avoiding this restrictive condition: in Merton [@M76] volatility was a deterministic function of time, in Cox and Ross [@CR76] was stock-dependent, Hull and White [@HW87] proposed a model where the squared volatility also follows a log-normal diffusion equation, Wiggins [@W87] considered underlying and volatility as a two-dimensional system of correlated log-normal random processes, in Scott [@S87], and also in Stein and Stein [@SS91], the instantaneous volatility follows a mean-reverting arithmetic Ornstein-Uhlenbeck process, and Heston [@H93] introduced correlation in the preceding model, just to name a few.

Another standard limitation of the Black-Scholes formula is that it is restricted to European derivatives: the option can be exercised at maturity only. However, most of the exchange-traded options are American: they can be exercised anytime during the life of the contract. Once again there is a clear connection with typical problems in physics: the earliest analysis of the issue of pricing American derivatives was formulated by McKean [@McK65] as a free boundary problem for the heat equation. Kim [@IJK90] provided an integral representation of the option price but, unfortunately, we have nowadays no explicit expression for the American counterpart of the Black-Scholes formula. The kernel of the problem is in that, in general, 