# Black-Scholes Equation

## Model Data

Type: classic pde

Physics Modes: custom equation

Keywords: finance equation editing validation

This is an example showing how to define a custom PDE equation model in the FEATool Multiphysics. In this case the Black-Scholes model equation, which is used in financial analytics to model derivatives and options pricing. The non-linear partial differential equation to be solved reads

$$\frac{\partial u}{\partial t} - \frac{\partial}{\partial x}\left( \frac{1}{2}\frac{\partial u}{\partial x} \right) - \frac{\partial u}{\partial x} = -u + ( (x-t)^5 - 10(x-t)^4 - 10(x-t)^3)$$

with boundary conditions u(0,t) = -t^5 and u(x,t) = (x-t)^5 on the left and right sides of the domain, respectively. The problem is time-dependent with initial condition is u(x,t=0) = x^5. For this problem an exact analytical solution exists, u(x,t) = (x-t)^5, which can be used to verify the computed solution.

This model is available as an automated tutorial by selecting Model Examples and Tutorials… > Classic PDE > Black-Scholes Equation from the File menu. Or alternatively, follow the linked step-by-step instructions.