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

Question

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?

Answer 1

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.