# 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.

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## 1 Answer

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.

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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