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!
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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.
