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Reconciling Wilmott’s Binomial Option Pricing Example

Article Quant Q&A · Author: ASN
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Summary

The document investigates a discrepancy in a textbook binomial pricing example for a European call. Given the stated stock price, strike, time step, interest rate, and volatility, the author cannot reproduce the book's up factor using the appendix formula and asks whether the printed value is a typo or results from another method.

One response suggests that limited precision in computing the square root could explain the mismatch, but notes that an alternative exponential up-factor formula also gives a different value. Another response identifies a different formula, involving the interest rate and volatility, that appears to reproduce the quoted up factor, with the down factor set as its reciprocal. The discussion offers competing explanations rather than a verified erratum or full derivation, so the precise source of the discrepancy remains uncertain.

Key ideas

  • The post compares a textbook's binomial up factor with the value produced by its stated formula.
  • A proposed explanation is rounding or truncation in an intermediate calculation.
  • An alternative formula for the up factor is reported as matching the book's example.
  • The reported down factor is the reciprocal of the up factor.
  • The discussion does not establish whether the textbook contains a typo.

Tags

OptionsDerivatives pricingBacktesting
Full text
# Failing to replicate Wilmott's results for binomial option pricing


# Failing to replicate Wilmott's results for binomial option pricing












I am working through Paul Wilmott introduces Quantitative Finance, 2nd ed. I am failing to reproduce one of his numerical examples and I would like to understand why.

I chapter 3, Wilmott introduces the binomial option pricing model. He gives a numerical example on p. 80 (section 3.17). The example is about pricing a European Call with Strike $K = 100$ and 4 months to expiry. The inputs are the following:

$S = 100$, $\delta t = 1/12$, $r=0.1$, $\sigma = 0.2$

Wilmott says "Using these numbers we have u= 1.0604, v=0.9431 and p'=0.5567" (p.80), where u is the up factor, v is the down factor and p' is the risk-neutral probability of the up movement.

I fail to reproduce these figures. Consider the up factor u. Wilmott says he is using the formulae given in the appendix to chapter 3, i.e. on p. 93. The formula for u given there is $u = 1 + \sigma \sqrt{\delta t} + \frac{1}{2}\sigma^2\delta t$ If I plug-in the figures, I get 1.0594, not 1.0604.

Am I missing something? Is this a typo in the book? If it is a typo, are you aware of a publicly available list of corrections (errata)?

Many thanks!

## Answer by Preston Lui (score 2)

https://quant.stackexchange.com/a/55022

I have checked the answer to my side, even using the alternative, the original formula $u=e^{\sigma \sqrt{\delta t}} $, I still get $1.05943$

I suspect it is not really a logic error, but more of a truncation error in calculating the $\sqrt{1/12}$

## Answer by Magic is in the chain (score 2)

https://quant.stackexchange.com/a/55025

Going through his code, seems like he has used this formula to come up with the figure 1.0604:

$u=\frac{1}{2}\left(e^{-r \delta t}+e^{(r+\sigma^2) \delta t}\right)+\sqrt{\frac{1}{4}\left(e^{-r \delta t}+e^{(r+\sigma^2) \delta t}\right)^2-1}$

and then $d=\frac{1}{u}$

Here is the relevant part of Wilmott’s code:

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.