Fixed income risk in terms of simple, intuitive factors

braincat · Post by **braincat** » Thu Sep 06, 2007 5:53 pm

I am looking at the risk of a large fixed income portfolio

that includes options (swaptions, caps, exotics) in addition

to simple products (e.g. treasuries). Currently the risk is

represented in a fairly standard way -- I have a delta strip

for the portfolio for a number of maturities, vega matrix (vol

sensitivity for each maturity, tenor), and spread risk (e.g.

swap spread to treasuries). I'd like to represent this risk in terms

of a small number of simple, intuitive factors.

For the rate part, I can just look at the three PCA factors (shift,

slope, curvature). What could be done to simplify vol

sensitivity? I am not sure running PCA on the whole vol matrix is

a good idea. Is there a way to represent ATM cap/swaption vol

matrix movement in terms of intuitively simple factors, similar

to the way PCA does it for rates? I am looking at the ATM vols

only, making an assumption that skew is more stable over time.

Thanks, Brain.

aaron · Post by **aaron** » Thu Sep 06, 2007 7:17 pm

I've seen this done in FX, but not IR. The basic factors are ATM vol, Strangle and Risk Reversal, say at 0.25/0.75 delta. You measure these at each tenor, then assume parallel shifts in each.

bsycheng · Post by **bsycheng** » Fri Sep 07, 2007 11:20 am

Interesting question brain.

It wasn't very intuitive, but we used to have a PCA for the ATM IR vol surface, with an East/West factor to show movements across underlying tenor. Main issue was that it took forever to run, don't remember the last time I looked at it, but will try to dig something up this weekend.

Aaron, was the FX PCA done on Comp vol or breakeven?

jungle · Post by **jungle** » Fri Sep 07, 2007 1:34 pm

EDIT: Maybe [url=/Show%20Post.aspx?PostIDKey=96818]this thread[/url] is of use?

I have seen PCA used for a swaption vol surface by CS; will have a look for the piece and send it on if I can find it.

braincat · Post by **braincat** » Fri Sep 07, 2007 2:24 pm

Thanks folks.

bsycheng, thanks, thinking of what you said -- it this somewhat equivalent

to having the first pca vector correspond to the parallel shifts

of the entire vol surface, and to have two more factors,

one expressing a slope in the tenor dimension, and the other

expressing a slope in the maturity dimension, so that

we have something like

sigma(t,T) = F1(t,T) + F2(t) +F3(T) ?

Jungle, aaron, thanks for your help.

brain.

bsycheng · Post by **bsycheng** » Fri Sep 14, 2007 12:16 pm

Couldn't find anything sorry, but it was a 3 factor model as you describe.

braincat · Post by **braincat** » Mon Sep 17, 2007 4:49 pm

bsycheng,

no worries, and thanks.

brain