in the volatility forecast is taken into account, the expected value of the option requires more than just plugging the point volatility forecast into the option valuation function

and this can be approximated by putting in the point volatility prediction for E(g) and the square of the estimated standard error of that prediction for V(o).

It is common practice for traders to plug a volatility forecast value into/(a) and simply read off the value of the option. But (5.7) shows that when the uncertainty in the volatility forecast is taken into account, the expected value of the option requires more than just plugging the point volatility forecast into the option valuation function. The extra term on the right-hand side of (5.7) depends on d2f/d<j2.20 For the basic options that are usually priced using the Black-Scholes formula, cP-f / 2 is generally positive when the option is OTM or ITM, but when the option is nearly ATM then d2f / 2 will be very small (and it may be very slightly negative).

We have already seen that Black-Scholes plug-in option prices for OTM and ITM options are too low and that this is one reason for the smile effect (§2.2.1). The adjustment term in (5.7) means that when some account is taken of the uncertainty in forecasts of volatility, the Black-Scholes plug-in price will be revised upwards. On the other hand, the Black-Scholes price of an ATM option will need negligible revision. It may be revised downwards, but only by a very small amount. We shall return to this problem in §10.3.3, when the volatility uncertainty will be modelled by a mixture of normal densities. The empirical examples given there will quantify the adjustment for option prices of different strikes and it will be seen that simple ATM options will require only very small adjustments, if any, because they are approximately linear in volatility.

However, the variance of the model value due to uncertain volatility will be greatest for ATM options. To see this, take variances of (5.7):

V(f(cj))*(df/dcj)2V(o).

(5.8)

This shows that the variance of the value is proportional to the variance of the volatility forecast, and it also increases with the square of the option volatility sensitivity, that is, the option vega (§2.3.3). Options that are near to ATM will have the largest contribution to the variance of the option portfolio value due to uncertain volatility, since they have the greatest volatility sensitivity.

The adjustments that need to be made to the value of a portfolio of options to account for uncertainty in the correlation forecasts are not easy to express analytically. A simple graphical method is to plot the value as a function of correlation and examine its convexity in the local area of the point correlation forecast. Figure 5.5a illustrates a hypothetical option value as a function of correlation. If the correlation forecast is p! then the value should be adjusted downwards for uncertainty in correlation because the function is concave at this point, but if the correlation forecast is p2 then the value should be adjusted

2092 9 2 is similar to the new Greek psi that was introduced by Hull and White (1997) to capture kurtosis sensitivity in options (§10.3.3).