Table 2.1: Prices, implied volatilities and volume of trade on call and put options

of implied volatility data show discrepancies between call and put implied volatility. Why should this be so?

Consider the data on option prices and associated implied volatilities for FTSE 100 index options shown in Table 2.1. A plot of these implied volatilities against strike is shown in Figure 2.1.

Notice that the call implied volatility is considerably greater than the put implied volatility for all strikes, as shown in Figure 2.2." Generally speaking, if call implied volatilities are significantly different from put implied volatilities it is because the evaluation model is not applied correctly. Probably it is a model based on spot price when the appropriate hedging instrument is a future, and the future is not quite in line with the spot price. In a rising market, or whenever the future trades above its fair or theoretical value, calls will appear more expensive than they should because market prices are based on a higher underlying price than the underlying price taken in the model. The way that the model is adjusted to account for the high market price is to jack up the volatility. Similarly, puts will appear less expensive than they should, so the implied volatility that is backed out of the model will be lower. The opposite is true in a falling market, or whenever the future trades below its fair value.

If call implied volatilities are significantly different from put implied volatilities it is because the evaluation model is not applied correctly

In the example above the closing value of the FTSE 100 index was 6451.2, so the theoretical fair value of the future was 6453.92. However, the FTSE 100 future closed at 6486.12 The call implied volatilities are too high because the

1 These are American options and they only have 3 days to expiry. Compared to the time value, the early exercise premium may have a large effect. However, the early exercise premium only increases the value of put options and so makes the point even more strongly in this example.

12The index closes 30 minutes later than the future, so the index had fallen back.

35 30 -25-

10 -5 -

-I-1-1-.-1-1-

6200 6300 6400 6500 6600 6700 6800

-----Calls -Puts

Figure 2.2 Implied volatilities on the FTSE 100 index option, 15 June 1999.

market price is based on a higher price than is assumed in the model, and put implied volatilities are too low.

This sort of anomaly would disappear with a model that takes into account the actual futures price. Whenever possible, implied volatilities should be taken from futures options rather than options on the underlying cash instrument, unless the cash market can be used for hedging. Then, when the model is applied correctly, the implied volatility in an ITM call should be about the same as the implied volatility in an OTM put of the same strike.

2.1.3 Differences between Implied and Statistical Volatilities

Implied volatilities should be viewed differently from statistical volatilities, even though they both forecast the volatility of the underlying asset over the life of the option. The two forecasts differ because they use different data and different models.13 Implied methods use current data on market prices of options, so implied volatility contains all the forward expectations of investors about the likely evolution of the underlying. The model for implied volatility assumes complete markets, no arbitrage and a GBM constant volatility continuous time diffusion process for the underlying asset price. Contrast this with statistical methods for generating volatility forecasts, which use historic data on the underlying asset returns in a discrete time model for the variance of a time series.

If the option pricing model were an accurate representation of reality, and if investors expectations were taken as correct, then any observed differences

13The model for the underlying price process in a statistical analysis should not differ from the model used for option pricing.

Tracking implied volatility shows that the option was overpriced here

Upper and lower 95% confidence imits for statistical volatility forecast

Time to maturity

Figure 2.3 Volatility cones.

between implied and statistical volatility would reflect inaccuracies in the statistical forecast. Alternatively, if statistical volatilities were accurate, then differences between the implied and statistical forecasts of volatility would reflect a mispricing of the option by the market.

Implied volatilities can be compared with the statistical forecasts of volatility over a horizon equal to the term of the option. This type of comparison has been used as a means of evaluating whether an option is cheap or expensive. Upper and lower confidence limits can be taken from statistical volatility forecasts (§5.2) and any substantial mispricing of the option in the market will show up when the implied volatility exceeds one of these limits. If the implied volatility exceeds the upper limit the market may be overpricing the option, and if it falls below the lower limit the market may be underpricing the option.

To trade on this type of observation, of course, one needs to gain some sense of the volatility history of the option. This is what is gained by looking at volatility cones. To construct a cone, estimate confidence limits for volatility forecasts of several different horizons, as described in §5.2. If implied volatilities are available then these should be used to construct the cones. Otherwise one might construct an empirical distribution of historical volatility from all / -period volatilities during the last few years and record the upper and lower 95% confidence limits. Repeat this for a number of different holding periods, from 1 day to 1 year say, and this will give the upper and lower limits of the cones, as in Figure 2.3.

Cones are used to track implied volatility over the life of a particular option, and under- or overshooting the cone can signal an opportunity to trade, as shown in the figure. However, if the cone is constructed from the confidence limits for statistical forecasts they should be used with caution, particularly if overshooting is apparent at the long end. This is because differences between long-term statistical and implied volatility are to be expected. In particular, transaction costs can be substantial for long-term positions when there is a real

This type of comparison has been used as a means of evaluating whether an option is cheap or expensive