formula tends to underprice OTM and ITM options (§5.3.1). The second example in this subsection shows how to modify Black-Scholes option prices using a normal mixture density that models uncertainty in volatility.

When excess kurtosis is present over the maturity of an option (so the option has maturity of no more than a few days) it will affect the price. Since the price of a European option is the expected value of a pay-off function over the probability density of the underlying asset price at maturity, if this probability density is expressed as a mixture of several density functions, then the price of the option is the weighted sum of the prices under each of the densities in the mixture. Thus the price of an option if returns are generated by a zero-mean normal mixture, is simply the probability-weighted sum of the normal option prices.21

In other words, the normal mixture option price is the expected value of the option over a distribution in volatilities. In mathematical terms, let /(a,) be the price of an option assuming normal returns with known volatility rj,. Then the normal mixture option price will be

/7,7(0-, ) + ...+/?„ ",,).

If we want to calculate the price of an option under the assumption that returns are generated by a normal mixture, how should we choose the parameters of the normal mixture? Normally we would choose parameters so that the variance and the excess kurtosis under the normal mixture are the same as those observed on empirical returns.

To illustrate results throughout this subsection we shall suppose that observations on daily returns indicate an annual variance of 0.04 (corresponding to an annual volatility of 20%), but that this estimate is very uncertain. Also the excess kurtosis of daily returns is 2.53. We choose to model the uncertainty in variance by a mixture of two zero-mean normal densities: with probability />, = 0.6 the annual variance is rj2 = 0.01 (annual volatility - 10%); with probability p2 - 0.4 the annual variance is rj2 = 0.085 (annual volatility = 29.15%). Thus the average variance is \ + p2aj = 0.04 and by (10.14) the daily excess kurtosis is 2.53. Thus our normal mixture matches the average variance and the daily kurtosis.

With this normal mixture a standard ATM 1-day European call option will have a value (as a proportion of the underlying asset price) of22

0.6 x 0.253 + 0.4 x 0.737 = 0.444.

21 In their paper Fitting volatility smiles with analytically tractable asset price models Brigo and Mercurio (2001) derive a price process with a lognormal mixture density of the form dSIS = rj dt + a(S,t)dZ where the local volatility is given by a(S, t)~ - Yp*g2: and p* - p;[§, ,(5)/r,(5)].The paper is available from www.fabiomercurio.it.

22 It follows from the Black-Scholes formula that the value of a 1-day ATM option V is approximately (1 /V2it)oS, where a is the daily volatility. So if annual volatility is 10%, V= 0.4(0.1/V250)S = 0.0025298S, and if annual volatility is 29.15%. V= 0.4(0.2915/7250) = 0.007374S.

The price of an option if returns are generated by a zero-mean normal mixture, is simply the probability-weighted sum of the normal option prices

For ATM options, which are approximately linear in volatility,23 the normal mixture option price obtained above is the same as the Black-Scholes price that corresponds to the expected volatility:

0.6 x 10% +0.4 x 29.15% = 17.66%.

It is not the same as the Black-Scholes price based on a volatility that is equal to the square root of the expected variance.24 In fact, the expected variance is 0.04, corresponding to the volatility 20% and this volatility gives the Black-Scholes option price as 0.506, which is very far from 0.444.

Therefore, when there is uncertainty over volatility or variance, it is the expected volatility (not the square root of the expected variance) that should be used for the Black-Scholes price, and this should be compared with the normal mixture price that is based on the whole distribution of volatility. If instead we were to use the expected variance for the Black-Scholes price, we would reach the conclusion that ATM options are very overpriced by the Black-Scholes model. Now, even though the Black-Scholes model is based on unrealistic assumptions, we should like to think that it can, at least, price a simple ATM option. Therefore, in Table 10.3, we shall compare normal mixture option prices with Black-Scholes prices that are obtained using the expected volatility and not the square root of the expected variance.

For ATM options the normal mixture option price is the same as the Black-Scholes price that corresponds to the expected volatility. It is not the same as the Black-Scholes price based on a volatility that is equal to the square root of the expected variance

The above example was based on a simple ATM option, and there was no difference between the Black-Scholes price and the normal mixture price that is based on the distribution of volatilities. But in general an option price is not linear in volatility. In that case it is very important to use a model price that is based on the distribution of volatilities because this price will not be the same as the Black-Scholes price, whether it is based on the expected volatility or the square root of the expected variance.

Standard options are convex in volatility except when they are ATM (see Figure 2.1); some exotic options, on the other hand, may be concave in volatility (e.g. knock-out barrier options). For these options, it is very important to use a model price that is based on the distribution of volatilities (such as the price we can obtain using the normal mixture model). Exotic options will not be priced using Black-Scholes model of course, but the general option value function shown in Figure 10.9 indicates that the same conclusion can be drawn: when the variance is highly uncertain one should always calculate the option price as an expected value over the distribution of volatility

23 When S = Ke-" then \n{S/Ke-") = 0 and a call option has price = S(<J>(oVt/2)- <J>(-oVt/2)). Thus / - (5 / ) ( / /2). where (-) is the normal density function. For small x, <j>(*) can be approximated by (0). so / % (5 / ) (0) which is a constant. Figure 2.1b also shows that the price of a near to ATM option is approximately linear in volatility.

24 The expected volatility is not equal to the square root of the expected variance, as was already noted in Chapter 3.

When the variance is highly uncertain one should always calculate the option price as an expected value over the distribution of volatilities, and not as a function of the expected

Figure 10.9 Option price under expected volatility compared with option price under distribution of volatilities.

volatilities, and not as a function of the expected volatility (or, for that matter, as a function of the square root of the expected variance).

Figure 10.9 illustrates why the option price using a distribution of volatilities will be different from the price using the expected volatility. When options are convex in volatility the option price under the distribution will be greater than the price under the expected volatility, as shown in the figure. Therefore if (Black-Scholes) option prices are calculated using the expected volatility they will be less than the (normal mixture) option prices that are calculated from the whole distribution of volatilities. Conversely, when options are concave in volatility the (normal mixture) option price under the distribution will be less than the (Black-Scholes) price under the expected volatility.

For longer-term options it is very unlikely that excess daily kurtosis will have any effect on pricing as it tends to disappear quickly. Therefore, our behavioural model of uncertainty in volatility will change. The model described in §10.3.2, with heterogeneous traders that have different short-term volatility forecasts, explains how excess kurtosis results from volatility uncertainty in the market as a whole, even though each individual has no uncertainty. However, there may still be considerable uncertainty in the volatility forecast that is held by one individual and, again, we shall model this uncertainty using a normal mixture distribution.

Suppose that we want to price a 30-day option and suppose the daily returns indicated an average annual variance of 0.04-that is, a 20% annual volatility - as before. Suppose also that our forecast over 30 days can be approximated by the following normal mixture distribution: with probability Pi - 0.5 the annual variance is a2 - 0.018 (annual volatility - 13.41%); with probability p2 - 0.5 the annual variance is a2, = 0.062 (annual volatility = 24.91%). Thus the expected variance is again p\a\ + 2< \ - 0.04 but now the expected volatility is 19.16%. Table 10.3 compares the value of 30-day European call options with an underlying asset price of 100, priced first according to the average volatility of 19.16% and second as an average of