# 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.