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Topic Title: What is convexity?
Created On Tue Feb 10, 04 11:36 PM

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Paul
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Tue Feb 10, 04 11:36 PM

...giving examples from equities, fixed income etc. Suggested by mrbadguy.

P

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GiacomoGalli
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Wed Feb 11, 04 07:35 AM

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 problem

mathematically:

duration =

where:

PVCF= the present value of the cash flow in th eperiod t discounted at the yield to maturity
PVTCF= total present value of the cash flow of the security determinated by the yield to maturity
k= number of payments per year

convexity

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doublebarrier2000
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Wed Feb 18, 04 08:47 PM

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

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exotiq
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Mon Mar 08, 04 12:32 AM

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.

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"Nobody goes there no more; it's too crowded", Yogi Berra

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jimmycarter
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Thu Apr 01, 04 04:03 PM

In John Hull, it says convexity adjustment is the difference between future and forward prices. Is this convexity the same as that convexity?

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Thu Apr 01, 04 10:38 PM

Yes it is similar

As Exotic rightly said , there comes into play a vol issue speaking of convexity

As 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

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Thu Apr 01, 04 10:56 PM

One can't mention convexity without a nod to Jensen's inequality:

Hence, options have value.

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Daddy, I just got cider in my ear.

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jasemin
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Wed Aug 18, 04 06:38 PM

Does anyone know how to fit the data to a convextity curve?

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piterbarg
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Mon Nov 15, 04 10:24 PM

Convexity, to an interest rate quant, is a situation when a payment measure is diffeernt from the martingale measure of the underlying rate
-V

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Sun Dec 12, 04 04:15 AM

Quite elegantly said this way...thanks cvz !

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Money
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Wed Jan 26, 05 01:21 AM

"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...

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exotiq
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Wed Jan 26, 05 08:17 PM

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.

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"Nobody goes there no more; it's too crowded", Yogi Berra

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hamb
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Tue Feb 01, 05 12:04 AM

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 here
a) 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.

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Here I am, still alive

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Thu Mar 31, 05 09:09 PM

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.

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Mon Apr 11, 05 05:09 PM

Referring to DoubleBarrier2000's comments on convexity adjustment and referring to Hull on Timing Adjustment/Convexity Adjustment/Natural Time Lag, can somebody outline clearly when what should be used/applied.

For LIBOR in arrear swaps, we have timing adjustment (natural time lag) only? But the convexity adjustment value turns out to be the same as timing adjustment.
For Normal in advance swaps, we have both effects cancelling each other or no effect at all?
For CMS, we have convexity adjustment (5yr rate paid every six months, but paid in advance so no timing adjustment)?

Pls elaborate.

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Joined: Dec 2003

Fri Apr 22, 05 10:54 AM

Correction
I slipped up in my previous posting. The third paragraph is not true. The slope coefficient dB/dY is a component of the formula for modified duration: D=(dB/dY)(1/B). It is the slope coefficient of tD (true duration) that lies between the slope coefficients of the modified durations D and D*. The statement in the paragraph does not apply to the durations themselves. My notation isn?t too hot either. Sorry.

Addition
Another way of looking at the formula for tD is as follows. Assume a fall in the yield from (1+Y) to (1+Y*) such that (1+Y)=(1+Y*+a). Further assume that there is one value of Y* but n values of Y and a, ie, (1+Y_i)=(1+Y*+a_i). It follows that the formula given in the previous posting can be rewritten as
deltaB/B=product[i=1 to n](a_i)/(1+Y*)^n
or
(deltaB/B)(1/a_1)=tD=product[i=2 to n](a_i)/(1+Y*)^n where a_1=orthodox deltaY=(Y_1-Y*).

Every element in the top line of this last formula for tD reaches into negative or complex territory and has been ignored by orthodox analysis to date.

