# How to gamma hedge and vega hedge an autocallable product?

I am pretty new in quantitative finance, and I am interested by the hedging of autocalls. Could you, please explain which financial products should be traded (specify the way, please) to delta hedge, gamma hedge and vega hedge an autocall? And why the latter strategies work?

• Not sure why this question is given a "-1", it's not straightforward to hedge an auto callable. Giving it a +1. Feb 23 '20 at 15:11
• Hello,thanks for your answer. Do you know some books that can help me to improve my understading of all the risks that matter to hedge an autocallble product? Apr 16 '20 at 11:24
• To understand autocallables you need to understand barrier options. So as a start if I were you I would study the pricing and hedging of barrier options. Here is also a link to a discussion of barrier option risk on this forum: quant.stackexchange.com/questions/37200/… Apr 16 '20 at 12:13

Well, it's a topic which actually should have its own book dedicated. Unfortunately, existing literature is rare or not practical enough. Let me at least try to provide some key ideas and challenges you should consider when hedging this kind of structure.

First, let's start with the question: How not to hedge an autocallable? What is not going to work is that you flatten the market risk for a given risk factor set and the corresponding greeks by permanently using the spot and derivatives market. Why? Because sensitivities are so dynamic for an autocallable note that this approach would eat up all the margin you passed to investors in maybe 2-3 days. Of course, you can charge more in order to cover your hedging costs, but you are in competition and need to have an eye on external pricings in order to win deals.

Now, what are the challenges. We have Delta and Gamma risk towards the spot.

Delta: Directional (as an issuer you are short and hence need to buy/sell stocks or futures)

Gamma: Has sign changes around barriers and will explode around them. Will make hedging very difficult, if not impossible depending on moneyness. Of course, you would need to buy/sell options in order to risk manage your Gamma from the structure

Vega: This is a tough one. Besides the question what kind of Vega we are speaking about, what really matters is that you have exposure across the whole term structure and need to measure this exposure. However, the exposures on different Vega buckets are so dynamic that Vega can be concentrated around one bucket today but then, depending on spot movement, may shift to other buckets tomorrow. So once you have flatten Vega in one specific bucket, you may be forced to buy your hedge back and buy/sell Vega in other buckets. Now you see, that all means crossing (costly) bid-offer spreads while hedging your Vega

Dividends: Same as vega except for the fact that the dividend market is clearly more illiquid, especially for single stocks. Typically, you will sell (exchange traded) dividend futures but then again, the quotes are so wide (if they exist at all) that you do not want to cross the spread permanently. But you will have to, because dividend sensitivity is as dynamic as Vega risk. It will change as spot moves. If no dividend futures exist, you can, e.g., try to flatten the risk via synthetics (calls and puts).

2nd order derivatives (Vanna, dividend sensitivity w.r.t to spot): These second order mixed partial derivatives are incredibly important as they will provide insight in how your (hard to hedge) Greeks Vega and Div sensitivity will change with spot. Therefore, traders will have a look at different scenarios: What will be my Vega if spot moves by +2%? What about dividend sensitivity then? You will, e.g., observe very intense convexity in your sensitivity towards dividends once spot moves significantly. How can you hedge this 2nd order Greek? Well, you could try to buy dividend options but there is actually no market for dividend options on single stocks. Alternatives? You could try an overlay hedge via Dividend Options in the Eurostoxx

There are so many other risks factors I even did not mention (third derivatives, EQ-IR correlation etc.). If you want to incorporate all risk factors and potential hedging costs, the fair value of the note would probably be infinity

Last but not least: Watch out for market directionality which is an issue for Index Autocallables rather than single stocks. As every issuer has the same risk position w.r.t Eurostoxx, they often need to start buying/selling Vega at the same time in order to hedge their books. In this case, you will have a hard time to find a reasonable quote in the interbank market as everyone needs to do the same thing (probably you will see very large bid-offer spreads). That's why Banks have a very critical impact to the volatility term structure of the Eurostoxx. This actually is a very famous example for what we call "Liquidity is never there when you need it most in hedging"

I know my answer is not a recipe, simply because there is no recipe to hedge this highly structured and dynamic risk product. It will differ from desk to desk depending on internal risk limits and trader psychology etc. What is more important is that you have a look at different scenarios which will lead to sharp changes in your greeks and find a framework and level of risk you are comfortable with as a trader.

