# Autocallable option Delta

There have been numerous exotic trading desk blow ups lately, related to various reasons. However, in particular, one bank had some issues where they were pricing autocallable notes with Local Volatility and not producing a Delta "true up" using Stochastic Volatility that is common among other banks. In other words, Delta of the autocallable notes is higher in magnitude under Local Volatility compared to Stochastic Volatility. Since the bank thought it was holding more (negative) Delta as a result of the Local Volatility model, they bought too much stock to hedge and had large losses when the market declined.

Can someone provide an intuitive explanation of why Delta is higher in autocallable products under Local Volatility compared to Stochastic Volatility? The price of the product is different under the two volatility models on account of vol-of-vol differences, but it's not entirely clear to me why the Delta difference is in this direction.

Thanks.

• Hi @ellie_cat, are you sure this is related to the delta itself and not the Vanna break-even levels when you delta-gamma hedge autocalls which are completely different under LV versus SV? There is an interesting chapter on that in Lorenzo Bergomi's Stochastic Volatility Modelling book. Teaser: when you sell autocalls, you are long Vega at inception and have to sell options to hedge. But as you move closer to the knock-out trigger, you have to buy back options and are thus very sensitive to how your model predicts $(S,\sigma)$ will move together (hence the sizeable Vanna). Sep 15 '20 at 7:08
• Thanks for the comment. Vanna is certainly an issue. At the same time though, I've observed for a fact that banks hold negative Delta\$ reserves to account for the deficiency of Local Volatility compared to Stochastic Volatility. Maybe this accounts for some effect of assumed volatility dynamics as you described, but I'm not sure. Sep 18 '20 at 18:09
• It's an interesting question. As indeed this could be some sort of shadow Delta, correcting for the inconsistent Vanna break-even levels predicted by a LV model. I wished I could help you more but I only have a theoretical perspective on this, having a practical answer from an exo trader would help Sep 18 '20 at 18:14