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How Binomial Option Prices Converge to Black–Scholes

Article Quant Q&A · Author: Andromeda
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Summary

The discussion asks whether the Black–Scholes option value is the limiting value of a binomial tree as the number of time steps grows, and whether implied volatility can be used in that convergence. The replies point to a study comparing the models and report that binomial prices converge toward Black–Scholes prices as the tree is refined. A further explanation connects the result to the convergence of a binomial distribution toward a normal distribution.

This provides an intuitive relationship between a discrete, stepwise pricing model and the continuous-time Black–Scholes framework. The material is brief and does not spell out the tree’s assumptions, parameter choices, or convergence rate, nor does it work through a numerical example. It also raises the distinction between European and American exercise, but the answers do not resolve early-exercise behavior or establish that the same convergence claim applies unchanged to every option specification.

Key ideas

  • A binomial tree with increasingly many periods can converge to the Black–Scholes price under suitable model assumptions.
  • The convergence has an intuitive connection to binomial distributions approaching a normal distribution.
  • The discussion cites a comparative study but does not detail its assumptions or convergence rate.
  • American exercise and early-exercise valuation are raised but not addressed in depth.

Tags

OptionsDerivatives pricingVolatilityStatistics
Full text
# Black Scholes vs Binomial Model


# Black Scholes vs Binomial Model












I'm trying to confirm my understanding of the 2 models. It is my understanding that the black-scholes is a special case of a binomial model with infinite steps.

Does this mean that if I were to start with a Binomial model with 1 step and increase steps towards infinity I would approach the same value concluded by the black-scholes?

If so does this mean I could use the implied volatility from Black-scholes formula derived from the market price of an option with the rest of the values (r, t, K, S, σ(IV) ) and approach the same market price from the black-scholes as # of steps approaches infinity? Would this only be the case for a European call with more disagreement on the value of American options with early exercise?

Thanks!

## Answer by Andromeda (score 7, accepted)

https://quant.stackexchange.com/a/11010

As anticlimactic as this may be, I'm going to answer my own question here..

I found this article that shows the connections between the two models..

http://epublications.bond.edu.au/cgi/viewcontent.cgi?article=1126&context=ejsie (mirror) that shows

that prices do converge as N Periods increases. Also they provide all the Excel formulas to recreate their work if anyones interested.

## Answer by user3264325 (score 2)

https://quant.stackexchange.com/a/12598

FYI, the binomial distribution converges to normal as n goes to infinity, which is a nice way of thinking about the relationship between BS and the binomial tree models.

see here

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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