Using ADI Methods for Multidimensional Black–Scholes PDEs

Summary

The document asks how to solve a three-dimensional backward Black–Scholes PDE with time- and state-dependent coefficients. Applying a Crank–Nicolson scheme across all three state variables at once creates a difficult coupled system. The response points to alternating direction implicit (ADI) finite differences, which split the multidimensional calculation into successive directional steps. Each step can be arranged as a tridiagonal linear system, making the computation more manageable than a fully coupled solve.

The document offers no derivation, comparison of ADI variants, numerical example, or stability and accuracy evidence. It also does not specify boundary conditions, grid design, or how to handle the mixed behavior and coefficients in the stated PDE. ADI is therefore a suggested approach rather than a complete implementation recipe; the choice of scheme and validation would depend on the particular model and payoff.

Key ideas

• ADI schemes split a multidimensional PDE calculation into directional steps.
• Each directional solve can retain a tridiagonal linear system.
• The suggested approach addresses the complexity of applying Crank–Nicolson across several state variables.
• The document does not provide implementation details or numerical validation.

Tags

# Numerical Solution to 3 Dimensional Backward BS PDE


# 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?

## Answer by StupidMan (score 1)

https://quant.stackexchange.com/a/55493

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.

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