Figure 5.5 (a) Option value as a function of correlation; (b) adjustment for uncertainty in correlation; (c) options with opposite correlation characteristics.

upwards for uncertainty in correlation because the function is convex at this point. Figure 5.5b shows that the amount of this adjustment depends on the degree of uncertainty in correlation and the shape of the value as a function of correlation. If correlation is known to be p then the option value is V(p). However, if correlation is unknown, suppose it takes the value p[ with probability p and the value p2 with probability 1 - p: in this case, because the option value is convex in correlation, the option valuepV(Pi) + (1 -p)V(p2) is greater than V(p). Thus the uncertainty in correlation, in this case, would lead to an upward adjustment in the option value.

Rather than making an ad-hoc adjustment in portfolio values, it may be preferable to hedge this correlation risk by taking a position with the opposite characteristics. For example, if value is a concave function of correlation this

Rather than making an ad-hoc adjustment in portfolio values, it may be preferable to hedge this correlation risk by taking a position with the opposite characteristics

portfolio could be hedged with a portfolio that has a value which is a convex function of correlation, as depicted in Figure 5.5c. Unfortunately, it is not always possible to find such products in the market, although recently there has been some growth in the markets for OTC products such as currency correlation swaps to hedge correlations between two currency pairs.

Before ending this section, it is worthwhile to comment that the method for adjusting portfolio values due to uncertain correlation that is depicted in Figure 5.5 should be applied to all parameters in the portfolio pricing model. Volatility is easier, because the adjustments have a simple analytic form described by (5.7) and (5.8). But for correlation, and all other parameters, it is worthwhile to investigate the degree to which they affect the value and, if necessary, make adjustments for uncertainties in parameter estimates.

5.3.2 Uncertainty in Dynamically Hedged Portfolios

How much does it matter if there are errors in the implied volatilities that are used to calculate deltas for dynamic hedging of an option portfolio? It is possible to answer this question with an empirical analysis, as the following example shows. Consider a short position on a 90-day call option that is delta hedged daily using an implied volatility of 15%. The process volatility is unknown, of course, but suppose you believe that it could take any value between 8% and 18% and that each value is equally likely. What is the error in the value of the delta hedged portfolio?

For a given process volatility we can estimate the value of the option in h days time, and so also the value of the delta hedged portfolio. Using 1000 Monte Carlo simulations on the underlying price process, with a fixed process volatility, we obtain an expected value for the option (taking the discounted expected value of the pay-off distribution as described in §4.4.2) and a measure of uncertainty in this option value (either the standard error of the option value distribution over 1000 simulations, or the 1% lower percentile of this distribution). In each simulation, the evolution of the underlying price and the fixed delta (corresponding to an implied volatility of 15%) allow us to compute the expected value of the delta hedged portfolio in h days time, along with a measure of uncertainty (standard error or percentile). We can do this for each process volatility x% ( - , 9, . . ., 18) and thus generate a curve that represents the expected value of the hedged portfolio as a function of the process volatility. Then, taking the time horizons h - 1, 2, . . ., 90 gives a surface that represents the expected value of the hedged portfolio as a function of the process volatility x and time horizon h.

For the uncertainty around this surface, the 1 % lower percentile of the value change distribution is a good measure since it corresponds to the 1% A-day VaR measure. Figure 5.6 shows the 1% VaR for each possible process

Figure 5.6 One per cent VaR of a dynamically delta hedged call.

Figure 5.7 The effect of using an incorrect hedging volatility.

volatility between 8% and 18% and for each holding period from 1 to 90 days. The figure shows that VaR measures can be high even when process volatility is low, if the option is hedged for a long period using the wrong hedging volatility.21 So much is obvious, but what is more interesting is that the VaR measures are much greater when the process volatility is above 15%. That is, when the (wrong) hedging volatility of 15% is underestimating the process volatility, some VaR measures can be very considerable indeed.

There is a simple explanation for the shape of this surface. With continuous rebalancing, the variance of a delta hedged portfolio due to using the wrong hedging volatility is approximately (ah - rj0)2 (vega)2, where rjh denotes the implied volatility used for hedging and rj0 denotes the process volatility.22 This is only a local approximation. In fact the standard error of the hedged portfolio value has the asymmetric shape shown in Figure 5.7.

21Many thanks to Chris Leigh for providing this figure.

22Discrete rebalancing increases the standard error of the hedged portfolio by a factor that depends on 1 /J(2n). where n is the number of rebalancings, and on the process volatility; the standard error curve still has an asymmetric smile shape.