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Binomial Model Log Returns and the Black-Scholes Limit

Article Quant Q&A · Author: user123124
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Summary

The document explains why a binomial stock-price model can be expressed using independent, identically distributed log returns. Writing the terminal log price as the initial log price plus the sum of period log returns makes the connection to the central limit theorem direct: as the number of periods grows, the sum approaches a limiting distribution under the model’s assumptions.

This convergence helps show how the binomial model approaches geometric Brownian motion, which underlies the Black-Scholes framework. The explanation is brief and focuses on the role of the log-return variable and the limiting argument. It does not detail the assumptions or derivation, so readers should treat the convergence as conditional on the model setup rather than a general claim about real asset returns.

Key ideas

  • The variables in the setup represent independent, identically distributed log returns.
  • The terminal log stock price is the initial log price plus the sum of period log returns.
  • The central limit theorem describes the limiting behavior of the accumulated log returns under the stated assumptions.
  • With many periods, the binomial stock-price model converges toward geometric Brownian motion and the Black-Scholes model.

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StatisticsEquitiesDerivatives pricing
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# A question on the binomial model


# A question on the binomial model












I dont understand the introduction and/or idea of the variable $X$ on page $80$ in the following handout.

http://www.maths.lth.se/matstat/kurser/fmsn25masm24/ht17/Ch3.pdf

Does someone know whats the deal with it, why do we change and what are the implications of the change?

## Answer by Kevin (score 1, accepted)

https://quant.stackexchange.com/a/47551

As Alex said, the $X_i$ correspond to i.i.d. log-returns. The benefit is that $$\ln(S_T)=\ln(S_0)+\sum_{i=1}^N X_i.$$ And for sums of i.i.d. random variables, we know the limiting distribution due to the central limit theorem. Then, you can easily show that the stock price in the binomal model converges to a geometric Brownian motion (and hence, you end up in the Black-Scholes model).

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.