$7\mathrm{.}2\mathrm{.}2$ Modeling volatility as a risk factor

Contradicting the Black$-$Scholes model, empirical tests show that $$ is not constant. One way

to adapt the model is to introduce a local volatility function $(T,t,S,E),$ as described in

$204$ Financial Engineering with Finite Elements

Table $7\mathrm{.}10$ Results for a capped call on a basket with gradient boundary conditions