# Barrier option (autocallable) Vega profile

I have a question about the Vega profile(graph) on an autocallable option. Generally for a regular option, the vega graph looks like a normal (kinda normal) distribution with the vega highest at-the-money. How does it change with a barrier option? My guess is that the left side of the graph looks the same, but as soon as it crosses the barrier vega becomes negative, since as soon as you break barrier vega starts to work against you(let's assume knock-in barrier). Can vega be both positive and negative in one graph?

Thanks

EDIT: Bonus if someone can help me draw the graphs for all the greeks (delta, gamma, theta) i understand how they look for a normal option, but for barriers i can't seem to graph them.

• As your chance to knock out increases, the value will decrease yes - so you will have the two effects fighting against one another. The question then becomes "which wins". And the answer to that depends on the specifics of your barrier option (ie where is the barrier, how long to expiry? Etc)
– will
Dec 4 '17 at 10:00
• when you say two effects fighting each other, how would the graph look like? lets assume 1 year to expiry call option with a barrier at 100, spot at 90, and strike a 95.
– Paul
Dec 5 '17 at 15:01

## 1 Answer

You have a multidimensional problem - there isn't an answer of "this is what the greeks look like" for all cases, because it depends on the various levels of the different parameters.

For example, if we limit ourselves purely to KO Call options, where the spot is 100, and there is no drift, with a time to maturity of 1 year (changing this is equivalent to just changing the vol really).

Now, we still have 2 other parameters to deal with: the option strike ($K$), and the barrier level ($H$). Each point on this surface will have its own greeks.

For example, here are the value,delta,gamma,vega,theta* for different strikes and barrier levels:

So you see, the relationship between all the different parameters is not so trivial. If you look at the back of each plot though (barrier at 180), these are approaching the vanilla values.

You also need to bear in mind that these are the greeks at a spot of 100 for each of the scenarios.

*Please also note that i have not priced barrier options properly here, I have used an approximation** - but for the purposes of demonstrating the way the specifics of the barrier option influence the value and greeks, I think it suffices, giving qualitatively similar results.

**$\mathrm{Call} \cdot \mathrm{P}(\mathrm{knock out})$

To address the three questions in the comments:

1. Taking your example, why do i see them going negative when the barrier is closer to the spot price? My assumption was that these are generally all positive if I am long a call option. There are two effects going on at the same time. When the price goes up, you expect to make more money from your call option, but it has also become more likely to knock out, by merit of being closer to the barrier. One of these is positive, the other negative, and their magnitudes depend on how close to the strike we are, and how close to the barrier we are. Here are the same plots as above, but with a strike of zero (i.e. just a barrier) Again we see the same theme as before, we the greeks tending towards those of a delta one product as the barrier is pushed farther away, as we would expect. But not that when the barrier is close, they are negative, since increasing the spot price makes the product more likely to knock out (delta), as we move closer to the barrier, the likelihood of knocking out increases more for the same increment as before (gamma), having a larger vol makes it more likely we will hit the barrier in any given day (vega), and as time goes on, if we have not hit the barrier already, then we now have fewer opportunities to do so, and so the value goes up (theta).

2. (a)Specifically taking a look at Vega, when barrier is at 180, they approach vanilla values (which i agree with, a kinda normal distribution with Vega highest ATM. However, when the barrier is at roughly 120, it looks like Vega is lowest at -120, and then Vega goes back up to -50 with barrier at 100. Why is it curved like this (If you see delta its not curved like this, ATM barrier delta is lowest) (Gamma is also curved like this but not as curved as Vega) How can Vega be negative? (b) If you see the barrier at 100, Vega is flat at 0 from strike 180 to 100, then it goes negative. Why is this like this..? We have no Vega exposure if we're at the barrier? There are two things here, but again they're related to the vega of the barrier. First off, if the strike is above the barrier, then it can never by in the money, so the value, and all the greeks are always zero when the strike is above the barrier. Secondly, if we look at the vega for just the barrier, for a variety of barriers from 101 up to 180, for different volatilities, we get the following: So, we first can say that as the barrier moves to higher strikes, the vega goes away (it will of course always come back if we increase the vol enough though) - this is because sending the barrier to infinity is like converting it into a forward. Secondly, we see there's a minimum - why is this? This is because when the barrier is very close, the vol doesn't really matter - knocking out is extremely likely, regardless of the vol (case and point, if the barrier is at 100, with a spot price of 100, then we have already knocked out and the trade is worth zero, and all of its greeks are zero too). As we move the barrier away though, changing the vol starts to have an effect, until we get too far away again and we collapse to a delta 1 product again.

3. Also last question kinda, what is a drift? I never heard this term before in options. Is this vols? You are presumably familiar with this: $\frac{\mathrm{d}S}{S} = \mu \mathrm{d}t + \sigma \mathrm{d}W$. $\mu$ is the drift, where it's made up of the risk free rate, $r$, the dividend yield, $q$, and the borrow cost, $b$: $\frac{\mathrm{d}S}{S} = (r - b - q) \mathrm{d}t + \sigma \mathrm{d}W$. These terms define how the (expectation of the) stochastic process drifts away from it's initial value.

• Thanks a lot Will! This was a lot more descriptive than I was looking for haha (i was only expecting a 2D graph at most). Now that I am looking at this, I do have a couple follow up questions I want to ask you. If I look at delta, gamma, vega, i see a very similar pattern. First question I have is: 1) Taking your example, why do i see them going negative when the barrier is closer to the spot price? My assumption was that these are generally all positive if I am long a call option.
– Paul
Dec 6 '17 at 17:54
• 2) Specifically taking a look at Vega, when barrier is at 180, they approach vanilla values (which i agree with, a kinda normal distribution with Vega highest ATM. However, when the barrier is at roughly 120, it looks like Vega is lowest at -120, and then Vega goes back up to -50 with barrier at 100. Why is it curved like this (If you see delta its not curved like this, ATM barrier delta is lowest) (Gamma is also curved like this but not as curved as Vega) How can Vega be negative? (related to first question)
– Paul
Dec 6 '17 at 17:59
• 3) Also last question kinda, what is a drift? I never heard this term before in options. Is this vols?
– Paul
Dec 6 '17 at 18:01
• Adding on my question 2, if you see the barrier at 100, Vega is flat at 0 from strike 180 to 100, then it goes negative. Why is this like this..? We have no Vega exposure if we're at the barrier? Not just Vega, gamma and delta is also flatline (almost) when barrier is close to 100.
– Paul
Dec 6 '17 at 18:14
• @Paul: See edit.
– will
Dec 6 '17 at 18:46