Numerical Solution to 3 Dimensional Backward BS PDE

Question

I have a three dimensional backward BS PDE.

$$ \frac{\partial V}{\partial t} + a(t) S \frac{\partial V}{\partial S} + \frac{1}{2} \sigma(t, S)^2 \frac{\partial^2 V}{\partial S^2} + b(t, M) \frac{\partial V}{\partial M} + c \frac{\partial V}{\partial \phi} - rV = 0$$

with the terminal condition $V(T, S, M, \phi) = g(S, M, \phi)$

If I try to apply a Crack Nicholson method

$$2 \frac{df}{dx} = \frac{f(t+1, x_{i+1}) - f(t+1, x_i)}{\delta x} + \frac{f(t, x_{i+1}) - f(t, x_{i})}{\delta x}$$

to $S$, $M$ and $\phi$ the equation gets too complicated to solve.

So, should I only apply this to the $S$ variable and approximate the derivatives for $M$ and $\phi$ only usingw backward time values? How would you approach this?

Answer 1

You may want to have a look on Alternating Direction Implicit for solving multi-dimension PDE on finite difference method. The linear system will still be tridiagonal matrix.