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

  1. Do a Taylor series expansion of $$V = V(S,t)$$
  2. Do a Taylor series expansion of $$V^{+} = V(u \cdot S, t + dt) \hspace{5mm}:\hspace{5 mm} u = 1 + \sigma \cdot \sqrt{dt}$$
  3. Do a Taylor series expansion of $$V^{-} = V(d \cdot S, t + dt) \hspace{5mm}:\hspace{5 mm} d = 1 - \sigma \cdot \sqrt{dt}$$
  4. 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:

  1. The three equations given have no flesh, i.e. there is nothing to take a derivative on.
  2. 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.
  3. It is unclear which point to do the Taylor expansion around.
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Check out Approximation of CRR as Black Scholes PDE. I show the derivation in my post there

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