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Deriving Black–Scholes from a Discrete Binomial Tree

Article Quant Q&A · Author: ensabahnur
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Summary

This discussion outlines how a discrete binomial pricing relation can lead to the Black–Scholes partial differential equation as the time step becomes small. It starts from a one-period no-arbitrage equation for the option value, then substitutes approximations for the up and down movements and Taylor expansions for the option value at the next time step. Combining those expansions and retaining first-order time terms produces the continuous-time equation.

The post gives the approximate forms for the up and down factors and sketches the expansion for the up-state option value; it says the down-state expansion is analogous. It does not show the algebra that combines the terms or fully state the resulting equation, so it is a concise pointer rather than a complete derivation. The argument relies on the stated small-step approximations and a no-arbitrage setup; readers seeking every intermediate step will need to work through the substitutions themselves.

Key ideas

  • The one-period no-arbitrage pricing relation provides the starting point for the derivation.
  • The up and down factors are approximated through terms involving the time step and volatility.
  • Taylor expansions express next-step option values in terms of current value derivatives.
  • Combining the expansions and retaining first-order time terms yields the Black–Scholes equation.

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Derivatives pricingOptionsStatistics
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# Breakdown of Wilmott's Binomial Tree derivation of Black-Scholes equation


# Breakdown of Wilmott's Binomial Tree derivation of Black-Scholes equation












Hi guys, I tried to follow the chapter of PWIQF on binomial model and got stuck when it derived the Black-Scholes (please see image). I tried to backtrack the said equations but couldn't trace back the derivation. I tried looking at Hull's derivation and it's totally diff. since he used the binomial distribution's equation instead of the binomial tree approach of Wilmott. Can somebody be so kind and show how to go from discrete binomial tree to black-scholes? I'm familiar with taylor series but I don't know what equations applied it to along the way. Thanks in advance!

## Answer by artemidorus (score 1)

https://quant.stackexchange.com/a/60229

Earlier in the chapter, Wilmott derives the equation

$$V = \frac{V^+ - V^-}{u-v} + \frac{uV^- - vV^+}{\left(1 + r\delta t\right)\left(u - v\right)}$$

from the non-arbitrage argument $\delta\Pi = r\Pi\delta t$, where $\Pi = V - \Delta S$.

If you then use the expansions for $u$ and $v$,

$$u \approx 1 + \sigma\sqrt{\delta t} + \frac{1}{2}\sigma^2\delta t \\ v \approx 1 - \sigma\sqrt{\delta t} + \frac{1}{2}\sigma^2\delta t$$

and expand $V^+$ and $V^-$ using the Taylor Series expansions, you will obtain the Black-Scholes Equation (to first-order in $\delta t$). The expansion for $V^+$ looks like (again to first-order in $\delta t$)

$$V^+ = V\left(uS, t + \delta t\right) \approx V\left(S, t\right) + \sigma S\sqrt{\delta t} + \frac{1}{2}\sigma^2S\frac{\partial V}{\partial S}\delta t + \frac{\partial V}{\partial t}\delta t + \frac{1}{2}\sigma^2S^2\frac{\partial^2V}{\partial S^2}\delta t.$$

The expansion for $V^-$ is similar.

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