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### Convexity adjustment in bootstrapping the yield curve

Posted: **February 2nd, 2011, 8:25 pm**

by **gorilla**

Hi,I am trying to bootstrap a yield curve using cash deposits, futures and swaps. In order to do that I need to make a convexity adjustment between futures and forward rate. In Hull's book this adjustment factor is given as 0.5 * sigma^2 * T1 * T2, where sigma is the volatility of the change in the short-term interest rate in 1 year, and T1 and T2 are the times to maturity and maturity of the underlying rate.To estimate sigma, I followed the method outlined in the paper "A practical guide to swap curve construction" as to use the implied volatility from interest rate caps. To calculate the implied volatility, I can get the market price for 1Y cap with 3 caplets and try to solve for the implied flat volatility. The problem is, to price a cap(let), I need the forward rate and the discount factor at each reset / payoffs points, which is what I am trying to solve in the first place (bootstrapping a yield curve). Isn't this just a chicken and egg problem?Is my understanding above correct? Or have I missed something? I am new to this area so any help would be greatly appreciated.Thanks.

### Convexity adjustment in bootstrapping the yield curve

Posted: **February 3rd, 2011, 3:08 pm**

by **DavidJN**

Yes, you are correct. I described the chicken and egg problem when I wrote about the same topic in the late 1990's in the context of the Hull-White interest rate model. In that model one requires a zero curve to calibrate the short rate vol and mean reversion parameters to the prices of traded options, but to derive the zero curve adjusted for futures convexity bias one has to know the short rate vol and mean reversion parameters. The solution is to estimate both the volatility parameters and the adjusted futures rates simultaneously using an iterative process.

### Convexity adjustment in bootstrapping the yield curve

Posted: **October 24th, 2011, 1:42 pm**

by **rockinsquat**

Hi David JN,Could you be more precise in your answer ? Maybe have so references to point to ? (That would be much appreciated !)Thx in advance,R

### Convexity adjustment in bootstrapping the yield curve

Posted: **October 25th, 2011, 2:01 pm**

by **DavidJN**

I have little more to offer you than what I've already said, which I think is pretty clear. It boils down to including the futures convexity adjustment in the calibration process and that is a fairly model-specific thing. It's a real fine point that some people seem to miss. You can see if the original paper (Risk, March 1997) has any more details but I doubt it. I believe Bloomberg adopted this specific convexity correction method for their HW yield curve model but I have no idea if they did a good job. You can also find this convexity adjustment in the free Quantlib source code project, but again I dont't know whether they went as far as simultaneous estimation.

### Convexity adjustment in bootstrapping the yield curve

Posted: **October 25th, 2011, 3:05 pm**

by **rmeenaks**

The paper can be found on the author's website at:

http://www.powerfinance.com/convexity/Cheers,Ram

### Convexity adjustment in bootstrapping the yield curve

Posted: **October 26th, 2011, 4:06 pm**

by **mathmarc**

QuoteOriginally posted by: rmeenaksThe paper can be found on the author's websiteA more recent presentation for more general one factor HJM and including options on futures can be found at Eurodollar Futures and Options: Convexity Adjustment in HJM One-Factor Model. The extension to multi-curves framework can be found at The Irony in the Derivatives Discounting Part II: The Crisis.This precise one way to compute the adjustment, but does not answer the original question of simultaneous calibration of the curves and the volatility parameters. In the one factor HJM, the "adjustment factors" (Z in the Kirikos-Novak article and \gamma in the above references) are model but not curve specific. So if you accept to calibrate your HJM model using a curve without futures, then you can decouple the model step and the bootstrapping step (at the cost of a tiny inconsistency as the curve used for model calibration will not be exactly the same as the one created by the bootstrapping with the futures).

### Convexity adjustment in bootstrapping the yield curve

Posted: **October 29th, 2011, 4:15 pm**

by **miltenpoint**

Is there a good proxy for volatility (and mean reversion) of the short rate?

### Convexity adjustment in bootstrapping the yield curve

Posted: **October 31st, 2011, 8:27 pm**

by **DavidJN**

miltenpoint, not that I am aware of (haven't studied this for quite some time) but in the HW constant parameter single factor model the level of short rate vol moves in step with lognormal Black vols typically used to calibrate. Increase the vol surface uniformly by 10% and the short rate vol moves the same amount proportionately. Must dig out that other paper I wrote on this stuff...

### Convexity adjustment in bootstrapping the yield curve

Posted: **November 15th, 2011, 6:57 am**

by **archlight**

I remember iteration stops when deposits, future and swaps overlap in a smooth way

### Convexity adjustment in bootstrapping the yield curve

Posted: **December 13th, 2011, 2:14 am**

by **manilla**

Have a look at Andrew Lesniewski's notes Lecture 4 section 5 Lecture 5 section 6http://

www.math.nyu.edu/~alberts/spring07/index.htmlAlso Vaillant articlehttp://

www.probability.net/convex.pdfPITERBARG ... ds/100.pdf

### Convexity adjustment in bootstrapping the yield curve

Posted: **August 27th, 2015, 2:30 pm**

by **studenttt**

I've found Andrew Lesniewski's notes in the links above particularly useful.