I am trying to "strip" (bootstrap) the LIBOR cap volatilities that are quoted by ICAP (VOLS<GO> on Bloomberg). The stripping methodology I'm following is described in the attached paper. Equations (6) and (7) describe how to convert the flat LIBOR cap volatilities (as quoted by the market) into cap prices. This requires applying the Black (1976) formula, as explained in the paper. The black formula takes the log of the forward rate between each caplet reset period (which is quarterly reset period in the case of LIBOR). So this means in order to make sense of the LIBOR cap quotes provided by the market, we must compute the logarithm of the quarterly forward rates using the current zero curve. Now, if the zero curve is sufficiently downward sloping then we end up with negative forward rates and we can no longer apply the Black '76 formula (since we cannot take the logarithm of a negative number). How is this problem dealt with in practice? Is there some quoting convention (e.g. replace negative forward rates with +1 bps or something)?Thanks in advance.

Another related issue ... In equation (9) a formula is given for the "forward cap price." It is the difference between the cap prices computed for two consecutive maturities. The two computations of these two cap prices are based on the corresponding "flat volatility" quotes from the dealer ... But what is to guarantee that the difference between these two prices will always be positive? A "hump shaped" volatility term structure could mean that the computed price of (say) the 20 year cap is LOWER than the computed price of the 15 year cap ... then the forward cap price (as defined in equation 9) between years 15 and 20 would be negative but a negative price doesn't make any sense at all.

How about stripping vols from a displaced Black model ?

normal instead of lognormal

The displaced Libor would be Lognormal. The distribution of the Libor would be somewhere between Lognormal and Normal, depending on the level of the displacement. But it would allow for the possibility of negative rates.Moreover you still have a tractable problem, since backing out the implied vol from a displaced model is no more difficult than inverting a pure Black model.You should be able to back out the displacement from the caplet smile.

Negative forward rates (implied by zero curve generation method) mean that the procedure is not arbitrage free and therefore you might consider reviewing it. There are I think "smooth forward" methods which ensure that forward rates are not negative. If there is a particular need for using a specific bootstrap algorithm then maybe the negative forward rate has to be manually over riddenOne of the bloomber staffers mentions in this presentation that they do the same:taipeibond.gretai.org.tw/cv/Bloomberg%20Mr.%20Lee(panel%203-1).pptRegarding the -ve TV for forward capA steep fall in caplet vol might probably be accompanied by a steepening zero curve, causing the implied forward rates to compensate and thus the forward cap TV might not be zero or negative. (May be a look at snapshot of yield curve for USD libor and USD cap vol in Dec 08 illustrates this).I cant think of what happens when the yield curve is inverted and the cap vols are falling.

Hi,Sorry for the bump but I have the same problem and after reading this post I can't figure it out yet.So the problem is: I'm trying to use the Black '76 formula to construct prices of at-the-money caps with implied vols but there are a few negative forward rates in the input data. The input data (thus with the negative forward rates) comes from my (gaussian) affine term structure model that I estimated using weekly data of USD LIBOR and swaps. I use the estimated model to construct the fair prices and forward rates needed to strip the caplets (thus with half year intervals). For a few data points in October and November 2010 the constructed forward rates at the short end of the curve (the 6m->12m, 12m->18m and 18m->24m forwards) are negative (max. -11 bps.).I've looked up the displaced Black/displaced lognormal model, but don't see how I can apply it to back out cap prices from vols. My most promising lead is Lee & Wang (2009), I think I have to transform the implied vols but don't know how. Simply using (strike-theta) and (forward-theta) with theta<0 doesn't give the right prices (of course).Thanks in advance

just use a Normal model... bpvol. you have to work out some expectations but it's not too hard.