Deriving the Linear Implied Volatility Skew from Local Volatility
Summary
The document explains a linear approximation for implied volatility in Emanuel Derman’s sticky implied tree model. The answer starts with local volatility modeled as a linear function of the underlying price, with slope determined by a parameter describing how volatility changes as the underlying moves away from its initial level. It then approximates an option’s implied volatility by the average local volatility between the current underlying level and the strike.
For a linear local-volatility function, this average equals the average of its endpoint values and simplifies to a linear expression in the strike and underlying. This yields the stated skew dependence on their combined displacement from the initial underlying. The explanation is an approximation, attributed to Derman’s treatment, and provides no numerical calibration, error analysis, or evidence about how accurate it is beyond the linear setting.
Key ideas
- The derivation assumes local volatility varies linearly with the underlying price.
- Implied volatility is approximated by averaging local volatility between spot and strike.
- For a linear function, the interval average equals the mean of the endpoint values.
- The resulting implied volatility varies linearly with the strike and underlying levels.
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# How is Emanuel Derman's implied tree model implied volatility skew derived?
# 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?
## Answer by Gordon (score 4, accepted)
https://quant.stackexchange.com/a/67925
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.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.