Bond Convexity Versus Gamma in Bond Options

Summary

This explanation distinguishes two meanings of convexity that arise in fixed-income markets. Bond convexity describes the nonlinear relationship between a conventional bond’s price and its yield. For a non-callable bond without other embedded options, the price response is asymmetric: a yield decline tends to raise the bond’s price more than an equal-sized yield increase lowers it. This concept concerns the bond’s price-yield curve.

Bond option convexity refers instead to gamma, the rate at which an option’s delta changes as the underlying bond or other reference value moves. Near the strike, a small market move can substantially change delta; close to expiry, an out-of-the-money option may have low delta that rises sharply if the underlying crosses the strike. The explanation is conceptual and gives no formulas or pricing framework. It also does not address how a bond’s own price-yield convexity may affect the value or risk of an option written on that bond.

Key ideas

• Bond convexity measures the nonlinear relationship between a bond’s price and its yield.
• For a non-callable bond without embedded optionality, price gains from falling yields can exceed losses from equal yield rises.
• Option convexity is gamma, which measures how quickly delta changes as the underlying moves.
• The two concepts involve different variable relationships even though both describe nonlinear behavior.

Tags

# Bond Option Convexity


# Bond Option Convexity












As part of my learning, I came across bond convexity and was wondering how that would apply to bond options, as in would it be the expectation of ytm ? How do we define it? Are there any good articles/ book chapters in can read that would explain it?

Thank you so much!!

## Answer by user68819 (score 1)

https://quant.stackexchange.com/a/80887

A bit more depth from my comment:

- Bond convexity is to do with the non linear relationship between price and yield. Namely, as yields fall the price of the bond (assuming its non callable and no other embedded optionality in it) will increase more than it would fall for a similar change in yield.

- Bond option convexity, also known as gamma is to do with the fact that the payoff of an option is nonlinear. That is, if you are hovering around the strike the change in your sensitivity ('delta') to the underlying can change substantially. For example if you are a few mins before expiry and your option is OTM, your delta would be very small. Now if the market rallies a few ticks taking you through the strike your delta goes to 100%. This rapid change in delta is gamma at work.

Both are due to convex relationships between variables, however, its due to different variables.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.