# 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!

• I use CRR upward formula and get the same answer as you, 1.0594. Jun 18 '20 at 10:41

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 $$\sqrt{1/12}$$

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:

• Many thanks for your reply. I think Wilmott reports this piece of code on p.85. In any event, using this formula I could also replicate the 1.0604.
– ASN
Feb 5 at 16:56