# help with derivation of equation 8 in Derman and Kani's binomial tree for local vol

in this paper "The Volatility Smile and Its Implied Tree" - Derman and Kani 1994 i understand the derivation of all equations up to 7. But eq 8 i cannot figure out how to derive! i have asked a quant at work who looked for 10mins and also couldnt figure it out, therefore i hope you guys can help? Note that i succeeded to derive EQ 6 which was a bit hard, but this one i can't figure out.

here is the equation: and

Let's start from (EQ 5) (introduce $$w$$ notation for wealth factor and $$C_i$$ for call price).

$$wC_i = \lambda_i (F_i -S_i)(S_{i+1}-S_i)^{-1} (S_{i+1}-s_i) +\Sigma$$

I have used (EQ 3) $$p_i = (F_i -S_i)(S_{i+1}-S_i)^{-1}$$.

This is equivalent to:

$$wC_i (S_{i+1}-S_i)= \lambda_i (F_i -S_i) (S_{i+1}-s_i) +\Sigma (S_{i+1}-S_i)$$

$$wC_iS_{i+1} -wC_iS_i=\lambda_iF_iS_{i+1}-\lambda_iF_is_i -\lambda_iS_i S_{i+1} +\lambda_iS_i s_i + \Sigma S_{i+1} - \Sigma S_i$$

$$wC_iS_{i+1} -\lambda_iF_iS_{i+1} - \Sigma S_{i+1} = -\lambda_iF_is_i -\lambda_iS_i S_{i+1} +\lambda_iS_i s_i - \Sigma S_i +wC_iS_i$$

This is further equivalent to:

$$S_{i+1}(\lambda_i F_i -wC_i +\Sigma)= \lambda_iF_is_i +\lambda_iS_i S_{i+1} -\lambda_iS_i s_i + \Sigma S_i -wC_iS_i$$

So the the denominator in (EQ 8) is correct and doesn't care about the extra relationship:

$$S_iS_{i+1} = s_i^2.$$

We now use the extra relationship to process our right hand side:

$$\lambda_iF_is_i +\lambda_iS_i S_{i+1} -\lambda_iS_i s_i + \Sigma S_i -wC_iS_i$$ $$= \lambda_iF_is_i +\lambda_is_i^2 -\lambda_iS_i s_i + \Sigma S_i -wC_iS_i$$ $$= (\lambda_iws_i^2 + \lambda_is_i^2) - (wC_iS_i + \lambda_iS_i s_i - \Sigma S_i)$$ (I used $$F_i =w s_i$$ in the last equality.)

The second parenthesis pair certainly resembles the numerator in (EQ 8), but I have no idea why subtracting it from the first parenthesis pair would produce the coveted numerator.

Any help from community (or different approach/answer) is more than welcome.

• thanks very much, you got further than i did! lets hope someone can show us how to do the numerator Aug 14 '20 at 9:45