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?
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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.