Nonlinear Black Scholes Modelling – FDM vs FEM
Black-Scholes model
In the classical Black-Scholes model, the financial parameters, like the volatilities and correlations, are assumed to be known. These are very strong assumptions that are unrealistic in the real world. Hence, a common extension of the Black-Scholes model assumes one or more of the parameters to be random. This leads to partial differential equations that include the financial parameters as unknowns.
Another approach is the uncertain parameter model, where the financial parameters are assumed to lie between some known upper and lower bound. Solving the problem for the worst case scenario leads to a non-linear generalisation of the classical Black-Scholes equation. Several works have addressed this type of problem. One asset problems with an uncertain volatility are discussed in [ALP95, AB99, Wil98, WO98, Top01a, PFV03]. The works [Wil98, Top01a, Top01b] analyze two assets problems where the correlation is only known to lie within a given interval. However, in these cases, the volatility of both asset is always assumed to be known. Finally, in [Wil98, WO98], problems with an uncertain risk-free interest rate are addressed. In all cases, the resulting problem is a non-linear generalization of the classical Black-Scholes equation. It reduces to the standard form when the upper and lower bound for the parameter are chosen to be equal. Although several of these works state that a generalization of these different types of problems to a problem where more than one parameter is uncertain is ‘straight forward’, there seems to be no work addressing this generalization, neither analytically nor numerically.
