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I am reading Emanuel Derman's paper Patterns of Volatility Change. The section, Implied Volatility In The Sticky Implied Tree Model has the linear skew approximation near the old underlying $S_0$ $$\Sigma(S,K,t)=\Sigma_0-b(K+S-2S_0)$$

A related passage is

In the linear approximation of the local volatility model you can write $\Sigma=f(S+K)$ with $\Sigma$ a function of $S+K$.

I am wondering how these are derived from the implied volatility tree model which I think is the tree version of the local volatility model. Can someone please shed light on this question?

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This was also discussed in Derman's book The Volatility Smile (see Chapter 16). Specifically, he approximated the local volatility by a linear function of the form \begin{align*} \sigma(S) = \sigma_0 -2 b(S-S_0), \end{align*} and then approximated the implied volatility $\Sigma(S, K)$ for an option with strike $K$ by the average of $\sigma(S)$ between S and K. That is, \begin{align*} \Sigma(S, K) &\approx \frac{1}{2}\big(\sigma(S) + \sigma(K) \big)\\ &=\sigma_0 -b(K+S-2S_0). \end{align*} This can also be treated as \begin{align*} \Sigma(S, K) &\approx \frac{1}{K-S}\int_S^K\sigma(S')dS'\\ &= \sigma_0 -b(K+S-2S_0). \end{align*} See Formula (14.16) in the above book.

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  • $\begingroup$ I suppose the same answers this related question quant.stackexchange.com/q/67909/6686? $\endgroup$
    – Hans
    Sep 14 at 20:41
  • $\begingroup$ Yes, this can answer your other question as well. $\endgroup$
    – Gordon
    Sep 14 at 23:01
  • $\begingroup$ But why does $\Sigma(S, K) \approx \frac{1}{2}\big(\sigma(S) + \sigma(K) \big)$? The equation that can closest explain this is Equation (15.63) $$\frac{\ln\frac KS}{\Sigma(\frac KS)}=\int_0^{\ln\frac KS} \frac 1{\sigma(x)} dx$$ in that book. But I fail to see a reasonable path to the desired arithmetic mean. The integration is from $0$ to $ln\frac KS$ rather than, say, from $\ln S$ to $\ln K$. Would you be able to shed some light? $\endgroup$
    – Hans
    Sep 15 at 4:04
  • $\begingroup$ Please see also Chapter 14 of his book. $\endgroup$
    – Gordon
    Sep 15 at 13:16
  • $\begingroup$ There seem to be several averages between the implied volatility and the local volatility: 1) mean of the squares 2) arithmetic mean 3) harmonic mean. When the variation is slow, all three are pretty close. Is that the rationale? $\endgroup$
    – Hans
    Sep 15 at 14:05

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