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To some derivatives professionals, Ito's Lemma is nothing more than a weighted averaging fix for the drift + vol binomial tree.

To me, Ito's Lemma is much, much, more.

An Ito process is a stochastic differential equation. A stochastic process is a mathematical structure used to model a dynamical system that is non-deterministic with an intrinsic randomness which makes prediction impossible. However, strong correlations may apply.

This model is based on binomial trees and Brownian Motion.

Basically, it's an aid to solve stochastic differential equations. Stochastic diff eq's don't exist in the same way that classical diff eqs exist because of their random component, which prevents the system from having a bounded measure. Which means that the idea of the mathematical 'derivative' does not exist. (Quite ironic for those in my field of work). Stock prices are stochastic in that changes in prices involves a deterministic part which is a time function and a stochastic part which depends on a random variable.

Ito's Lemma is similar to a 'chain rule' for stochastic functions, yet dissimilar in that you can't solve the stochastic differential equation by applying it-- you need to solve it using guesswork and intuition and only then can you use Ito's lemma to check to see if you have the right answer. Here's how to derive it.

  1. Start with a generalized Weiner process differential equation, which in Finance is known as a random walk asset model.
  2. Expand it into a Taylor series with regards to the change in the variable (stock price) and t (time).
  3. substitue the original equation for d(stock price).
  4. Expand out all of the quadratics
  5. Take the limit as dt goes to zero, and drop out some terms.
  6. Rearrange the terms and separate the dx terms (where x is the random variable) from the dt terms.
And there you have it.

This shows that the function of the asset price with regards to time has a lognormal probability density function.

Isn't that just beautiful?

The actual differential equation for Ito's Lemma is beyond the ASCII scope of this writeup, and is left as an exercise in derivation to the noder.

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