Skip to content
All library documents

How Binomial Risk-Neutral Probabilities Converge to Black–Scholes

Article Quant Q&A · Author: Preston Lui
Read the original

Summary

The document explains a limiting relationship between risk-neutral probabilities in a finite-step binomial model and those in the Black–Scholes framework. For a given event, the risk-neutral probability generally differs between the two models when the binomial tree has a finite number of steps.

As the number of binomial steps increases and the multiplicative price process converges to geometric Brownian motion, the risk-neutral probability of the same event converges to its Black–Scholes counterpart. This gives a connection between discrete-time replication and the continuous-time model. The answer states the convergence claim without deriving it or specifying assumptions about tree parameters, event regularity, or convergence details, so it is a conceptual explanation rather than a proof.

Key ideas

  • Finite-step binomial and Black–Scholes models can assign different risk-neutral probabilities to the same event.
  • As the binomial tree is refined, its multiplicative process can converge to geometric Brownian motion.
  • Under that limit, event probabilities under the binomial risk-neutral measure converge to the Black–Scholes probabilities.
  • The explanation states the relationship but does not provide a derivation or conditions for convergence.

Tags

OptionsDerivatives pricingStatistics
Full text
# What are the relation between the risk neutral measures in binomial tree and in Black Scholes model?


# What are the relation between the risk neutral measures in binomial tree and in Black Scholes model?












I appreciate that both are the direct result of constricting a replicate portfolio using stock and bonds.

Are there deeper relationship between the two?

## Answer by Peter Carr (score 5)

https://quant.stackexchange.com/a/54389

There is a deeper relationship between the two risk-neutral measures. Take any event in the binomial model with a finite number of steps and calculate the risk-neutral probability of it. Take the same event in the Black Scholes model and calculate the risk-neutral probability of it. For most events, the two probabilities are different. Now let the number of steps in the binomial model become infinite and have the multiplicative binomial process converge to geometric Brownian motion. As a result, the risk-neutral probability in the binomial model converges to the risk-neutral probability of the same event in the Black Scholes model, no matter what the common event is. This is a deeper relationship.

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.