Hi Sunil,

It’s a *truly* astute observation. There is a subtle, superficial discrepancy between option gamma and bond convexity that manifests in Ch 13 (as I look at it, I only noticed this as you’ve pointed it out)

Absolutely a bond can have negative convexity. Jorion means a plain bond without embedded derivatives will always have positive convexity (and, yes, IMO he also means a short plain vanilla bond will always exhibit positive convexity). To achieve negative convexity, you only need to get the price/yield curve to “bend back” (2nd derivative < 0) at low yields with a callable bond or at high yields with a puttable bond. Last year, the FRM assigned Tuckman’s chapter 22 on mortgage-backed securities and MBS are good examples of classically negative convexity. At low yields, refinances are akin to callable bonds. At high yields, defaults cause prepayments which act (sort of) like puttable bonds. (Sections of the price/yield curve can be variously concave/convex.)

But in the “bond context” duration/convexity are instrument-specific; the statements hold regardless of long/short. UNLIKE Jorion’s option gamma discussion, which gives rise to the superficial discrepancy.

The reason is due to the definitional (semantic) difference between gamma and convexity: gamma is the second-derivative (i.e., the slope of the tangent line), convexity is not exactly the 2nd derivative; it’s is the 2nd derivative divided by bond price. (Just like duration is, contrary to what many people think, not the slope of the tangent line. It’s the slope divided by, or “infected” with, bond price). Because convexity divides by price, in the short position, you do in fact (just like the option) switch to a negative slope (2nd derivative) but it is canceled by the negative price of the short position!

Put another way, if convexity were defined as gamma (as purely 2nd derivative), those charts in Jorion Ch 13 would apply to bonds, too, and a short bond would similarly have negative gamma.

I notice Jorion writes gamma is “similar to” the concept of convexity, so as usual, he is precise. The difference is subtle, in summary:

* As Jorion shows, gamma is a function of long/short
* But due to canceling effect, duration/convexity are features of the instrument (not functions of long/short). Plain vanilla will always give positive convexity, but generally adding embedded options will introduce negative convexity (for portions; again, a bond can be positively convex passing thru an “inflection point” to negatively convex)

I hope that’s not too much, I think your observation is very interesting…David

David,

The formula for convexity in your slides has two (change Y)^2 one in the measure and one in the adjustment. Is it correct that these can be different numbers?

Frank

Hi David,
Holding yield constant , the bond with lower coupon would have higher duration and greater convexity.
But I read somewhere that by keeping both yield and duration constant, the bond with lower coupon has lower convexity.Convexity is a measure of dispersion of cash flows.Could you please explain the same.
If possible could you upload an excel working for this if possible.

Regards,
Sunil

@ sunil: sorry for delay, i trust i’ve answered here: http://www.bionicturtle.com/forum/viewthread/750/
@ Chinquee: it shouldn’t matter, you shouldn’t be asked about the convexity measure (the meaningless interim step). I suppose it would be safe to follow Tuckman and compute the convexity “metric” without a (2) in the denominator; and call that convexity. Then just make sure the (1/2) is used to estimate price change.

David

They are typically different. I agree with you if the shock = ultimate yield change you are estimating. But, more typical is something like: use 10 or 20 basis points for the shock, then use 1% (100 basis points) for the convexity adjustment. So like: [V+ - V- - (2V0)]/[2*V0*0.001)^2 for Convexity measure, then [Convexity measure]*(1%)^2 * 100.

I guess you have a point if, for example, both are 100 bps, then (1%)^2 cancels, but then your convexity adjustment is only good for one yield change. I sort of see you point, maybe this is just to given the adjustment flexibility.

David