The lognormality of the asset price distribution, which underlies the Black-Merton-Scholes [BMS] derivation, is not a satisfactory assumption. In fact, it is documented that equity prices do not follow such a distribution. Nonetheless, the BMS methodology results into a formula that is intuitive and very easy to implement in practice, and therefore it is widely used both for academic and practical purposes. In addition, the fact that the volatility of the underlying asset and the risk free rate of return are assumed constant, simplifies the exposition, by forcing the markets to be complete, but on the other hand reduces the empirical performance of the BMS model.

Testing the BMS model gives rise to many theoretical and practical problems. When one uses actual option prices, she cannot distinguish between the potential misspecifications of the pricing formulae and the market inefficiencies. The joint hypothesis that the correct model is used and that the markets are efficient is necessarily tested.

In addition, at any time, the only parameter of the BMS model which is not directly observed is the volatility of the underlying asset; therefore it is unknown which value is the appropriate one to be used. One approach is to use historical estimated volatilities, resulting into the problem of choosing the appropriate window for estimation. Alternatively, traders use implied volatilities from other contracts to construct volatility matrices, and then interpolate according to the moneyness level (or the Delta) and the maturity of the particular contract to be priced, in a fashion that tries to mimic a term structure of implied volatilities.

A third problem arises from the possible asynchroneity of the equity, bond and option markets. If trading does not take place simultaneously, or the market are very thin, it is questionable if the assumption of completeness is satisfactory. Market microstructure issues can heavily bias the quoted prices in such instances. Practitioners try to overcome this problem by extracting the implied forward price and the implied rate of retrurn from quoted option prices, rather than using market quotes for these values.

Researchers, straight after the publication of the BMS results were interested in inverting the theoretical option price, in order to retrieve this unobserved implied volatility from traded options, across different levels of moneyness and maturity -see for example the papers of Rubinstein (1985, Journal of Finance), Jackwerth and Rubinstein (1996, Journal of Finance) and Rubinstein (1994, Journal of Finance). In Rubinstein (1985) different patterns of impled volatilities seem to emerge, depending on the particular period that was used, while in Jackwerth and Rubinstein (1996) implied volatilities tend to be higher for in-the-money call options and lower for out-of-the-money calls, assuming that BMS price at-the-money contracts correctly. The emerging pattern of implied volatilities with respect to moneyness is often encountered in the literature as the implied volatility smile, skew or smirk.

Stylized facts about the distribution of asset returns has been well documented in Bollerslev, Engle and Nelson (1994, Handbook of Econometrics IV), Ghysels, Harvey and Renault (1996, ''Handbook of Statistics 14''), and others. They include:

1. Leptokurtosis
2. Volatility clustering
3. Leverage effect
4. Implied volatility skew

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