Deriving Equation 8 in Derman and Kani’s Implied Tree
Summary
The document works through an algebraic attempt to derive equation 8 in Derman and Kani’s implied binomial tree for local volatility. Starting from an earlier call-price relation, the answer substitutes the risk-neutral probability, rearranges terms to isolate the next tree node, and checks the resulting denominator against equation 8. It then uses the tree’s geometric relation between adjacent prices and the forward-price relation to simplify the numerator.
The derivation identifies why the denominator has its stated form and narrows the numerator to two grouped terms. However, the answer does not resolve how those terms combine into the numerator in equation 8; the author explicitly leaves that step unanswered. Thus, this is a partial algebraic analysis useful for locating the sticking point, not a complete proof. It assumes the equations and notation from the cited paper, which are not reproduced in full, so readers need that context to verify the manipulations.
Key ideas
- Rearranging the earlier call-price equation isolates the next node in the implied tree.
- The risk-neutral probability substitution leads to the stated denominator in equation 8.
- The relation between adjacent tree prices simplifies terms in the numerator.
- The proposed manipulation does not complete the derivation of equation 8.
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# help with derivation of equation 8 in Derman and Kani's binomial tree for local vol
# 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
## Answer by ir7 (score 2)
https://quant.stackexchange.com/a/57273
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.