The BGM takes as an input vols implied by the Black's model. Interest rate caps are usually quoted in terms of flat vols, so that cap flat vols are always available. To implement BGM, one must strip them to get the caplet vols (get a reasonable set of caplet vols so the sum of the caplets gets the same price as the cap). Caplet vols go into BGM.

How should one extract the caplet vols from the cap flat vols? Is this done inductively using Newton-Raphson?
Say, flat vols are the following: fv1=15% and fv2=18%. Therefore caplet1_vol is 15% and caplet2_vol is X% which has to be found solving the following equation:
Cap_price(18%, T=1,2)=Caplet_price(15%, T=1)+Caplet_price(X%, T=2).
Am I right?

Another issue is the calibration of the multifactor BGM. Could anybody briefly explain how to estimate covariance matrix of vols implied by Black's model?


Thanks.

Caplet vols: Newton-Raphson should do the job.

Most people directly "strip" the caplet vols from flat cap vols. This is a little tricky for the following reasons:
a) one is only given flat vols for (say) 1y, 2y, 3y, 4y, 5y, 7y, 10y caps; and each year contains 4 caplets (except by market convention, the caps do not containt the first 3m caplet). One has to "postulate" an interpolation formula (like piecewise constant, or piecewise linear volatilities) to relate the four caplets to each measurement;
b) the cap vols may be quoted at different strikes, which tangles the skews/smiles up in stripping process;
c) one can also use "sturctures" like the 3x5 (which is all caplets between years 3 and 5) to simplify the stripping process. I believe these are liquidly quoted.
d) Newton Raphson is good, but add the "global Newton" trick makes it a bit better and safer. This is explained in Num Recipes in C.

A potentially cleaner approach is to take a short rate model, and calibrate it to all the caps, and then use it to price the caplets ... but this removes the advantage of the BGM model that it "doesn't need" calibration.

A potentially cleaner approach is to take a short rate model, and calibrate it to all the caps, and then use it to price the caplets

I am a bit confused. At the beginning, one should calibrate a spot rate model to all the caps. As a result, the spot rate model is consistent with the cap prices. Can the caplets be priced automatically with the model estimated or or there is the second calibration needed?
What does the sigma of the spot rate model mean: the spot rate vol, the cap flat vol, the forward rate vol, or something else?

... but this removes the advantage of the BGM model that it "doesn't need" calibration.

As I guess, some of IR exotics are priced with the BGM model and some are priced with a spot rate model. When is it preferable to use BGM and when a spot rate model (which)?

What are the pros and cons for using the BGM and the spot rate models?


Thanks.

Hello!I've got a question concerning the Black's model for caps/floors pricing. In the Black's model, one should input the cap rate Rx and the forward rate Rf which are expressed with a compounding frequency equal to the frequency of resets.So that one should1) Interpolate the market yield curve to obtain the spot rates.2) Find the discount factor. Should one discrete or continuous compounding such asdiscountfactor[t]=exp(-resetdate[0,t]*rateinterpolated[0,t]) ?3)Calculate the forward rates using forwardrate[t,t+1]=(discountfactor[0,t]-discountfactor[0,t+1]) / (tenor[t,t+1]*discountfactor[0,t+1])4) Calculate the compound cap ratecapratecompound[t,t+1]=(1+tenor[t,t+1]*caprate)^(1/tenor[t,t+1])-1I'm not sure with 2) and 4). Are these correct? What is the market practice?Thanks.