Volatility Models: Non-constant volatility models
There is plenty of evidence that returns on equities, currencies and commodities are not normally dis-tributed, they have higher peaks
and fatter tails than predicted by a normal distribution. This has been cited as evidence for non-constant volatility.
Before the market crash in 87 the Black-Scholes model provided a good description of the Vanilla market: the strike dependency of the
implied volatility was negligible. As the graphs show in today’s markets out-of-the-money (otm) put options are more expensive (in
volatility terms) than otm calls.
Figure 2: implied volatility surfaces for the key equity indexes
The volatility smile cannot be explained in the framework of normal distribution with constant volatil-ity.
Another reason for introduction of non-constant volatility models is the pricing and hedging of exotic options. Exotic options can be
hedged with vanilla options with different strikes and maturities - as-suming a continuum of strikes and maturities, and therefore they
can only be priced and hedged if the model is consistent with the price of options for all strikes and maturities. Within Black-Scholes we
can get the correct price of an option for only one strike and only one maturity. A time-dependent volatil-ity (Merton’s model) leads to the
correct price of an option for only one strike and all maturities. The need for modelling the volatility in a more general setting is clear,
and mainly three distinct types of continuous models have been developed: local volatility models, jump-diffusion models, and
stochas-tic volatility models.
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