Reconciling Local Volatility Smile Shifts and Upward Flattening

# Local Volatility smile dynamics


# Local Volatility smile dynamics












In The Local Volatility Surface (2008), Emanuel Derman analyzes the dynamics of the local volatility smile. He observes that, in a negatively skewed market, the evolution of the index is characterized by lower local volatility as the index level increases and higher local volatility as the index level decreases. This behavior causes the local volatility smile to shift in the opposite direction of the underlying asset's price movements. To illustrate this phenomenon, Derman includes the following image (Evolution of the smile: the smile at a variety of index levels, assuming an initial index level of 100. The line labeled 100 is the initial smile. Other lines represent the implied tree’s fair skews at different market levels six months in the future, as indicated by the corresponding labels.), which depicts how the local volatility smile systematically decreases with increasing index levels and increases with decreasing index levels:

In contrast, Hagan et al., in their seminal work Managing Smile Risk, identify an additional dynamic of the smile within the local-volatility framework. They highlight the following behavior:

> Local volatility models predict that the market smile or skew shifts in the opposite direction of the underlying asset’s price. Moreover, the implied volatility curve not only moves in the opposite direction of the underlying but also shifts upward (and flattens) regardless of whether the underlying price increases or decreases.

This behavior is illustrated in the following images:

How can these two perspectives on smile behavior—the directional movement observed by Derman and the upward shift noted by Hagan et al.—be reconciled?

## Answer by carry_and_pray (score 0)

https://quant.stackexchange.com/a/85553

Derman and Hagan are describing the same local volatility mechanism but at different levels of detail. Derman's figure is obtained by rolling the calibrated implied tree forward and then reading off the fair future smile conditional on being at a given future index level.

He states explicitly that Figure 12 shows the smile at different market levels six months in the future, computed from the corresponding future subtrees. In a negatively skewed equity surface, those subtrees have lower local vol on up nodes and higher local vol on down nodes so the future fixed-strike smile is lower after an up move and higher after a down move.

Hagan's discussion is the precisely the same phenomenon written analytically. After calibrating a local-vol model to today's smile let's call $\sigma_B^0 (K)$ he shows that to first order $\sigma_B(K, f) \approx \sigma_B^0 (K + f - f_0)$ where $f_0$ is todays forward and $f$ is the moved forward.

if $f$ rises, the smile shifts left in the strike space otherwise, if $f$ falls, it shifts right which is exactly the statement "smile moves opposite to the underlying" effect that Derman's picture shows.

Hagan also adds a next-order correction. For a smile initially curve, the new smile isn't just a pure translation but there's also a $O((f - f_0)^2)$ term so the curve is pushed upward and tends to flatten regardless of whether or not the underlying moved up or down.

Derman essentially shows the dominant first-order transport effect while Hagan is pointing out the local also implies a 2nd order lift/flattening of the smile. Derman's chart is a family of future conditional smiles at different nodes of the implied tree whereas Hagan's is a comparative-statics picture around the current smile which is why Hagan's upward shift is easier to isolate visually while Derman's mainly highlights the anti-correlation between smile and index level