Implied volatility

-0.2 Moneyness

0.2 0

Maturity

Figure 2.5 Black-Scholes smile surface for FTSE 100 index options, December 1997.

implied volatility against strike price17 and maturity, which is called a volatility smile surface. In most markets there is little trading in long-term or far-OTM options, therefore considerable skill is required to fit a smile surface to the available market data.

For example, the prices of the FTSE 100 Index European options on 15 June 1999 are shown in Table 2.2. Although some prices are quoted for the longer-term expiry dates, these options are very sparsely traded and the quoted prices will be rather unreliable. In general, only a few data points will be available at the long end of the smile surface, and it is necessary to interpolate and extrapolate between these points to get a smooth surface such as that shown in Figure 2.5.18

In this figure the smile and skew effects are much more noticeable in short-term options, and this is often found to be the case. If an OTM option has a long time to expiry, there will still be a good chance that it will end up ITM even with a relatively low level of volatility. However, short-term OTM option prices must reflect higher volatility expectations than short-term ATM option prices. With only a few days left to expiry an OTM option will require a big change in the underlying price - that is, a high volatility - if it is to be worth something at maturity.

The volatility smile surface over the (K, x) or the (x, x) domain should not be confused with the volatility surface - they are quite different. The volatility smile surface o(K, x) is a surface of the implied volatility (and usually the Black-Scholes implied volatility) viewed as a function of (K, x). On the other hand, the volatility surface a(S, t) is a specification of the process volatility as a function of the underlying asset price S and time t. Of course, if the process

1 It is also common to use a moneyness metric instead of a strike metric for the smile surface, as in Figure 2.5.

bThis smile surface was fitted using the cubic spline method from Press el ah (1992). Many thanks to Chris Leigh for providing this.

volatility is constant this surface is flat; it is only when the process volatility is assumed to be non-constant that the question of the specification of the volatility surface arises.

To calibrate an option pricing model, which may include a volatility surface, the analyst will need to use market data on option prices. But is it really possible to specify a volatility surface by assuming a functional form and then to use market data on option prices to estimate the parameters of this function? The implied volatilities that are backed out from option prices are forecasts of the process volatility, so these could be used to estimate the parameters of a volatility surface. However, it is not consistent to use Black-Scholes implied volatilities in this way, because the Black-Scholes model assumes constant volatility.19 Therefore, what is often done is that the Black-Scholes implied volatility for a particular option (usually ATM) is regarded as a forecast of the average volatility of the process over the lifetime of the option. If options are available for many different maturities then the time dimension of the volatility surface can be calibrated in this way.20 The space dimension, S (i.e. the underlying price dimension), is more difficult, as we shall see in the next section.

2.3 The Relationship between Prices and Implied Volatility

Implied volatilities are derived from market prices. Therefore, you may ask, if the underlying price changes to some level, how will the implied volatilities change? An answer to this question will indicate how to calibrate the volatility surface in the space dimension21 (§2.3.1). It will also tell us how to construct scenarios for prices and implied volatilities for use in risk management (§2.3.2). This is also a very interesting question for option traders, because the answer will give us the volatility sensitivity to price term, /dS, which is an important determinant of the option delta (§2.3.3).

2.3.1 Equity Prices and Volatility Regimes

This subsection follows the work of Derman (1999) that is based on binomial trees with local volatilities rather than volatility surfaces.22 Derman (1999) shows how local volatilities may be fixed by the strike of the option in some

!9To be consistent, one should really use the implied volatilities that are backed out from market prices using a model with a time- and space-varying volatility process. However, this leads to a circular argument: the volatility process cannot be specified before one knows the parameters of the volatility surface, so one cannot back out the implied volatilities corresponding to that process from market prices.

2t)For example, if 1-week ATM volatility is 0,(1) at time t. and 2-week ATM volatility is cj2(t) = vKiC)2 + °i(t + D2)/2], then cj](r + 1) = [(2 2(02 - Oi()2]- tn a Brownian process, the variances are additive, not the volatilities.

2!Assuming that Black-Scholes implied volatilities are forecasts of the average of a time-varying process volatility.

22A binomial tree is a discretization of a price process in space and time that is calibrated so that it will fit a deterministic volatility surface. The trend in nodes over time is equal to the risk-free rate and the local volatility of a node is a measure of the dispersion at that node.

Figure 2.6 One-month fixed strike volatilities, at-the-money volatility and the index level.

circumstances (the sticky strike model). That is, there will be a different tree relevant for pricing each option with a different strike, and all the local volatilities in this tree will be the same. In other circumstances (in fact when markets are trending but stable) the tree should have local volatility that is determined by the delta of the option, rather than the strike (the sticky delta model). When markets are jumpy there is a single tree that can be used for all options, with local volatilities that really do vary from node to node (the sticky implied tree model).

Figure 2.6 shows the 1-month implied volatilities for European options of all strikes on the FTSE 100 index for the period from 4 January 1998 to 31 March 1999.23 The bold grey line indicates the ATM volatility and the bold black line the FTSE 100 index price (on the right-hand scale).

Look at the movements in the index and the way that ATM volatility is behaving in relation to the index. In the spring of 1998 the index was trending upwards while ATM volatility remained relatively constant. From mid-July until the end of August 1998 the index fell by over 1000 points in the wake of the Russian crisis. ATM volatility rose from 18% to 44% around this time, and

23The fixed maturity implied volatility data used in this section have been obtained by linear interpolation between the two adjacent maturity option implied volatilities. However, this presents a problem for the 1-month volatility series because data on the near maturity option volatilities are often totally unreliable during the last few working days before expiry. Therefore, the 1-month series rolls over to the next maturity, until the expin date of the near-term option, and thereafter continues to be interpolated linearly between the two option volatilities of less than and greater than 1 month.