Demonstration of Ito's correction term/lemma in binomial tree

I am preparing an undergraduate QuantFinance lecture. I want to demonstrate the ideas of Ito's correction term and Ito's lemma in the most accessible manner.

My idea is to take the "working horse" of Quantitative Finance, the binomial model and demonstrate both concepts there. Unfortunately I haven't found any references and am encountering unanticipated difficulties myself in combining both views.

When these concepts can be found in the continuous version they must be hiding in the discrete version too - can anybody please demonstrate them this way or give some reference.

EDIT
I found the following demonstration of a skewed Galton board which results in a lognormal distribution here: It is described in this article too (p. 343): http://stat.ethz.ch/~stahel/lognormal/bioscience.pdf

I think that - if anywhere - Ito's lemma/correction term must hide here. But this has to be made exact!

• @Mark Joshi: Would you like to take a stab at it? I would love to see your answer here! – vonjd Feb 27 '15 at 6:35
• hmm, you might find my paper ssrn.com/abstract=928186 helpful. I look at how the trees converge and see where some of the terms come from. – Mark Joshi May 22 '15 at 1:08

I doubt you can do this. Correction term appears in Ito because Brownian motion has infinite variation (non zero quadratic variation). In discrete and therefore finite models you cannot observe this phenomenon.

• Thank you. Yet I don't think this is the full truth. You can at least pinpoint analog effects in numerical simulations (which are per definition discrete). Have e.g. a look here: people.maths.ox.ac.uk/richardsonm/SDEs.pdf p. 6-7 and 21. I adapted the code to R, but I don't know how to transfer this onto binomial models. – vonjd Jan 25 '12 at 16:27
• You can run simulation for binomial model. And as there is no correction term in the definition of stochastic integral in discrete time, you will not need it in numerical simulation. – Alexey Kalmykov Jan 25 '12 at 17:35
• But the more you get to the limiting case of your binomial tree (which is the continuous case) it must show up somewhere - as it shows up in the simulation in the paper the more randomness you include in your process (increasing $\sigma$). It shows up in the sense that it is different compared to the non-random case and in the choice of the endpoint (left=Ito) of the intervals. – vonjd Jan 25 '12 at 18:16

One way to start thinking about this is to work out a couple of

Discrete versions of Ito's lemma

1. Øksendal (6th edition) Example 3.1.9: almost surely, $$B_t^2 - t = \int_0^t 2B_s dB_s$$

This has a discrete version which holds everywhere: let $X_n=\pm 1$ and $S_n=\sum_{i=1}^n X_i$, then $$S^2_n-n = 2\sum_{i=0}^{n-1} S_i X_{i+1}$$ To verify just note that both sides increase by $2S_{n-1}X_n$ when going from $n-1$ to $n$.

1. Øksendal's exercise 4.2: $$B_t^3 = \int_0^t 3B_s ds + \int_0^t 3B_s^2 dB_s$$

Here the discrete version is not a perfect analogue: $$S_n^3 - S_n = 3\sum_{i=0}^{n-1} (S_i + S_i^2 X_{i+1})$$ The extra term $S_n$ seems related to the fact that $(dB_t)^3 = 0$.

Actually it is quite simple to demonstrate Ito's correction term in a binomial tree.

Details can be found in my new paper (p. 8-10):
von Jouanne-Diedrich, Holger: Ito, Stratonovich and Friends (April 21, 2017)

Abstract
This exposition should provide you with the bigger picture of stochastic calculus, especially stochastic integrals. It heuristically and pedagogically develops key concepts and intuitions of one of the most important fields of applied mathematics today, namely quantitative finance. It demystifies ideas that a normally either too starkly dumbed down or hidden under highly technical details, so this text tries to fill a missing link in the literature where there seems to be no middle ground as of today. Additionally, the paper gives two results which cannot (to the best of my knowledge) readily be found in the classical literature: an illustration of the Ito correction term within binomial trees and a Taylor expansion for the Stratonovich integral.

Here I only give a summary of the general idea:

We start with a simple binomial tree with $n$ steps and $p=\frac{1}{2}$. Then we transform this tree with a convex function, e.g. with the quadratic function.

After that we compare the expected value of this transformed tree with the square of the expected value of the original tree - the difference is Ito's correction term.

All of this leads to a well known identity: $$\mathbb{E}[X^2]=\mathbb{E}[X]^2+Var[X]$$

So in this case the variance can be interpreted as Ito's correction term - a nice correspondence to the well known $\frac{1}{2}\sigma^2$-term in the mean of the log-normal distribution.