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: eq8



1 Answer 1


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.

  • 1
    $\begingroup$ thanks very much, you got further than i did! lets hope someone can show us how to do the numerator $\endgroup$
    – Randor
    Aug 14, 2020 at 9:45

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