Until now, all the dynamics we use to price assets have been characterized by a Brownian term in the form of
The volatility array was assumed to be deterministic, referring to one particular asset, such as one stock, or one forward rate, and one maturity.
Unfortunately, this Brownian noise does not reflect the way assets prices actually react to broad upward or downward moves. For the needs of a more reliable risk assessment, this basic model had to be enhanced or, at least, amended.
At the inception of the options industry, a smile effect was added to the volatility term structure in the form of positive or negative spreads according to the type of securities. For instance, regarding put equity options, out-of-the-money implied volatilities were higher than at-the-money ones. As a matter of fact, volatility curves became volatility surfaces. The convexity of these surfaces determined the smile effect.
Some authors have established that market prices of European-style options determined one unique diffusion process. Dupire (1994) stated the following relationship mapping instantaneous volatilities to option prices:
A further step is taken when considering the volatility as randomly distributed. As a result, some additional dynamics related to the ...