# Example of Black-Scholes model

The Black-Scholes model gives the dynamic price estimate V(t) of a European stock option depending on time t, the underlying asset price S(t), and the risk-free interest rate r(t). The mathematic foundation of the Black-Scholes model is a partial differential equation.

The NMPC-Graph allows to nonlinearly generalize this partial differential equation. Based on this, a NMPC-model synthesized by CYNERELO™ does not only support the regressive adaption to historic economic data, but also its numeric solution.

The multiplications, divisions, and constants in the Black-Scholes differential equation are replaced by modulators within a NMPC-Graph and this way they become generalized without giving up on the original effect of the algebraic operations.

Since CYNERELO™ basically covers only ordinary differential equations, the logic of partial differential equations – like for this model – needs to be explicit. Thus, the NMPC-Graph becomes more complex than usual and is spread over four parts interconnected with each other.

However, for the conceptual understanding the first figure is sufficient.

The derivation by S(t) in the original differential equation is modelled by small dynamic difference values in the NMPC-Graph. Doing so, the parameters S1(t) and S2(t) are designed to have values being very close to each other. This way, the values of V1(t) and V2(t) also result very close to each other in favour to the approximation of the derivation delta_V / delta_S.

The variables S2(t) und S3(t) also are meant to have close values. Thus, also the values of V2(t) and V3(t) result close to each other and support the approximation of the second derivative delta2V / delta_S2.

The predefined step constant delta_SGrid has a much smaller value than the interval between minimum and maximum value of the underlying asset price S(t).

This part of the NMPC-Graph can be simplified without significant information loss, since modulators usually are not necessary if one of their inputs has got a constant value.

This way, the forth part of the NMPC-Graph may be expressed with less components which for a NMPC-Model generated from it would allow less computing effort and more precise results applying the same data input compared to the non-simplified NMPC-Graph.