 Paul
Topic Author
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Joined: July 20th, 2001, 3:28 pm

### What is convexity?

...giving examples from equities, fixed income etc. Suggested by mrbadguy.P GiacomoGalli
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Joined: December 17th, 2003, 11:20 am

### What is convexity?

Convexity is a measure of the curvature in the relationship between bond prices and bond yields.for large changes in yields, duration provides an approximation, so convexity is the right measure for more accurate estimations. the true relashionship between a change in price and a change in interest rate for option free securities is not linear.If we use a straight line approximation this will create error that grow with the magnitude of the interest rate changes. For small changes in yield duration does a good job in estimating the actual price, but for larger changes duration becames less accurate and the error become larger. Convexity is what we use to correct this problemmathematically:duration = where:PVCF= the present value of the cash flow in th eperiod t discounted at the yield to maturityPVTCF= total present value of the cash flow of the security determinated by the yield to maturityk= number of payments per yearconvexity doublebarrier2000
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Joined: July 14th, 2002, 3:00 am

### What is convexity?

from FI products, convexity comes into play when the payout from your product is not in line with the tenor of the underlying rate.eg. Constant Maturity Swaps - say we swap 6m LIBOR for the 3yr Swap rate paid every 6 months. Here the 3yr rate is being paid out 6 months after it fixes.Back Set swaps, CM caps/floors exotiq
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Joined: October 13th, 2003, 3:45 pm

### What is convexity?

Convexity generally means exposure to some form of volatility.The simplest example is the gamma of an option. When long an option, a large move in either direction will result in a larger gain or smaller loss than a linear position in the delta of the option. The degree to which the convexity increases the price (or decreases the yield) of a gamma position is directly linked to the market's perceived likelihood of such a large move.Similarly, a long position in interest rate convexity, such as in a 25-year zero-coupon bond, when delta hedged by shorting a lower-convexity bond, provides an added return in the case of a large interest rate move, paid for by an inferior total return should rates not move so much. jimmycarter
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Joined: July 14th, 2002, 3:00 am

### What is convexity?

In John Hull, it says convexity adjustment is the difference between future and forward prices. Is this convexity the same as that convexity? fc
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Joined: May 9th, 2002, 8:45 am

### What is convexity?

Yes it is similarAs Exotic rightly said , there comes into play a vol issue speaking of convexityAs futures are fixed in arrears, we have to take into account a convexity adjustement that make them worth less than the corresponding FRA and corresponding to the uncertainty at which the future will be fixed cvz
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Joined: January 7th, 2003, 9:20 pm

### What is convexity?

One can't mention convexity without a nod to Jensen's inequality:Hence, options have value. jasemin
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Joined: August 17th, 2004, 7:59 pm

### What is convexity?

Does anyone know how to fit the data to a convextity curve? piterbarg
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Joined: October 29th, 2002, 6:42 pm

### What is convexity?

Convexity, to an interest rate quant, is a situation when a payment measure is diffeernt from the martingale measure of the underlying rate-V Jezza
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Joined: September 24th, 2004, 3:49 am

### What is convexity?

Quite elegantly said this way...thanks cvz ! Money
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Joined: September 6th, 2002, 4:00 pm

### What is convexity?

"Convexity, to an interest rate quant, is a situation when a payment measure is diffeernt from the martingale measure of the underlying rate"don't understand this statement from q' perspective... exotiq
Posts: 1888
Joined: October 13th, 2003, 3:45 pm

### What is convexity?

Covexity, to an interest rate quant, is what they get for talking in terms of rates instead of bond prices. Bonds have no convexity with respect to bond prices, only to rates, which are an artificial measure designed to be convinient to the eye, but convex by the 1/r, 1/(1+r)^T, or exp(-rT) definitions.Financially, convexity is what you expect to lose on a delta-hedged position due to volatility. hamb
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Joined: December 29th, 2004, 3:12 pm

