# Autocallable pricing under stochastic vs. local volatility

I am interest in the reason why an Autocallable (structured product) is cheaper under local volatility compared to stochastic volatility.

I thought this was due to the following:

• when thinking in terms of vega hedge, the seller of the Autocallable (long volatility) is selling volatility on the trade date to hedge his vega (let's say the max maturity is 5 years with yearly observations).

• If the Autocall gets early redeemed after, say, 1 year, then the seller would have to buy back the volatility he sold to close his hedge. That is, he is buying back volatility on the 1 year forward skew, maturity 4 years.

• The forward skew flattens under local volatility vs. stochastic volatility, thus the volatilities he buys back are higher under stochastic vol (thus his hedge costs more under stochastic vol hence the autocall's price is higher under stochastic vol).

I was then told this was not the exact reason. This may not be due to the skew forward exactly. However I can't seem to think of any other reason. If anyone has the answer, that would be much appreciated, thanks.

I put this as an answer because it's too long for a comment. It will probably also be quite rambly - feel free to edit and clean it up.

If the underlying is dependant on the vol in a non linear fashion, then you need to get the distribution correct in order to correctly price the product. When the LV surface is created, it is calibrated to products that do have vol of vol exposures, but the lack of the LV model's ability to accurately model these means that any difference is projected onto the local vol surface.

The reality is that the LV model is not correct, it is an approximation. If we calibrate LV to vanilla options, then we can correctly price vanila options. We can also correctly price anything that can be approximated by a linear combination of vanilla options. If you now take a different class of derivative, and try and price that, then we are effectively extrapolating. And with that, there will likely be some error. This is a projection error. If our model is appropriate for the derivatives we are pricing, then the projection error will be small (and we can consider what we are doing more an interpolation than an extrapolation). If we try to price something that depends on something we're not actually modelling (i.e. vol of vol) then we shouldn't be surprised if we have a larger error.

Another reality is that SLV is not correct either, but what it does do is give us flexibility to calibrate a dynamic for the vol of vol. This allows us to reduce the projection error when we're dealing with products that have this dependency - but we need to calibrate to products that do have this dependency, otherwise we will just get meaningless parameters (i.e. you can't calibrate a model to predict whether or not your wife is angry at you to if your only input is the phase of the moon...).

As for why autocallables price differently under LV and SLV, it's because of vol of vol yes, but this is mainly because a side effect of that is the forward vol / conditional vol dynamics. LV models give very flat forward skew, while SLV allow a fwd skew more easily. Because of this, things like options conditional on not knocking out in one year at 100% price differently under the two models.

Typical reasons why the SV price is higher than the LV price for a given product are 1-forward skew dynamics as you mentioned, 2-product volga ie convexity of premium vs implied volatility.

In the latter, the implication is that the product is sensitive to the volatility of volatility. Only the SV model features vol of vol, by definition.

Autocalls and in particular if they embed a down and in put, exhibit significant (short) vol of vol exposure for the seller.

• your second statement isn't quite true - an LV model will have a vol of vol, purely from the spot path moving randomly accross the local vol surface. That it's deterministic does not matter, what is important is that the distribution of the volatility is correct. – will Apr 28 '18 at 18:16
• Thank you both for your answers. If I understand correctly, LV as well as SV models show vol of vol, except it is deterministic in the first case. Problem is, distribution of the vol is wrong under LV. Consequence of this, in particular when it comes to Autocallables, is the modelling of vol of vol exposure (that the Autocall seller is exposed to) under LV and SV, which is a consequence of the way the forward skew dynamics is modelled under LV and SV. Is that correct? – Alex Apr 30 '18 at 8:18
• If so, is there an example to show this vol of vol exposure and impact on Autocall price in practice (like the vega hedging example showing the forward skew dynamics impact)? – Alex Apr 30 '18 at 8:23
• @alex, yes. You can calibrate local vol and slv with vol of vol fixed at X to the same options. When x=0 you should get a very similar price. As you vary x your price should change. – will Jun 28 '19 at 15:38