Contrary to popular belief, bonds and stocks are non linear derivatives, just as options are. They are just less obviously so.
Stocks can be thought of call options on the value of a company with a strike of zero.
Bonds can be seen as short put options on the value of the company with a strike of zero also.
If a company goes bankrupt and the value of equity is wiped out the company's bonds are exercised and bond holders claim the remaining value of the company.
In effect the outstanding debt on bankrupted companies is converted to equity (which makes sense, as the 'fixed income' part of the bond ain't so fixed anymore!).
Over time, we should see bonds exhibit negative convexity with respect to the value of the company (upside more constrained than the downside).
Equity should exhibit positive convexity over long periods, the downside being more constrained.
This is not intuitive.
Over short periods of time we actually see equity returns exhibiting negative skew of course.
And, most of the time when people mention convexity and bonds in the same sentence they refer to interest rate convexity, which is almost always positive.
The downside shows that the high yield bond index (HYG) beta is 0.45 with respect to the Russell 3000. The upside is 0.4.
A -0.05 swing in the wrong direction! The index gobbles up more risk when things are bad!
Now let's look at a junk bond index (JNK).
We should expect higher volatility, and therefore our 'short put option' should be an even greater risk, if things go well we hit upside limits easier, but the downside is wide open.
And indeed we se ever so slightly more negative convexity, 0.38 vs 0.32. I.e a -0.06 spread in the wrong direction.
Note that the volatility for HYG and JNK is 14% and 15%; which isn't that great a difference, especially put in context with the volatility of the Russell 3000 (22%) and an investment grade index (LQD) 10%.
The LQD shows no negative convexity, although if I had enough data and could test over very long periods of time, perhaps it would exhibit some.