### What is convexity?

I came across this definition and classification of convexity from a paper:The notion of convexity refers to different situations, which can be sometimes seen as having almost nothing in common. Sometimes used as the gamma ratio for interest rate options, as an indicator of risk for bonds portfolios, as a measurement of the curvature of some financial instruments or as a small adjustment quantity for a wide variety of interest rate derivatives, convexity has become a synonym for small adjustment in fixed income markets, related somehow to the notion of mathematical convexity and more generally related to a second order differentiation term. There are two types of convexity particularly in interest for practitioiners, 1. the bias due to correlation between the interest rate underlyng the financial contract and the financing rate. Example, the bias between forward and future contract.2. the modified schedule adjustment. Two subtypes are defined herea) One period interest rate (money market rates, zero coupon rate) and bond yield. An example is the difference between plain vanilla products and in-arrear ones. b) Swap rates, as in the case of CMS. fa
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Joined: December 10th, 2003, 11:07 am

### What is convexity?

There is another way of looking at convexity.Assume a vanilla bond with known price B, coupon C, and number of periods N. If the equation is solved for the YTM, (1+Y), there are actually N solutions, an entire constellation, call it S. Under normal circumstances, only one (real) value of S is calculated, i.e., the orthodox rate that is produced by an HP calculator or Excel. The other (N-1) values of (1+Y) in S are either negative real numbers or complex conjugate pairs positioned off the real number line. Associated with this set-up, there will also be a value for modified duration, D.If the YTM changes from (1+Y) to (1+Y*), there will not only be a new bond price B*, but also a whole new constellation of interest rates S*, and a new value for modified duration, D*.The true (accurate, precise) value for duration is somewhere between the two modified durations, D and D*. And the differences between the true value, and one or other of the modified durations on either side of it, can be regarded as kinds of convexity. Therefore these differences repay investigation.According to the formula for modified duration, D, the change in the price of a bond is dB/B=D.dY where d is the differential operator. It is only approximately true for small values of dY. Assume that there is a version of duration that gives accurate, precise results for any change in the yield; call this duration tD. In this, necessarily, discrete version, the change in the price of the bond is deltaB/B=tD.deltaY. The critical point is this: a formula for tD does exist, but it can only be obtained by going into the complex plane and considering the entire constellations of interest rates S and S*.Let the old constellation of values be (1+Yi)=Zi and the new constellation (1+Y*i)=Z*i. The distances in complex space between any two of these yields can be defined as abs(Zi-Z*i)=abs((1+Yi)-(1+Y*i))=abs(Yi-Y*i). That is to say, distances in complex space between the zeros of the bond pricing equation are differences between interest rates. For example, the absolute value of the orthodox difference deltaY is equal to abs(Z1-Z*1)=abs((1+Y1)-(1+Y*1))=abs(Y1-Y*1) where Z1 in S and Z*1 in S* are the orthodox yields.It can be shown that the following equation holds true: (modified duration * dY) = dB/B={product[for i=2 to N] of (abs(Zi-Z1))}*abs(Z*1-Z1)/(Z1^N). Compare it with the following tidier formula: deltaB/B={product[for i=1 to N] of (abs(Zi-Z*1))}/(Z*1)^N. The latter formula encompasses tD, and it shows that an accurate formula for duration involves every interest rate that is usually thrown away or ignored in the orthodox analysis. It is worth drawing figures in complex space to ‘see’ the formulas.In words, the formula for tD is found as follows: plot the zeros of the bond pricing equation in the plane, join them with straight lines to the new orthodox yield (1+Y*1), multiply the lengths together and divide the result by (1+Y*1)^N.Earlier, it was assumed that convexity can be defined as the difference between dB/B (an approximate value) and deltaB/B (an accurate value). The new formulas for these concepts can be compared because they are in the same language of ‘products of distances in the plane’. The comparison suggests that convexity is the result of a lack of clarity. The confusion is about which yields change position, and which stay in the same location, within the constellation, S, when the orthodox yield is adjusted in the bond pricing equation. Optron
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Joined: October 10th, 2004, 1:21 am

### What is convexity?  