imbalance between supply and demand, and these are implicit in implied volatilities but not in the statistical volatilities (Swidler and Diltz, 1992).

2.2 Features of Implied Volatility

We have just seen that the constant volatility assumption does not fit the market prices of options very well. One consequence of this is that, if model prices are to be close to market prices, then different volatilities for options of different strikes and maturity on the same underlying asset must be used. This section describes, and attempts to explain, some well-known patterns in the behaviour of Black-Scholes implied volatility as the strike and the maturity of the option change.

2.2.1 Smiles and Skews

The smile effect refers to the empirical fact that for most underlying assets a plot of implied volatility against strike has a smile shape.14 Implied volatility is usually higher for OTM puts and calls than for ATM options. Why should this be so?

if returns are fat-tailed then large price changes will be more likely, and consequently an OTM option will have a higher chance of becoming ITM, than is assumed in the Black-Scholes model. The only way that the Black-Scholes price can match the market price is to increase the volatility

Most options are priced using the Black-Scholes formula, but it is well known that the assumptions upon which this formula is based are not justified empirically. In particular, return distributions may be fat-tailed (§10.2) and their volatility is certainly not constant (§4.1). Therefore, it is not really appropriate to model the underlying price as a geometric Brownian motion. If prices are not governed by a geometric Brownian motion then large price changes may be observed empirically with a frequency that is greater than that assumed in the Black-Scholes model. If returns distributions are normal but volatility is stochastic, or if volatility is constant but returns are fat-tailed - or, indeed, both - then large price changes will be more likely, and consequently an OTM option will have a higher chance of becoming ITM, than is assumed in the Black-Scholes model. Therefore, the Black-Scholes model price will be less than the market price for an OTM option. Now the only way that the Black-Scholes price can match the market price is to increase the volatility - this is the only parameter that the model is free to change. Thus the implied volatility for OTM options will be greater than the ATM implied volatility.

Symmetric smiles where both OTM call and put options have higher implied volatilities than ATM options are commonly observed in foreign exchange markets. However, in equity markets the smile has a negative skew appearance, with higher implied volatilities for the low strike options and lower implied volatilities for the high strike options. This can only be due to the fact that the market price of low strike options (ITM calls and OTM puts) is much higher than the Black-Scholes model predicts. The main reason for this is that equity

It is also common to plot a volatility smile as implied volatility against moneyness instead of strike, in §2.3.3 we see that the moneyness. as measured by x, and the Black-Scholes delta of an option have a simple relationship: ABS(>S, a) = ( ). Therefore it is also common to plot volatility smiles with respect to delta.

markets are not symmetric. A price fall is bad news for the shareholders, whereas a price rise is good.15 The negative skew is often very noticeable in the index market, where a very large price fall could precipitate doom and gloom for the economy as a whole. Traders will therefore use a high volatility to price an OTM put, and they are able to do this because there is a high demand for the insurance offered by these puts from risk-averse investors. This highly priced supply of OTM puts will normally be met by an equal demand as long as investors hold pessimistic views about the possibility of a market crash. It is notable that the skew in equity markets has only been pronounced since the 1987 global crash in equity markets.

A secondary reason for the negative skew in equity markets is that equity markets become much more turbulent after a large price fall than they do after a price rise of the same magnitude. This so-called leverage effect is discussed in §4.1.2. The market price of an ITM call (or equivalently an OTM put) reflects the fact that if there is a large fall in price (so the ITM call option becomes OTM) the underlying asset volatility will remain high for some time. Then the prices of all options will increase and, in particular, the call option will have a high probability of becoming ITM again because of the increased volatility. The higher than expected market price of ITM calls and OTM puts can therefore be attributed to more than just the doom and gloom in the economy - the leverage effect also plays a secondary role.

2.2.2 Volatility Term Structures

Now consider the relationship between maturity and implied volatility for a fixed strike option. For example, reading across a single row in Table 2.2 gives the market prices of different maturity options of the same strike, and backing out the volatilities implied in these prices will give a term structure of volatility. Figure 2.4 shows the implied volatility, plotted as a function of the option maturity, which corresponds to the call prices given in the second row of Table 2.2 (K= 6325).

The volatility term structure converges to the long-term average volatility level, which is about 21% in the figure, and this is a typical feature of implied volatility. It is a consequence of the mean-reverting behaviour of volatility; the fact that volatility comes in bursts or clusters (§4.1.1). If the current period is very volatile, short-term volatilities will be well above the long-term average. Expectations may be that volatility will be high over the next few days, but over a longer period it is usually expected that volatility will fall back to its average level. On the other hand, if markets are relatively tranquil then short-term volatility will be below the long-term average and volatility term structures will converge from below.

The volatility term structure converges to the long-term average volatility level. This is a consequence of the mean-reverting behaviour of volatility; the fact that volatility comes in bursts or clusters

15The opposite is usually the case in commodity markets: price falls are good news and price rises are bad news. The volatility skew in commodity markets is normally the other way around, that is, OTM calls (high strike options) have higher volatilities than OTM puts (low strike options).

Table 2.2: Market prices of FTSE 100 call and put options with different strikes and

maturities

Expiry Jun Jul Aug Sep Dec

end: - - - - -

Strike Call Put Call Put Call Put Call Put Call Put

366.5 182

332.5 197.5 397 244 582.5 373

300 215

269.5 233.5 333 279 517.5 405.5

240 254

213 276 272 316.5 272 448

187 300

163.5 326 219 362 219 495.5

Source: Financial Times.

0.5 0.6

Time (years)

Figure 2.4 Term structure of implied volatilities on 15 June 1999; 5 = 6451.2, 6325, r = 0.055.

GARCH models are statistically tractable models of volatility that give convergent volatility term structure forecasts. However, if GARCH models are used there are many implications that will warrant careful consideration, and these will be discussed in Chapter 4. For example, the Black-Scholes delta (or gamma or vega) will no longer be appropriate, and pricing and hedging become uncertain because one can no longer assume the risk-neutrality hypothesis.16

2.2.3 Volatility Surfaces

Combining the smile (or skew), which is a cross section of implied volatilities from different strike options of the same maturity, with a volatility term structure for each of these strikes, one obtains a three-dimensional plot of

,6The risk neutrality hypothesis rests on the completeness of markets and the arbitrage that results when it is possible to perfectly hedge any asset. However, when volatility is stochastic it will not be possible to hedge every type of asset perfectly (§4.4.2).