Table 10.3: Black-Scholes option prices with normal mixture distributions

options priced at the low and high volatilities of 13.41% and 24.91%. For simplicity we assume zero interest rate and no dividends.

When = 100 the option price is approximately linear in volatility so the two prices are the same.25 For other strikes the mixture prices are reflecting the uncertainty in the volatility forecast, so the mixture prices are higher than the price under the average forecast. The relationship between Black-Scholes and normal mixture prices for simple calls and puts can be investigated using the normal mixture option pricing spreadsheet on the CD.

For other strikes the mixture prices are reflecting the uncertainty in the volatility forecast, so the mixture prices are higher than the price under the average forecast

One of the reasons why the Black-Scholes formula tends to underprice OTM Much of the observed

options is that large price movements have a higher probability than is assumed by a normal density and so OTM options have a greater chance of ending ITM than assumed in the Black-Scholes framework. The only way the Black-Scholes model can account for an observed market price that is greater than the model price is to jack up the volatility used in the model price. This is why we observe a smile effect in Black-Scholes volatilities (§2.2.1).

convexity in implied volatilities can be explained by an uncertain volatility that is captured by the normal mixture model

The last column of Table 10.3 shows the Black-Scholes implied volatilities that match the normal mixture option prices. The implied volatilities display a symmetrical smile as observed in many option markets. Note that the implied skew (negative for equities, positive for commodities) has different causes, but much of the observed convexity in implied volatilities can be explained by an uncertain volatility that is captured by the normal mixture model.

We shall end this chapter with a discussion of the implications of non-normal densities for dynamic hedging. When there is excess kurtosis dynamic delta

25 This is the case whatever the parameters of the normal mixture density and not just for the parameters chosen in this example. Equality of these prices is due to the fact that ATM options are linear in volatility, as shown above.

It is really the excess kurtosis in the return distribution that increases uncertainty in the value of the delta hedged portfolio

hedging may be a very uncertain activity. This is because the standard error of a delta neutral portfolio will increase with kurtosis (as well as with the volatility of the underlying). The change in MtM value of an option portfolio has a nonlinear relationship with the change in the underlying, AS, and because of this the change in MtM value of a delta hedged portfolio as the underlying price moves will have a standard error that depends on the kurtosis as well as the volatility of the underlying returns. In fact if P is a delta hedged portfolio then

( ) 2( + 2)o-4S4A,2,

(10.18)

where is the excess kurtosis and is volatility of the price process. Assuming the option gamma (y) and the volatility are constant over a small price range and time interval, formula (10.18) shows that it is really the excess kurtosis in the return distribution that increases uncertainty in the value of the delta hedged portfolio. Thus estimates of the portfolio value change can become very unreliable when returns are fat-tailed, and therefore it may be necessary to rebalance the portfolio very frequently indeed.

An optimal delta rebalancing strategy is normally designed to achieve a tradeoff between the increased transaction costs that are incurred by frequent rebalancing, and the costs that may (or may not) result from uncertainty in the portfolio value. If there is much uncertainty in the value of the hedged portfolio, as there will be when returns have a large excess kurtosis, the natural tendency will be to increase the rebalancing frequency. However, this may not work, because kurtosis usually tends to increase when rebalancing intervals become shorter.

Perfect hedging may be impossible because excess kurtosis would lead to an infinite transaction volume

For example, suppose the portfolio is delta hedged twice a day and that the excess kurtosis of 12-hour returns is approximately 2. By (10.18) the standard error of the hedged portfolio will be proportional to 2, and if either the gamma or the volatility is large this standard error could be a problem. It may present too much uncertainty in the value of the hedge, so one would prefer more frequent rebalancing, say four times a day. But then the standard error will become even larger because the excess kurtosis will increase.27 Therefore there will be even more uncertainty about the value of the hedge. This argument leads to the conclusion that perfect hedging may be impossible because excess kurtosis would lead to an infinite transaction volume.

The hedge uncertainty It is also possible that the hedge uncertainty may not be reduced indefinitely by may not be reduced an active rebalancing strategy; it may have a positive minimum value. To see indefinitely by an active recall (footnote 18) that if returns are uncorrelated, the excess kurtosis

rebalancing strategy

26To see this, use a delta-gamma approximation dAt + + y(AS)2, where 8 0 for delta hedging over short time intervals. Then V(AP) = E((Ap)2) - (E(AP))2 %\y2{E(AS)4 - (E(AS)2)2}. Using AS « Sc-jAt gives V(AP)ly2(K + 2)aiS4At2.

27 If the kurtosis of 6-hour returns is around 4. the standard error of the hedged portfolio will be proportional to (3/2) ~2, which is bigger than it was before (and the problem is compounded by the fact that the volatility will also increase).

observed over one time interval should revert to zero when observed over many time intervals. Conversely, if one reduces the rebalancing interval of a hedging strategy, the excess kurtosis will increase: for example, the total hedge uncertainty over a period (0, t) with rebalancing every / would be proportional to 2 (2 + ).28 Hull and White (1997) have suggested a possible solution to this problem. Instead of trying to cope with kurtosis by increasing the frequency of the delta hedge they suggest hedging with a new Greek which captures the sensitivity of option prices to kurtosis in the underlying.

28 From (10.18) and footnote 17. V(AP) == 2((/ /)(( / <) + 2) <2 = 2 ( + 2 ).