Volatility smile
In economics, the term volatility smile (also called volatility smile , English smile means 'smile') is understood to mean that the implied volatility - this is that which, according to the Black-Scholes model, must be present in order for the current market price of an option to come about comes - the lower the more the option is “at the money”.
Frequently, especially with currency options, the implied volatility increases both with exercise prices below and above the current market price; so it has its minimum at exercise prices “at the money”. The name of the term comes from the fact that the implied volatility as a function of the exercise price results in a curve that is reminiscent of a smiling mouth.
The shape of the volatility curve depends on the respective market and the type of option. In many cases it is also observed a skewness (English skew ), in which the implied volatility increases at low strike prices and falls at higher strike prices.
While the phenomenon of the volatility smile had been observed for a long time with currency options, it only occurred for stock options after the stock market crash of 1987 .
Causes and Modeling
There are various competing explanations for the occurrence of volatility smiles, about which there is no consensus. Since the Black-Scholes model assumes constant volatility, it cannot explain the occurrence of volatility smiles.
An attempt to explain from behavioral economics is that the players following the 1987 crash of fear of another crash put options that are far out of the money, preferred, since they represent a favorable hedging against currency crashes. This explains a higher implied volatility at low exercise prices. Since this explanation suggests that the market does not price options rationally, it is rejected by proponents of the market efficiency hypothesis .
Other explanations suggest that Black-Scholes' model assumptions are overly simplistic. If the volatility is not assumed to be constant, but rather depends on the current price of the underlying asset as well as on the time, this is called local volatility . Important models for this are the discrete-time Derman-Kani model (an extension of the binomial model ) and the continuous model by Bruno Dupire .
Another approach to explaining the volatility smile is to describe volatility as a variable. Well-known models with variable volatility are the Heston model and the GARCH models .
Another extension is to replace the continuous Wiener process, which is assumed for the logarithm of the base value in the Black-Scholes model, with a stochastic process with jumps. This leads to jump diffusion models like that of Robert Carhart Merton , which can also be used to model volatility smiles.
A model extension that can explain the above-mentioned volatility skew is the inclusion of default risks in the option price model.
literature
- Jim Gatheral: The volatility surface. A practitioner's guide , Wiley (2006)
- Paul Wilmott: Paul Wilmott on quantitative finance , Wiley, 2nd edition (2006)
- Neil A. Chriss: Black Scholes and beyond . McGraw-Hill Professional (1997)
Individual evidence
- ^ Mark Rubinstein : Implied Binomial Trees. Journal of finance, 1994
- ↑ Hersh Shefrin : A behavioral approach to asset pricing . Academic Press, 2005.
- ↑ Emanuel Derman , Iraj Kani: Riding on a Smile RISK, 7 (2), 1994, pdf
- ^ Bruno Dupire: Pricing with a Smile , Risk (1994)
- ^ Robert C. Merton: Option pricing when underlying stock returns are discontinuous. Journal of financial economics, 1976
- ↑ Jim Gatheral: The volatility surface. A practitioner's guide , Wiley (2006) Chapter 6