How is Emanuel Derman's implied tree model implied volatility skew derived?

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.

• I suppose the same answers this related question quant.stackexchange.com/q/67909/6686?
– Hans
Sep 14 '21 at 20:41
• Yes, this can answer your other question as well. Sep 14 '21 at 23:01
• 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?
– Hans
Sep 15 '21 at 4:04