Splitting the Three-Asset Black-Scholes Operator
Summary
The document compares two ways of decomposing a three-asset Black-Scholes partial differential operator for finite difference methods. Both decompositions allocate the drift, diffusion, correlation cross-derivative, and discounting terms across component operators, but they distribute the mixed derivatives differently. In one formulation, each pairwise correlation term is split between the operators associated with its two assets; in the other, all mixed derivative terms are divided among all three operators.
The source poses the question of whether these decompositions are equivalent or which should be used, but provides no answer, derivation, convergence analysis, or numerical comparison. Their sum appears intended to recover the full PDE operator, so practical choice depends on the specific operator splitting scheme and how each decomposition affects its discretization and stability. The document is therefore useful mainly as a formulation question; it does not establish that either decomposition is generally preferable.
Key ideas
- Operator splitting decomposes a multidimensional Black-Scholes PDE into component operators for numerical solution.
- The two formulations distribute pairwise correlation cross-derivative terms differently across those operators.
- Both formulations also divide the discount term across components while assigning each asset’s drift and variance terms.
- The document does not derive equivalence or compare numerical accuracy or stability.
- The appropriate decomposition depends on the chosen splitting scheme and its discretization.
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# Operator splitting method on three assets black scholes equation
# Operator splitting method on three assets black scholes equation
Currently I am studying finite difference method on derivatives with three (or more) underlyings and little bit confused on operator splitting method because two papers have different result.
For the follwing differential equation, $$ u_\tau(x,y,z,\tau) = rxu_x+ryu_y+rzu_z+\frac{1}{2}\sigma_x^2x^2u_{xx}+\frac{1}{2}\sigma_y^2y^2u_{yy}+\frac{1}{2}\sigma_z^2z^2u_{zz}+\rho_{xy}\sigma_x\sigma_yxyu_{xy}+\rho\sigma_y\sigma_zyzu_{yz}+\rho_{zx}\sigma_z\sigma_xu_{zx}-ru $$
First paper, Comparision of numerical schemes on multi-dimensional black-scholes equations ended up with
$$ \frac{\partial u}{\partial \tau}=L_xu+L_yu+L_zu $$ where $$ L_xu = \frac{1}{2}\sigma_x^2x^2u_{xx}+rxu_x+\frac{1}{2}\rho_{xy}\sigma_x\sigma_yxyu_{xy}+\frac{1}{2}\rho_{xz}\sigma_z\sigma_xu_{xz}-\frac{1}{3}ru \\ L_yu = \frac{1}{2}\sigma_y^2y^2u_{yy}+ryu_y+\frac{1}{2}\rho_{yx}\sigma_y\sigma_xyxu_{yx}+\frac{1}{2}\rho_{yz}\sigma_y\sigma_zu_{yz}-\frac{1}{3}ru \\ L_zu = \frac{1}{2}\sigma_z^2z^2u_{zz}+rzu_z+\frac{1}{2}\rho_{zx}\sigma_z\sigma_xzxu_{zx}+\frac{1}{2}\rho_{zy}\sigma_z\sigma_yu_{zy}-\frac{1}{3}ru $$ (Original paper took derivative wrt $t$ and I changed to $\tau = T-t$ here)
and the second paper, A practical finite difference method for the three-dimensional black-scholes equation defined them as $$ L_xu = \frac{1}{2}\sigma_x^2x^2u_{xx}+rxu_x+\frac{1}{3}\rho_{xy}\sigma_x\sigma_yxyu_{xy}+\frac{1}{3}\rho_{yz}\sigma_y\sigma_zyzu_{yz}+\frac{1}{3}\rho_{xz}\sigma_x\sigma_zxzu_{xz}-\frac{1}{3}ru \\ L_yu = \frac{1}{2}\sigma_y^2y^2u_{yy}+ryu_y+\frac{1}{3}\rho_{yx}\sigma_y\sigma_xyxu_{yx}+\frac{1}{3}\rho_{yz}\sigma_y\sigma_zyzu_{yz}+\frac{1}{3}\rho_{xz}\sigma_x\sigma_zxzu_{xz}-\frac{1}{3}ru \\ L_zu = \frac{1}{2}\sigma_z^2z^2u_{zz}+rzu_z+\frac{1}{3}\rho_{xy}\sigma_x\sigma_yxyu_{xy}+\frac{1}{3}\rho_{zy}\sigma_z\sigma_yzyu_{zy}+\frac{1}{3}\rho_{zx}\sigma_z\sigma_xzxu_{zx}-\frac{1}{3}ru \\ $$
So I wonder which derivation I should use or they are just indifferentShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.