# 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

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.

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

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.