Stylized facts

The lognormality of the asset price distribution, which underlies the BS derivation, is not a satisfactory assumption. In fact, it is documented that equity prices do not follow such a distribution even from Bachelier's (1900, Théorie de la speculatión) era. Nonetheless, the BS 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 BS model.

Testing the BS 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 --see Hull (2000, Futures, Options and other Derivatives). In addition, at any time, the only parameter of the BS 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.^7.3 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.

Researchers, right after the publication of BS 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 options and lower for out-of-the-money options, assuming that BS 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: It has been long observed that asset returns follow a distribution which is far from normal, in particular one that exhibits a substantial degree of excess kurtosis --Fama (1965, Journal of Business). Merton (1976, Journal of Financial Economics), among others, notes that mixtures of normal distributions exhibit fat tails relative to the normal, and therefore models that result in such distributions can be used in order to improve on the BS option pricing results.

2. CLUSTERING: ARCH and stochastic volatility models have been used in the literature to mimic volatility clustering --Engle (1982, Econometrica), Ghysels, Harvey and Renault (1996) and the references therein. Financial time series exhibit periods where the volatility is consistently low that alternate with periods of consistently high volatility. This variation of volatility can be linked to the arrivals of information, and high trading volume --see Mandelbrot and Taylor (1967, Operations Research), and Karpoff (1987, Journal of Financial and Quantitative Analysis), inter alia. One can argue that trading does not take place in a uniform fashion across time: new information will result in a more dense trading pattern with higher trading volumes, which in turn result in higher volatilities.

3. LEVERAGE EFFECTS: Black (1972, Journal of Business) suggests that volatilities and asset returns are negatively correlated, naming this phenomenon the leverage effect. Falling stock prices imply an increased leverage on firms, which is presumed by agents to entail more uncertainty, and therefore volatility. This is also referred to in the literature as the Fisher-Black effect.

4. LONG MEMORY: Although volatility seems to follow a cyclical pattern, there seems to be a very high degree of persistency, usually modeled through an IGARCH or a FIGARCH specification--see Baillie, Bollerslev and Mikkelsen (1993, Journal of Econometrics).

5. VOLATILITY SMILE: Many of the above stylized facts are visualized in the literature through the volatility smile. The features of the smile are well documented in the literature and include the following:
   1. The $\cup$-shaped relationship between the implied volatility and the moneyness level, with a minimum around-the-money, although ''smirks'' and ''frowns'' are also encountered --see for example Marsh and Kobayashi (1998, Univ of Tokyo). This is usually attributed to the fat tails of the returns of the underlying asset.

   2. The volatility smile is often symmetric, although documented asymmetries might exist due to the leverage effects which result into returns with negative skewness--see for example Heston (1993, Review of Financial Studies), or liquidity issues since the more expensive contracts --which are far in-the-money-- are documented to be the least liquid ones.

   3. The amplitude of the smile decreases with time to maturity. Short maturity options tend to exhibit more acute volatility smiles, that tend to die out for long maturity contracts.