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Changing from a Forward Measure to an Annuity Measure

Article Quant Q&A · Author: adam
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Summary

The document explains a measure change used in a derivation of constant-maturity swap convexity. It considers the expectation of a swap rate under a forward measure associated with a bond maturing at the payment date, then rewrites it under an annuity measure. The annuity is the present value of the swap’s scheduled payments, and the Radon–Nikodym derivative between the measures supplies the bond-price and annuity terms.

The answer gives the derivative at the swap’s observation time and substitutes it into the expectation, showing how the initial bond and annuity values move outside the expectation. This accounts for the additional zero-coupon bond price factor in the displayed identity. The explanation is algebraic and relies on the stated measure-change relationship and consistent notation for dates and annuity values. It does not develop the broader CMS convexity adjustment or discuss numerical implementation, so readers need the surrounding derivation to connect the identity to pricing consequences.

Key ideas

  • The derivation converts an expectation under a bond forward measure to one under an annuity measure.
  • The density relating the measures contains both a bond-price ratio and an annuity ratio.
  • Substitution of that density explains the bond price factor outside the expectation.
  • The identity depends on using consistent dates and annuity definitions.

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Fixed incomeDerivatives pricingStatistics
Full text
# Girsanov theorem in CMS convexity derivation


# Girsanov theorem in CMS convexity derivation












I am going through the derivation of CMS convexity from the notes of Lesniewski

There is a transformation from $T_p$ forward measure to annuity measure $Q$ as

$$ P(0,T_p)E^{Q_{T_p}}\left[S(T_0,T)\right]=A(0,T_0,T_n)E^Q\left[S(T_0,T)\frac{P(T_0,T_p)}{A(t,T_0,T_n)}\right] $$

where $A(t,T_0,T)=\sum_{1\le j \le n} \alpha_i P(t,T_i) $ is price of annuity at t paying $\alpha_i$ at $T_1,...,T_n$

Why is there an additional $P(T_0,T_p)$ term (zero coupon price of a bond maturing at $T_p$ and starting at $T_0$O in above equation?

Edit 1:I guess his notation is not clear. Right hand side can be written as $E[S(T_0;T_0,T)D(0,T_p)]$ where $D(t,T_p)=E\left[e^{-\int_t^{T_p}r dt}\right]$ and $S(T_0;T_0,T)$ is the swap spread at $T_0$ for the period from $T_0$ to $T$. $E[S(T_0;T_0,T)D(0,T_p)]=E^{Q_{T_p}}[S(T_p;T_0,T)]P(0,T_p)$ assuming $T_p$ is after $T_0$. At the same time $E[S(T_0;T_0,T)D(0,T_p)]=A(0;T_0,T_n)E^Q \left[S(T_0;T_0,T)\frac{D(T_0,T_p)}{A(T_0;T_0,T_n)}\right]$ through measure change

## Answer by Gordon (score 3)

https://quant.stackexchange.com/a/14371

Note that $$\frac{dQ_{T_p}}{dQ}|_{T_0} = \frac{P(T_0, T_p)}{P(0, T_p)}\frac{A(0, T_0, T_n)}{A(T_0, T_0, T_n)}$$. Then $$E^{Q_{T_p}}\big(S(T_0, T_n)\big) = E^Q\bigg(S(T_0, T_n) \frac{P(T_0, T_p)}{P(0, T_p)}\frac{A(0, T_0, T_n)}{A(T_0, T_0, T_n)}\bigg) \\ = \frac{A(0, T_0, T_n)}{P(0, T_p)} E^Q\bigg(S(T_0, T_n) \frac{P(T_0, T_p)}{A(T_0, T_0, T_n)}\bigg).$$ That is, $$P(0, T_p)E^{Q_{T_p}}\big(S(T_0, T_n)\big) = A(0, T_0, T_n) E^Q\bigg(S(T_0, T_n) \frac{P(T_0, T_p)}{A(T_0, T_0, T_n)}\bigg). $$

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