Serving the Quantitative Finance Community

 
BerndSchmitz2
Topic Author
Posts: 12
Joined: December 20th, 2021, 7:56 am

Which expiry interpolation method for cap surfaces

March 30th, 2023, 8:26 am

Hi,

I want to bootstrap an (implied volatility) caplet surface from quoted caps on a fixed strike grid. I'm thinking about either using a left-continuous or a linear interpolation in expiry dimension. If one just looks at the smiles of each quoted cap expiry left-continuous (2nd picture) seems to be preferrable as the linear smiles (1st picture) are more spicky. However, a left-continuous interpolation will produce theta jumps. The jump always happens for a (standard) cap in the portfolio on the day when it's schedule aligns with schedule of the caps used to bootstrap the surface.

Therefore, out of these 2 candidates, I would clearly go for the linear expiry interpolation. Does anybody have an opinion on that?

Thanks,
Bernd

ps: I still have to implement a linear interpolation in variance. Is this still considered to be the best method out of the straightforward ones?
pss: I know that there are more fancy interpolation methods out there (e.g. using SABR smiles with interpolation on the SABR parameters) but I want to go for a simple interpolation as a starting point
Attachments
LinearCapSurface.png
LeftContinuousCapSurface.png
 
BerndSchmitz2
Topic Author
Posts: 12
Joined: December 20th, 2021, 7:56 am

Re: Which expiry interpolation method for cap surfaces

June 20th, 2023, 9:51 am

Nobody an opinion on that?
 
User avatar
pcaspers
Posts: 30
Joined: June 6th, 2005, 9:49 am
Location: Germany

Re: Which expiry interpolation method for cap surfaces

July 19th, 2023, 6:42 pm

Hi, 

is "left-continuous" the same as  "piecewise constant" / "backward flat"? And the interpolation variable is implied volatility, at least initially? And later on you talk about interpolation in variance, is that sigma^2 T, because I guess linear interpolation would then be the same as backward flat in sigma^2 = backward flat in sigma.

Anyway, if you want to choose between these two only, I think backward flat is more robust in the sense that it's less susceptible for zig/zags and self-amplifying oscillations. But linear - if it works - generates nicer shapes and better risks.
 
BerndSchmitz2
Topic Author
Posts: 12
Joined: December 20th, 2021, 7:56 am

Re: Which expiry interpolation method for cap surfaces

August 25th, 2023, 9:25 am

Yes, I'm directly interpolating on implied caplet volatilities. By left-continuous in expiry dimension I mean that implied caplet volatility is sigma_0 [0,1M], sigma_1 in (1M,3M] and so on. I assume backward flat is the same.
What would be your criteria for "if it works". Not too much zig-zagging - whatever this means? Or are there any objective criteria?
 
BerndSchmitz2
Topic Author
Posts: 12
Joined: December 20th, 2021, 7:56 am

Re: Which expiry interpolation method for cap surfaces

August 25th, 2023, 10:21 am

Yes with interpolation in variance I mean V(t) = sigma(t)^2 * t where simga is the implied volatility. Probably you meant something else but to me linear in implied variance is not the same as left-conintinuous in implied volatility. Just take the simple example sigma(2y)=20%, sigma(4y)=40% which yields V(2y)=0.08 and V(4y)=0.64. Linear interpolation gives me V(3y)=0.36 and, thus, sigma(3y)=34.64%. This is obviously not the same as 40% which I would get if I apply left-continious interpolation to sigma itself
 
User avatar
pcaspers
Posts: 30
Joined: June 6th, 2005, 9:49 am
Location: Germany

Re: Which expiry interpolation method for cap surfaces

August 25th, 2023, 2:21 pm

Sorry, I confused myself w.r.t. the linear in sigma^2 T interpolation.

As for the objective criterion. Maybe the total variation of the caplet volatility curve should be minimal?