Forward Skew and Volatility of Volatility in Local Volatility Models

Summary

The note describes how a local volatility surface can generate a conditional forward skew. Its suggested diagnostic is to start from a future point, simulate spot paths under the local volatility dynamics, reprice a grid of options along those paths, and infer the resulting volatility patterns. This distinguishes path-conditioned skew from simply reading the implied skew for options with future expiries on today’s surface.

Local volatility has no separate stochastic factor for volatility. Spot moves across the surface, producing path-dependent changes in local volatility that can look like volatility of volatility, but this effect is tied to spot and is not independently controlled. The answer sketches a way to imitate additional randomness by switching among local volatility surfaces, while cautioning that this does not reproduce the spot-volatility relationship of a stochastic volatility model. No quantitative comparison or implementation details are provided.

Key ideas

• Conditional forward skew can be examined by simulating future spot paths and repricing options along them.
• The local volatility surface makes the volatility experienced on a path depend on where spot moves.
• This path-dependent variation can resemble volatility of volatility, but the model has no independent volatility factor.
• Switching among multiple local volatility surfaces can imitate some volatility randomness but does not fully replicate stochastic volatility dynamics.

Tags

# Forward Skew in the Local Volatility Model


# Forward Skew in the Local Volatility Model












How does the local volatility model cause a forward skew? How is this different to the skew observed for future tenors in the vol surface?#

Also how do LV models underestimate vol of vol?

## Answer by will (score 1, accepted)

https://quant.stackexchange.com/a/39764

To see the conditional skew created by an LV surface, just start at some point in the future, and diffuse. Price all the options in a grid on the paths, imply vols, and you can observe the conditional slew created by your surface.

As for the vol of vol, there is no explicit vol of vol included in the model, instead, you end up with a pseudo random vol for each path, as it moves around the LV surface each time step (ie if there is a skew, and you move from 100 at one time point to 101 the next, it's likely the local volatility will be different). Because the move in spot is random, you'll see what appears to be a vol of vol - but you can't really control it with the model params*.

*this is more in depth, you can fake a vol of vol in LV by intersplicing multiple LV surfaces, ie if your spot ends in 0.1 you get surface 1, 0.02 surface 2, etc. Then you can make it appear there is a randomness of vol, but you can't control the spot/vol correlation like in a proper stoch vol model - you'll get more fwd convexity, but not see a fwd skew so much like in a stoch vol model.

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.