When Stochastic Differential Equations Resist Recombining Trees
Summary
The note considers whether a general stochastic process can be represented by a binomial tree, focusing on stochastic differential equations whose drift and volatility depend on state or time. It explains that a tree built directly for such a process will generally fail to recombine: different paths can reach distinct states even after the same number of steps. Stochastic volatility is cited as an example, with a related multinomial-tree approach as a reference.
A second answer invokes the time-change representation of continuous local martingales as Brownian motion, which in principle permits a standard tree after changing the time variable. Finding the needed transformation may be difficult or impossible in practice. Interpolating nodes onto a recombining grid is suggested as a possible workaround, but the note observes that practitioners may instead use PDE solvers. It provides conceptual guidance rather than a general construction or comparative evidence about accuracy.
Key ideas
- State- or time-dependent drift and volatility generally prevent a directly constructed binomial tree from recombining.
- Stochastic volatility is an example where a multinomial recombining-tree method has been studied.
- A continuous local martingale can be represented as time-changed Brownian motion, but the required time change may be hard to determine.
- Node interpolation may impose recombination, while PDE solvers are another practical approach.
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Full text
# Binomial representation of stochastic processes # Binomial representation of stochastic processes It is common knowledge that a random walk can be represented in the form of a binomial process. Is it possible to represent any generic stochastic process (including non-linear) of the form $dX=adt+bdW(t)$ (where $W(t)$ is a Wiener process) as a binomial process? Are there any papers which have dealt with this topic? Thanks. ## Answer by lehalle (score 4, accepted) https://quant.stackexchange.com/a/18222 For a martingale $dX=a(X,t)\,dt+b(X,t) dW(t)$ where $a$ and $b$ are not constant, your tree will not recombine in general [edit]. This is the main issue. See for instance: Florescu, I. and F. G. Viens (2008, March). Stochastic volatility: Option pricing using a multinomial recombining tree. Applied Mathematical Finance 15 (2), 151-181. It deals with the case of stochastic volatility, i.e. $dS=aS\,dt+b(Y_t)S dW(t)$ $dY=\alpha(\nu − Y_t)dt + \psi(Y_t)dZ(t)$ ## Answer by Brian B (score 1) https://quant.stackexchange.com/a/18229 Yes, this is trivially true once you know that every continuous local martingale is a time-changed brownian motion. Therefore, if you change your time variable $t$ in $dX=a\,dt+b\,dW(t)$ to the right $t^\prime$ you can get a standard tree representation. Now, the correct time change may be difficult or impossible to figure out, so this theorem is of limited use. One then becomes tempted to try making a tree adapted to the original SDE. As @lehalle notes, you are then likely to end up with a tree that does not recombine. You may be able to overcome that flaw by interpolating nodes back onto a recombined tree and applying some reasonable boundary conditions, but at that point most people just go with PDE solvers instead.
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