196 MODEL RISK AND FINANCIAL DERIVATIVESpricing model for options.^3 This implied volatility figure can then be used
as an input to other models, for instance to value more complicated options
on the same underlying asset for which we have no market price or illiquid
markets, typically over-the-counter derivatives.
In theory, there is nothing wrong with this type of approach. In fact,
derivative prices observed at any given time should contain forward looking
informationonvolatility. Thismeansthatthemodelsusedtopriceandhedge
derivatives must be determined partially from econometric information and
partially by solving “inverse problems” (in the sense of partial differential
equations) that reflect current market prices. However, the problem starts
when (a) we have several option prices available from which we can obtain
an implied volatility, and (b) different options on the same underlying asset
display different implied volatility. This phenomenon was originally called
the smile, because a graph of the implied volatility against the strike of
the corresponding option would typically look like a smile, deep in and
out of the money options having higher implied volatility. After the 1987
crash, the smile disappeared and usually became a frown, with implied
volatility generally decreasing as the strike price increase. As a result, in the
money call options and out of the money puts tend to have prices that are
above their respective theoretical Black and Scholes values, while out of the
money calls and in the money puts are priced below their respective Black–
Scholes values. Rubinstein (1994) attributed part of this phenomenon to
crash-phobia, that is, investors valuing more out of the money puts because
they fear a new crash. In addition to the frown, market participants often
observe a term structure effect: options with the same strike price but with
different maturities also tend to display systematic volatility patterns with
respect to time to maturity.
In order to account for these biases with respect to the Black and Scholes’
constant volatility assumption, market participants started to build up mod-
els that accept a volatility surface rather than a single volatility number as
an input. That is, depending upon the maturity date, the degree of money-
ness of their options and whether it was a put or a call, market participants
were considering different levels of volatility. These different levels were
obtained by inverting the Black and Scholes model to yield a local volatility
figure (see Figure 10.1).
From a practical perspective, the models that use an implied volatility
surface are extremely convenient. They can explain and quantify the skew-
ness and kurtosis in the empirical distribution of stock returns. They are
consistent with different types of stochastic processes for the underlying
asset. Moreover, they can be recalibrated to market data several times a
day and their pricing errors are extremely small, which gives a false sense
of security to their users. However, from a theoretical perspective, implied
volatility surface models are by definition unsatisfactory. If the Black and
Scholes assumptions hold, the volatility surface should be ... flat. And if the