Distributing the Rate Term in ADI Option Pricing Schemes
Summary
The document asks how to handle the negative rate-times-value term in a Heston–Hull–White partial differential equation discretized with an alternating direction implicit (ADI) scheme. The paper described by the question allocates this term evenly among the operators associated with the asset price, variance, and interest-rate directions. A related thesis is cited as distributing a similar term across two operators, but the document provides no answer explaining the intended finite-difference implementation or why this allocation is used.
The material is therefore a formulation question rather than a worked method. It identifies the operator-splitting issue and points to two references that use even distribution, but supplies no derivation, computational comparison, stability evidence, or guidance on implementation. Readers would need to consult the cited paper or thesis and the relevant ADI scheme to determine how the term enters each operator and whether the split affects computational cost or numerical behavior.
Key ideas
- The Heston–Hull–White pricing PDE includes a negative interest-rate term proportional to the option value.
- The paper described in the question distributes this term evenly across three directional operators in an ADI scheme.
- A cited thesis describes an even distribution across two operators.
- The document poses the implementation and computational-cost questions but does not answer them.
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Full text
# Dealing with the ru term in an ADI Finite Difference Scheme
# Dealing with the ru term in an ADI Finite Difference Scheme
I'm trying to code up the algorithm from this paper. The paper presents an ADI algorithm for pricing options in the Heston-Hull-White model.
The starting point is the Heston-Hull-White PDE, given below:
$$\frac{\mathrm{d}u}{\mathrm{d}t} = (\text{partial derivative terms})-ru\text{.}$$
The full PDE can be found in display (1.3) of page 2 of the linked paper. The (partial derivative terms) are split up across four matrices $A_0$, $A_1$, $A_2$, and $A_3(t)$. $A_1$ deals with partial derivatives in the $s$-direction. $A_2$ deals with partial derivatives in the $v$-direction. $A_3(t)$ deals with partial derivatives in the $r$-direction. $A_0$ deals with the mixed partial derivatives.
That leaves the $-ru$ term to be dealt with. On page 6, second paragraph from the bottom, the paper says the following:
> The $ru$ term in (1.3) is distributed evenly over $A_1$, $A_2$, $A_3(t)$.
I assume the author is alluding to some technique used in ADI methods for the Black-Scholes model, because I do not know what this means.
Questions:
- Can you tell how the author intends for us to deal with the $ru$ term in the finite difference scheme?
- Why is the $ru$ term split across matrices, possibly increasing computational costs, rather than being dealt with at once?
Edit: I have found a second mention of the $ru$ term in this thesis ( https://scripties.uba.uva.nl/document/656101 ) on page 25 (page 31 of the pdf). The quote is
> The term $−rV$ is spread evenly over the operators $L_2$ and $L_3$.
with no further explanation.Shown 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.