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Deriving the Black–Scholes PDE from a Binomial Model

Article Quant Q&A · Author: A.L. Verminburger
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Summary

The document outlines a proposed route from a one-step binomial option-pricing model to the Black–Scholes equation. It identifies the value function at the current stock price and time, then at up and down moves, and suggests expanding those values with Taylor series before substituting them into the discounted risk-neutral pricing relation.

The note gives the up and down factors and a risk-neutral probability involving volatility and the risk-free rate, but it does not carry out the expansions or derive the resulting differential equation. Instead, it points readers to a separate derivation of the Cox–Ross–Rubinstein approximation to the Black–Scholes PDE. It is therefore a signpost to the method rather than a self-contained explanation. The source does not state assumptions about the option payoff, boundary conditions, or convergence, so those details must be obtained from the referenced derivation or another text.

Key ideas

  • Taylor expansions of the option value at up and down stock moves can be substituted into a binomial pricing equation.
  • The suggested risk-neutral probability depends on the risk-free rate, volatility, and time step.
  • Taking a continuous-time limit of a binomial model can lead to the Black–Scholes partial differential equation.
  • The document refers readers elsewhere for the actual derivation and leaves its details unstated.

Tags

OptionsDerivatives pricingVolatilityStatistics
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# How to derive Balck Scholes from the Binomial Model?


# How to derive Balck Scholes from the Binomial Model?












The book gives the following recipe, but no further details:

- Do a Taylor series expansion of $$V = V(S,t)$$

- Do a Taylor series expansion of $$V^{+} = V(u \cdot S, t + dt) \hspace{5mm}:\hspace{5 mm} u = 1 + \sigma \cdot \sqrt{dt}$$

- Do a Taylor series expansion of $$V^{-} = V(d \cdot S, t + dt) \hspace{5mm}:\hspace{5 mm} d = 1 - \sigma \cdot \sqrt{dt}$$

- Stick the three expansions into: $$V = \frac{1}{k} \cdot \bigg(p' \cdot V^{+} + (1-p') \cdot V^{-}\bigg) \hspace{5mm}:\hspace{5 mm} k \text{ is some discounting factor, } p'=\frac{1}{2} + \frac{r \cdot \sqrt{dt}}{2 \cdot \sigma} \text{ (r -- risk free rate)}$$.

However a number of things are impeding me doing the Taylor expansion:

- The three equations given have no flesh, i.e. there is nothing to take a derivative on.

- Even if we assume that $V()$ is the same as the one where we will be plugging back our expansions it is not clear to me where exactly $S$ and $t$ inputs go.

- It is unclear which point to do the Taylor expansion around.

## Answer by ZRH (score 1)

https://quant.stackexchange.com/a/48885

Check out Approximation of CRR as Black Scholes PDE. I show the derivation in my post there

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.