
I believe that this topic has been posted on a decent amount before but I'm looing for some help with the formula and notations. I'm just looking for the simple black model to price a cms curve cap option. I have derived the curve cap vol from swaption vols for each forward tenor and correlations between the forward tenors. I have also calculated the convexity adjustment on the two rates. I'm pricing a 6month forward 10yr/30yr atm curve cap. For the inputs i have: (these numbers are approximate) 6m10yr 2.05 and vol of 60 annualnormvol 6m30yr 2.30 and vol of 55 annualnormvol spread of 25bps with a convexity adjustment of 1.5bps so atm strike of 26.5. I used about 96% correlation and got a vol of 18bps. In addition, let's say the 6month discount rate is 1.40%.
So can someone help me to calculate the curve cap price using these inputs? The actual calculation and not just the formula is what im looking for since that's part of my issue. Of course i'll be doing this in excel and the inputs given here may not tie out to what i say i've calculated since I was just trying to remember them but they can be taken as correct. Thank you! 




Cheng


Total Posts: 2838 
Joined: Feb 2005 


Can you pls delete the nonrelevant of the two posts? Double posting is frowned upon. 
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I'd be happy to but i do not see an option to do that. 





I think common practice is to get the mid curve vol by writing the 20y fwd as a linear combination of the 10y and 30y (using relevant annuity factors) and then use the standard formula for covariance of a sum to derive the mid curve vol.
Once you have the convexity corrected fwd, and the vol,just use the usual Bachelier pricing formula.
I don't know if the midcurve market is liquid enough that people price it in a more elaborate way (using correlation / correlation skew together with the skew of the underlyings, with maybe a replication (CMS style) argument for calculating the convexity corrected forward) but if you're just trying to sanity check a counterparty price, start as I've outlined above.
Actually, have you posted a termsheet? I have just assumed you're talking about a standard midcurve option but then you mentioned a cap? 




silverside, I think the OP is pricing a CMS curve cap, rather than a midcurve. 
Insofar as I may be heard by anything, which may or may not care what I say, I ask, if it matters, that you be forgiven for anything you may have done or failed to do which requires forgiveness... 




yes, it's a cms curve cap not a midcurve. thanks 




Ok fine
Same idea applies though? You have the vol and the forward (and the annuity/discount factors) so just use the usual cap pricer...? 





Yeah, I concur w/silverside...
In fact, Your Imperial Majesty, there's an old post somewhere where I was debugging a vanilla cap pricer in Excel. You could try and follow a similar procedure, if you're seeing something that you find strange. 
Insofar as I may be heard by anything, which may or may not care what I say, I ask, if it matters, that you be forgiven for anything you may have done or failed to do which requires forgiveness... 


mtsm


Total Posts: 206 
Joined: Dec 2010 


Looks like, you have really gone almost all the way to price the option making a bivariate normal assumption on the CMS rates. Whether right or wrong, all you need to do is to plug in the resulting parameters into a normal (Bachelier) option pricer.
Here are some old notes:
http://janroman.dhis.org/finance/Swaptions/normal%20swaptions.pdf https://www.princeton.edu/~rcarmona/download/fe/sirev.pdf 





Okay thank you guys i've looked at your attachments but that's kind of my question. I'm a bit blackScholes/mathematical notationally challenged so i'm kind of looking for help with the formula for my inputs. 




If it's at the money then the mid is something like 5bps of notional times 15 (or whatever the annuity factor for the cms rate is)





mtsm


Total Posts: 206 
Joined: Dec 2010 


I cannot believe you are talking with so much ease about some pretty hairy concepts, such as forwards, convexity adjustment, correlation and a normal spread vol, yet do not know how to apply the formula.
I think you have everything you need to use formula 2 or 3 in the first note. You reported you had the rate r, the forward F, the volatility sigma, and for 6M that's roughly 0.5 for Tt. 




I know shocking! But here we are. So do tell, how can i price such an option with unknown present value? I've heard of mysterious copula caplet formulas but would settle for a simple respite on the blackened scholes of the waters edge. 




mtsm


Total Posts: 206 
Joined: Dec 2010 


Not sure what you mean actually. You have already assumed bivariate normality on your rates and prepared everything for formula 2 or 3 in the first note I sent. Just plug in the numbers as I said in my last message.
Then, if you are unhappy with the price, apply a spread to the vol and adjust the price as you see fit. 


