Exploring Alternative Distributions in Ito's Lemma for Financial Engineering

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SUMMARY

This discussion focuses on the derivation of Ito's Lemma as presented in Hull's "Options, Futures, and Other Derivatives" (2008 edition). The primary inquiry is whether the lemma's assumptions hold true if epsilon is a discrete random variable taking values of +1 or -1 with equal probability, rather than being normally distributed. Participants agree that this alternative approach simplifies the derivation process, suggesting that the variance of the distribution could be linked to delta T, while also considering the implications of volatility in financial modeling.

PREREQUISITES
  • Understanding of Ito's Lemma in stochastic calculus
  • Familiarity with Hull's "Options, Futures, and Other Derivatives" (2008 edition)
  • Knowledge of Wiener processes and their properties
  • Basic concepts of probability distributions and variance
NEXT STEPS
  • Research the implications of discrete random variables in stochastic calculus
  • Study the relationship between volatility and variance in financial models
  • Explore advanced topics in financial engineering related to Ito's Lemma
  • Learn about alternative distributions in financial modeling
USEFUL FOR

Financial engineers, quantitative analysts, and students of financial mathematics seeking to deepen their understanding of Ito's Lemma and its applications in derivative pricing.

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For any financial engineers here who understand Hull's derivative pricing book. I've gone through chapter 12 (Wiener Processes and Ito's Lemma, 2008 edition).

The derivation of Ito's lemma assumes epsilon is normally distributed with a mean of zero and a variance of 1. I have a hard time filling in steps with this assumption.

Would the derivation also work if epsilon was a random variable that could take on only values of +1 or -1 with 50-50 probability? It would make the filling in steps much easier.
 
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I think the variance of the distribution actually delta T. What is 1 is the volatility.

I don't have Hull in front of me, but what you say makes sense. Imagine mini-epsilon that could take a value of +mini-epsilon or -mini-epsilon. Now imagine that you iterate that a large number of steps so that you get epsilon. At that point, what you get is a normal distribution with variance of epsilon.
 
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