Volatility smile

Lower volatility smile (English smile - smile, mutatis mutandis: " smiling volatility" ) is understood in the economics of the context that the implied volatility - this is the one that must be based on the Black -Scholes model to the current market price an option comes about - is lower, the more the option " in the money " is.

Frequently, particularly in currency options, the implied volatility increases both Strikes (exercise prices) below and above the current market price; so it has its minimum at Strikes " in the money ". The name of the term comes from the fact that this correlation, plotted in a graph, is reminiscent of a smile.

The shape of the volatility curve is dependent on the market and the type of option. In many cases, it is also observed a so-called skew (English skew - skew ), in which the implied volatility increases at low and higher Strikes Strikes falls.

While the phenomenon of volatility smile was observed in currency options for some time, it came for stock options only after the stock market crash of 1987.

Causes and modeling

For the occurrence of volatility smiles, there are several competing explanations about the absence of consensus. Since the Black- Scholes model assumes constant volatility, it can not explain the occurrence of volatility smiles.

An explanation of behavioral economics is that the market participants in 1987 from fear of another crash put options that are after the crash far out of the money, estimated in increased amounts, since they represent a convenient hedge against price crashes. This explains a higher implied volatility at low strikes. As this statement suggests that the market priced options are not rational, it will be rejected by representatives of the efficient market hypothesis.

Other explanations suggest that the Black-Scholes model assumptions are too simplistic. If volatility is not assumed to be constant, but depends on the current price of the underlying asset and by the time we speak of local volatility. Important models for this are the discrete-time Derman - Kani - model (an extension of the binomial model ) and the continuous model of Bruno Dupire.

Another approach to the explanation of the volatility smile is to describe the volatility as stochastic variable. Known models with stochastic volatility are the Heston model, the CEV model or GARCH models.

Another extension is to replace the continuous Wiener process, which is assumed for the logarithm of the underlying in the Black- Scholes model, having jumps by a stochastic process. This leads to Jump diffusion models such as the Robert Carhart Merton, which can be used also for the modeling of Volatility Smiles.

An extension of the model that can explain volatility skew is the inclusion of credit risk in the option pricing model.