Pricing

For listed and standardized products, the first visible consequence of model risk is a
price discrepancy: theoretical prices diverge from effectively quoted prices. Such
behaviour is rapidly detected for standard liquid instruments, but not for over-thecounter
or structured products, since similar assets are generally not available on
the market. Sometimes, for a particular product, the firm’s trading desk itself is even
‘the’ market!
Despite a bewildering large set of models, relatively little work has been done to
examine how these models compare in terms of their ability to capture the behaviour
of the true world. Because of a lack of a common framework, most studies have
focused on testing a specific model rather than on a comparison across models.
Noticeable exceptions are the studies of Bates (1995), Bakshi, Cao, and Chen (1997)
and Buhler et al. (1998), where prices resulting from the application of different
models and different input parameters estimations are compared to quoted market
prices in order to determine which model is the ’best’ in terms of market price
calibration.
It is worth noting that the pricing bias is not necessarily captured by the price, but
by an alternative parameter such as the implied volatility. For instance, it is now
well documented by various academics and practitioners that using the Black and
Scholes model to price options results in several biases:
Ù A bias related to the degree of moneyness of the option. This moneyness bias is often referred to as the ‘smile’, the ‘smirk’ or the ‘skew’, depending on its shape.
Typically, since the October 1987 crash, out-of-the-money puts trade at a higher
implied volatility than out-of-the-money calls. For options on dollar prices of
major currencies, the most common pattern is a smile, with an implied volatility
at-the-money lower than the implied volatility in- or out-of-the-money.
Ù A ‘volatility term structure’ effect, that is, biases related to the time to expiration
of the options of at-the-money options. Generally, for currency options, one can
observe a lower implied volatility for at-the-money short-maturity options. These
variations are small, but they are nevertheless a recurring phenomenon.

Ù A ‘volatility’ bias, where typically the Black and Scholes model over-estimates
options on stocks with high historic volatility and under-estimates options on
stocks with low historic volatility.
These volatility smiles and other patterns are often attributed to departure from
normality in the logarithm of the price of the underlying asset or to market frictions.
To ‘live with the smile’, most practitioners have adopted a rather pragmatic approach
and use an ‘implied volatility matrix’ or an ‘implied volatility surface’. Depending on
the time to maturity and the moneyness of the option, the volatility level is different.
This contradicts one of the fundamental assumptions of the model, namely, the
constant level of the volatility. It is nothing else than a rule of thumb to cope with an
inadequate model.