How to explain the path dependency in binomial tree model to price options?

Question

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 1

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.