# Numerical Solution to 3 Dimensional Backward BS PDE

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?