SERVING THE QUANTITATIVE FINANCE COMMUNITY

Randor
Topic Author
Posts: 80
Joined: October 8th, 2005, 8:33 am

### sensitivity of american digital (binary) options to changes in volatility skew

I am trying to think what is the sensitivity of american digitals (henceforth binary or one-touch (OT)) options to a) implied volatilities between atm and the barrier (henceforth ITM vols), b) implied vols on the other side (henceforth OTM vols) , c) the slope of the volatility smile (henceforth skew).

to me, intuitively, since implied vols are the average of local vols , then increasing the ITM vols should cause the chance of hitting the barrier to rise, thus should push the price up. And decreasing OTM vols too should push the price up. So increasing the slope of the vol surface, so that vols at strike=barrier go up and vols on the other side of the spot go down, should increase the price of the option.

Is this reasoning correct , or are there scenarios where it is not true?

I cannot see what is wrong with it, HOWEVER, i know that if the barrier is quite far away, then the binary price converges to the european digital (ED) price , and the ED price's sensitivity to itm or otm vols can be positive or negative depending on various things.

BUT, if the barrier is not far away, then the price can be more than twice the ED price and so it may not behave like an ED, so what can we say about it there?

Alan
Posts: 10322
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: sensitivity of american digital (binary) options to changes in volatility skew

I think you should start with the case of all vols the same (flat smile) and examine how the one touch value changes with vol. If this simple case is not monotonic, then the case with a skew is not going to be simple.

Randor
Topic Author
Posts: 80
Joined: October 8th, 2005, 8:33 am

### Re: sensitivity of american digital (binary) options to changes in volatility skew

Why would that simple case not be monotonic?

Alan
Posts: 10322
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: sensitivity of american digital (binary) options to changes in volatility skew

Because the ED is not monotonic

Randor
Topic Author
Posts: 80
Joined: October 8th, 2005, 8:33 am

### Re: sensitivity of american digital (binary) options to changes in volatility skew

Hmmm..
ED is monotonic in its Local skew.
Can you elaborate in more detail?

Btw whats a sinple way to price the OT given you have prices of EDs? As that perhaps may help here

Alan
Posts: 10322
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: sensitivity of american digital (binary) options to changes in volatility skew

Re the ED, assume it out-of-the money, no cost-of-carry parms, and plot its (GBM/Black-Scholes) value vs $\sigma$ for $\sigma \in (0,\infty)$. Should rise from zero and then fall back to zero. That's what I mean by non-monotonic in vol.

I don't know what 'monotonic in its local skew' means.

Having the prices of ED's is equivalent to having the prices of all the Euro-style puts and calls, which is equivalent to having an implied (or local) vol surface. But, I don't think any of those suffice to fix the price of a OT. For example, the local vol process is a diffusion, and I recall there are diffusion bounds on any OT price; but the market price might lie outside of those bounds.

Randor
Topic Author
Posts: 80
Joined: October 8th, 2005, 8:33 am

### Re: sensitivity of american digital (binary) options to changes in volatility skew

How about if we are considering a LOCAL VOLATILITY model, as opposed to other possible models, then is the reasoning from my OP true?

That is weird about the ED! Sounds strange , i will test that out!

By Monotonic in local skewci meant that since an ED price = -1×( dc/dk) = -1×( pc/pk + dc/dvol × dvol/dk) ( where p means partial derivative)
So as local skew = dvol/dk evaluated at the strike of the ED, goes up, then ED price goes down.

Alan
Posts: 10322
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: sensitivity of american digital (binary) options to changes in volatility skew

If the OT behaves like the ED, I think your last paragraph in your last post contradicts
" then increasing the ITM vols should cause the chance of hitting the barrier to rise, thus should push the price up"

Anyway, I don't know for sure. I just strongly suspect that the behavior may be more complicated, even with a local vol model. Perhaps take a simple (non-flat) local vol model, like CEV model, and see if your intuition holds there?

Randor
Topic Author
Posts: 80
Joined: October 8th, 2005, 8:33 am

### Re: sensitivity of american digital (binary) options to changes in volatility skew

Do you by any chance have VBA for that that i could test ? Or some algo that i can try implement to test it?

Alan
Posts: 10322
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: sensitivity of american digital (binary) options to changes in volatility skew

No, sorry. There are quasi-analytic solutions, but you would probably need something like Mathematica to implement them. Perhaps just coding a good Monte Carlo will suffice. Achieving 4-digit accuracy should be straightforward, and you can check against the results in the paper.

Randor
Topic Author
Posts: 80
Joined: October 8th, 2005, 8:33 am

### Re: sensitivity of american digital (binary) options to changes in volatility skew

I have no idea how to do montecarlo for americans! Need some brownian bridge i guess but with local vol...

Thanks for the paper llooks very complicated but i will try that out!

Randor
Topic Author
Posts: 80
Joined: October 8th, 2005, 8:33 am

### Re: sensitivity of american digital (binary) options to changes in volatility skew

i realise why the ED is not monotonic in vol, its because , for an OTM ED, as vol goes up, it at first gets more likely to be ITM, but then, it gets less likely because the middle of the distribution (=X=F*exp(volsqT) ) recedes away

Alan
Posts: 10322
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: sensitivity of american digital (binary) options to changes in volatility skew

Yes, I think of it similarly: competition between a vol effect (spreading out the density), and a negative drift effect (pulling the particle away from an upper barrier).

BTW, Paul Glasserman's Monte Carlo book has some discussion of Brownian bridges for improving barrier hit detection in general 1D diffusion MC's. (Ch 6 in my edition)

Randor
Topic Author
Posts: 80
Joined: October 8th, 2005, 8:33 am

### Re: sensitivity of american digital (binary) options to changes in volatility skew

That CEV looks very hard to understand and implement for me.
I think that implementing a tree thats calibrated to option prices may be the simplest approach for me to start with. I have Dermans GS papers that i can try. But i think perhaps Rubinsteins approach may be quicker and easier with less chance of mistakes? What do you think?
I think rubinsteins uses just the terminal distribution, and i guess thats why it would be quicker/easier

Alan
Posts: 10322
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: sensitivity of american digital (binary) options to changes in volatility skew

In the end you are working with (tree) approximations to diffusions, since every local vol model is a diffusion.

Since there are OT bounds for all diffusions, you might learn more general truths (about diffusions) by exploring the sensitivity of those bounds to changes.

Just don't expect the market to necessarily agree.