Why Autocallable Prices Differ Under Local and Stochastic Volatility
Article Quant Q&A · Author: Alex
Summary
The discussion explains why an autocallable can receive different prices under local volatility (LV) and stochastic or stochastic-local volatility (SV/SLV) models, even when the models are calibrated to vanilla options. LV can reproduce vanilla prices but may not capture volatility-of-volatility exposures that matter for path-dependent products. When that missing dynamic affects the payoff, the LV calibration projects the mismatch onto its surface, creating pricing error for products outside the calibration set.
The answers point to forward or conditional volatility dynamics, including skew evolution, and to the product’s sensitivity to volatility convexity (volga) as relevant mechanisms. Autocallables, especially those with knock-out conditions or embedded down-and-in puts, can expose these differences. SLV adds flexibility to represent volatility-of-volatility, but requires suitable calibration instruments. The explanations are qualitative: the size and even practical relevance of the difference depend on the payoff, calibration, implementation, and market data; one answer notes that a vanilla-style autocall can sometimes be close to market under LV after correlation adjustments.
Key ideas
- Local volatility calibrated to vanillas can still misprice products whose payoffs depend on volatility dynamics beyond those vanillas.
- Volatility-of-volatility can affect autocallable prices through both forward skew behavior and premium convexity.
- Stochastic-local volatility provides a way to model volatility-of-volatility, but it needs relevant calibration data.
- Knock-out conditions and embedded down-and-in puts can make autocallables particularly sensitive to these modeling choices.
- Price differences depend on the product and implementation, so the discussion does not establish a universal premium gap.
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# Autocallable pricing under stochastic vs. local volatility # 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. ## Answer by will (score 4) https://quant.stackexchange.com/a/39484 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. ## Answer by Ivan (score 1) https://quant.stackexchange.com/a/39474 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. ## Answer by AKdemy (score 1) https://quant.stackexchange.com/a/63564 Risk.net: Calling out autocallable pricing offers an intuitive explanation of modelling issues with autocallables with baskets. Since the vast majority of AC are based on baskets, this is very relevant. LV is simply insufficient. I like this tweet, it's funny and accurate at the same time. The same guy has a rant that explains issues with AC pricing quite well. Why LV is cheaper than SV? That is a general observation that holds for all barrier and touches. That said, if you have a somewhat vanilla AC paired with reliable vols (they frequently tend to mature when you are out of the liquidity range for listed options), and tweak you correlation matrix a bit, you will find that you actually are not too far from market quotes with LV either. All dependent on your exact implementation obviously. ``` [1]: https://www.risk.net/our-take/6348336/calling-out-autocallable-pricing ```
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