Why Binomial Price Steps Use Exponential Volatility Scaling

Summary

The document explains the rationale for setting the up and down multipliers in a binomial asset-price tree to exponential functions of volatility and the square root of the time step. The scaling makes the size of each log-price move proportional to volatility and to the square root of elapsed time, matching the diffusion scale of a continuous-time price model.

Its central argument is convergence: as the binomial time step shrinks, the price process is said to converge in distribution to the geometric Brownian motion underlying the Black–Scholes model. The cited terminal-price expression includes both a Brownian shock and a drift adjustment. This gives the formulas a statistical connection to continuous-time option pricing rather than treating them as arbitrary choices. The note is brief and points to an external derivation for details; it does not establish the convergence proof or discuss alternative tree parameterizations, risk-neutral probabilities, or practical calibration.

Key ideas

• The up and down factors are exponential in volatility times the square root of the time step.
• This scaling is intended to match the size of log-price changes in a diffusion process.
• As the time step approaches zero, the binomial price process is described as converging in distribution to the Black–Scholes underlying process.
• The note gives the intuition but leaves the derivation and parameterization alternatives to a cited source.

Tags

# What's the logic behind binomial model ups and downs?


# What's the logic behind binomial model ups and downs?












I want to understand what is the underlying logic in the calculation of u and d in a binomial model.

$$ u = \exp\Bigl(\sigma \sqrt{\Delta t} \Bigr), \quad d = \exp\Bigl(-\sigma \sqrt{\Delta t} \Bigr) $$

I don't know if i'm explaining myself correctly, but why are those formulas used to calculate the two possible values for an asset price given the volatility and a time step? what's the mathematical/statistical logic behind it?

## Answer by Cettt (score 6)

https://quant.stackexchange.com/a/45299

one of the most fundamental results states that the binomial model converges towards the Black Scholes model if the step size $\Delta t$ converges to zero.

The Black Scholes model is an option pricing model where the underlying is given by

$$ S_T = S_0 \cdot \exp \Bigl(\sigma W_T - \frac 12 \sigma^2 T \Bigr). $$

By choosing $$ u = \exp(\sigma \sqrt{\Delta t}), \quad d = \exp(-\sigma \sqrt{\Delta t}) $$

the price process converges in law (i.e. weak convergence) to the process $S_T$. More details can be found on page three of this document.

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.