Modelling dependence in finance using copulas - Thierry Roncalli's ...

4.1 Coherent valuation of multi-asset options
The Black-Scholes model
In the BS model, we have
dS (t) = rS (t) dt + σS (t) dW (t)
under Q. The price of an European option is then a function of the
volatility σ. However, when we compute the implied volatility from
the option prices for different values of the strike K, it is not
constant. This is the volatility smile effect.
Option models in banks Banks have then developped
sophisticated models (e.g. stochastic volatility models) to take into
account the smile effect.
To this day, therefore, the BS model continues to be used, out of
analytical and computational convenience, for contingent claims
based on different assets.
Modelling dependence in finance using copulas
New pricing methods with copulas 4-2