Why Binomial Trees Usually Do Not Track Option Price Paths
Summary
The document clarifies a misconception about path dependence in binomial option pricing. Backward induction through time does not itself mean the model represents the full sequence of market states along a path. A tree compresses information into states that preserve what is needed to calculate conditional expectations, and may not retain the history needed to price path-dependent payoffs.
Monte Carlo simulation instead generates market scenarios over time, making it suitable for instruments whose payoff depends on the route taken, though it can be more computationally demanding. The answer also notes that a stochastic tree can be used to simulate paths consistent with its transitions. The discussion is conceptual and does not give a specific pricing example or compare computational costs quantitatively; whether a tree can price a given payoff depends on whether its states preserve the relevant information.
Key ideas
- Backward induction in a binomial tree does not mean the model explicitly tracks complete market paths.
- Tree states must preserve sufficient information to compute conditional expectations.
- Path-dependent payoffs may require richer states or explicit scenario paths.
- Monte Carlo can generate market scenarios for path-dependent pricing but may require more computation.
- A stochastic tree can also support simulations consistent with its transitions.
Tags
Full text
# How to explain the path dependency in binomial tree model to price options?
# How to explain the path dependency in binomial tree model to price options?
I'm new to quantitative finance, so I'm confused with the so-called path dependency in binomial tree model.
Originally I thought the path dependency exists because in binomial tree model, we will price the options in a back-propagating fashion. We evaluate the price at time T, and then go backwards to calculate the price at time T-1. The dependency is from $P_{T - 1}$ to $P_{T}$. When I went over the Monte Carol Simulation approach to price options, I found that in Monte Carol Model, the price of option at time T depends on that at time T-1. The difference is that this dependency is from $P_T$ to $P_{T-1}$. However, per my understanding, we cannot say Monte Carol Simulation is a path dependent algorithm.
So how could I understand/explain the path dependency in binomial tree model when pricing options?
## Answer by Michaël Le Barbier (score 5, accepted)
https://quant.stackexchange.com/a/9448
In binomial tree models, there is no such a thing as a path. The binomial tree represents information about the distribution of the zero-curve at a given time and preserve enough information between different times to let you compute conditional expectations. Generally, you can not price path-dependant instruments in a model based on trees—because there is no path. In some sense, Monte-Carlo simulation are much more naive, because they produce market scenarios for the zero-curve. They are suited for the pricing of path-dependant instruments but are computationally more intensive than tree processes.
If you are keen on this question, you can try to express the $\sigma$-algebra filtration with respect to which the price process $V$ of a derivative has to be measurable in order to be priced correctly by a tree model. (In the case of a Monte-Carlo simulation, this filtration is the filtration associated to the Brownian motion driving the simulation, so that there is no real limitation.)
If you interpret your tree as a finite automaton with stochastic transitions, you can use it to run path-simulations compatible with the tree.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.