The adjustment to the yield can be rewritten multiplicatively instead of additively, ie, (1+Y_i)=(1+Y*+a_i)=(1+Y*)(1+m_i) where m is the multiplicative adjustment. The latter equation can be manipulated into product(a_i)/(1+Y*)^n=product(m_i). The left hand side of this equation is deltaB/b and so deltaB/B=product(m_i). An accurate, precise figure for a change in the value of a portfolio of bonds that does not require convexity is equal to the product of the multiplicative increments to ALL possible yields, including the highly negative and complex yields. The last expression is very general; it applies to many more financial situations than just fixed income.

The conclusion is that convexity is a patch to cope with the inaccuracy that results from ignoring the bulk of the solutions for the yield in the core pricing equation.

Question
How can you make this stochastic? The above deals with a single, instantaneous move (time does not change). The shift in the principal, real yield needs to be made random (taking into account what happens, or doesn?t happen, elsewhere in the plane). Then there needs to be a sequence of such moves with time changing too, and the number of zeros in the plane steadily diminishing. If the time increment is short then the polynomials will be of large order with many zeros padding the coefficients. My math is not up to it.

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Sun May 01, 05 06:54 PM

Further thoughts

Thought one ? On reality
Objections can be raised about the use of complex rates of interest. Not real. Don?t exist. But there appears to be no other way of calculating an accurate yield sensitivity that does not require convexity and the other terms of a Taylor series expansion. If the ?complex plane? approach works, then it must possess an element of reality. Many readers will be physicists and electrical / electronics engineers; surely there will be no objection from that quarter.

Thought two ? On practicality
Take a portfolio of vanilla bonds; say 100 of them, of all tenors, coupons, issue dates and market conventions. Compare two programming problems.

A - Program the orthodox duration and convexity of each bond and find the weighted average of all of them.

B - Take the vector of cash flows for the entire portfolio as the coefficients of a single polynomial, call it V; it will consist of irregular, variable coefficients, and will be padded with zeros on dates where no cash flow takes place. Then find product(abs(roots(V)/(1+Y*)-1) ), or whatever is needed in the math program of choice. It calculates the accurate interest sensitivity in one fell swoop.

Which program is easier to code?

Thought three ? On elegance
Compare the inaccurate formulas for duration and convexity with the accurate formula deltaB/B=product(m_i) embodied in the algorithm above. Would Dirac approve?

Thought four ? On past and future work
If all solutions for the yield in the constellation S are necessary to calculate an accurate interest rate sensitivity in the context of a one-shot move, is there not a question mark to be raised over the current stochastic approaches that follow the multiple period course of the one real yield (that actually, behind the scenes, stands for all yields at once). Is there not a ?stochastic convexity? issue? I don?t know what it all means. As I said before, my math is not up to it. Just raising the question.

I?m not the first one to raise the point. It has been raised before in the context of capital budgeting ? see Robert Dorfman (1981) The meaning of internal rates of return, The Journal of Finance, Vol. 36, No. 5. He examines the case where the proceeds of an investment are serially reinvested in projects of the same type, and the process is continued indefinitely. He shows that the growth path of a dollar placed in such an investment depends on all the roots of the internal rate of return equation, complex as well as real.

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Wed Jun 08, 05 11:44 AM

normally,convexity can be used in some perference in economic theories,but in finance ,does it work?

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xtang1
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Fri Sep 23, 05 07:40 PM

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Originally posted by: piterbarg
Convexity, to an interest rate quant, is a situation when a payment measure is diffeernt from the martingale measure of the underlying rate
-V

is that the same thing as saying that there is drift?

Thanks.

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johnself11
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Tue Oct 04, 05 08:09 AM

ok i'm not a quant and i think this question has gotten too theoretical...forget formulas and martingales for a moment... convexity is an attrbute of a term interest rate security which measures how the securites' price changes given a change in its duration.... it is the second-order derivative of the price/yield function of a bond, analogous to the gamma of a contigent premium (aka option) security.... the longer the duration of a bond, the more more convexity....

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