• Could you explain why you buying div future ? As an issuer, aren't you already long stock / index for the hedging so already long dividend ? thanks Dec 8 '20 at 11:56
• You are right, of course. Just edited it above.
– SI7
Dec 23 '20 at 18:19

In addition to SI7 answer, and more specific to the vega / gamma dynamics of an autocall (closely related to its hedging strategy, as you would try to neutralize exposures as they change):

The standard autocall payoff has several coupon/autocall dates and a down and in put at maturity.

• The coupons/autocalls for the client are like a strip of long digital calls, because he wants to be above the coupon/autocall barrier to get the coupon / to autocall.
• The client is short a DI put at maturity.

The trading desk is short the autocall, so it has the reverse position: short a strip of digi calls, long a DIP. So it has vega/gamma exposures from all these.

An additional consideration is that all these exposures are conditional to the product not having autocalled. So the trading desk is not exposed to 100% of the strip of digi calls and 100% of the DIP. But rather to a fraction of it (the probability of the product arriving to expiry, i.e. not disappearing/autocalling before its payoff).

So imagine the DIP has 100,000\$vega, but the autocall model gives only a 5% probability for the product to survive (i.e. not to autocall) until its maturity, then the trading desk will see it is long only 5,000\$ vega from the put (5% * 100,000\\$).

With these ideas in mind, we can have an idea of the vega/gamma exposure of the autocall:

• At initiation, client sells a put to get big coupons. So the exposure from the put dominates (survival probability of the put is relatively high). The trading desk is long vega and gamma from the put.
• If the underlying goes down, this survival probability of the put increases even more. So the trading desk becomes even longer vega (i.e. needs to sell vega to hedge itself).
• If the underlying goes up, the survival probability of the put decreases, so it loses vega. Plus, the underlying becomes closer to the strip of digi calls. So the trading desk picks up some short gamma from its short strip of digi calls.
• If the underlying rallies further, the digi call becomes ITM. So this gamma exposure flips and the desk becomes long gamma again. From the ITM digi call this time, not from the put (which has a value close to 0 as the product is very likely to autocall).

Basically, the big vega exposure comes from the put at maturity. This exposure is dynamic because it changes with the probability of survival of the put. And on top of that there are effects from the strip of digi calls.

Back in a past life, I used to work in one of largest issuers of autocallables... and have a catch-up with their equity derivatives desk almost daily doing my rounds.

I cannot pretend to understand the precise mechanics of all the greeks in forensic detail. As an index jockey, all I needed to know was that the hedging of these was always in the vanilla underlying; and this became "non-smooth" whenever one came close to the barriers. So the derivs jockeys would either be telling me autos were a live issue (and anything could happen, as the hedging flows could swing massively on a dime) or something else (cliquets, "sticky strikes" etc.) was the hot button du jour.

Knowing just little enough to be really dangerous, this came down to sign flips at the barriers. Theoretical hedging might entail infinite greeks at the "singularity" of the barrier; but we all had to trade in the real world.

So the "model" always seemed to model the exposures at -5%, -2%, -1% and +1%, +2%, +5% of spot and average these into something smooth that was never as crazily volatile as the actual Black-Scholes at any point in time. "Seemed" because the derivs guys were always a little deliberately vague on these kinds of issue ;-) Plus they did not want to become captive to systematic hedging flows that might arise because their model was the same as everyone else's across the street! The general assumption was that those with larger autos business relative to their cash, futures and options business in the same underlyings would always have to the "first to blink" hedging this. Those with deeper non-autos business in the same could afford not to immediately react (given their other flows in the same underlying).

So the hedging game didn't just reflect the greeks... it reflected market depth in other lines of business, that would allow one to avoid being a forced buyer/seller around the barriers. That generally favoured the big buys.