We study the Black-Scholes equation in stochastic volatility models. In particular, we show that the option price is the unique classical solution to a parabolic differential equation with a certain boundary behaviour for vanishing values of the volatility. If the boundary is attainable, then this boundary behaviour serves as a boundary condition and guarantees uniqueness in appropriate functio...

in which CBS is the usual European call price of an option without dividends. The function φ(S0, S, td) is the log-normal no dividend density function. The integral representation given in (1) does not have an explicit solution. It can be approximated with a numerical integration method, which may be complicated due to the right side boundary at infinity. Another possibility is to solve the Bla...

In this paper, a set of functions were constructed that transforms Black-Scholes partial differential equation into weak formulations. The analytical solutions: existence, uniqueness and other estimates also obtained in form with the use boundary conditions to establish effects its financial implications Sobolev spaces. regularity problem considered which coefficients, domain are all smooth fun...