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?


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.

  • $\begingroup$ 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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