Higher-Order Bond Price Sensitivities and Taylor Approximation Terms
Summary
The document discusses terminology for higher-order terms in a Taylor approximation of bond price changes as interest rates move. It relates the first and second derivatives of price with respect to rates to delta and gamma, respectively, and gives “speed” as a name sometimes used for the third derivative, describing how gamma changes with rates. The response traces that usage to derivatives-pricing literature.
A second answer cautions that higher-order Taylor terms do not all have universal mathematical names. Beyond familiar labels used in particular applications, they can be described by their derivative order or as successive polynomial corrections. The material is brief and does not develop a full bond risk framework: it leaves choices such as tenor-bucket representation and whether gamma is expressed as a matrix or a single aggregate measure largely contextual. Its main lesson is to distinguish conventional application-specific terminology from general Taylor-series language.
Key ideas
- Bond price Taylor expansions use first and second derivatives to describe linear and quadratic rate sensitivity.
- The third derivative is sometimes called speed because it measures the change in gamma as rates move.
- Derivative-based labels can depend on the application and representation of interest-rate risk.
- Higher-order terms can be named by their polynomial order when no broadly accepted term applies.
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Full text
# Are there names from the third term onwards in the Taylor approximation for bond pricing? # Are there names from the third term onwards in the Taylor approximation for bond pricing? The first terms are duration and convexity, but are there common names for the terms beyond this? ## Answer by Dimitri Vulis (score 2) https://quant.stackexchange.com/a/73967 I don't like duration. If the interest rates delta is the first derivative of the bond price with respect to the interest rates, probably by tenor bucket, and gamma is the second derivative of the bond price with respect to the interest rates, probably not a matrix but a risk-weighted sum as one number, then the interest rates speed would be the third derivative of the bond price with respect to the interest rates, or equivalently the rate of change in the interest rates gamma with respect to the interest rates. Source: Espen Gaardner Haug. The Complete Guide to Option Pricing Formulas, 2E, pp. 47ff, in turn citing Mark B. Garman "Charm School," Risk Magazine, 5(7), pp. 53-56 (1992). ## Answer by Draman (score 1) https://quant.stackexchange.com/a/75468 There is no Mathematical jargon for such terms. However in particular applications a good understanding of what those terms do may warrant applications specific names such as the one mentioned by the previous poster in bond market. First order = linear approximation of whatever function or price you have. The rate of change being the Constant coefficient Second order term = quadratic correction. Constant coefficient being the rate of change of the previous Constant coefficient 10 th order term = 10th order polynomial approximation. Constant coefficient being the rate of change of the previous (9th order) coefficient
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