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Why Forward Rates Can Begin at the Observation Date

Article Quant Q&A · Author: Sithered
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Summary

The document asks whether a rate written as F(a,a,c) is a forward rate or a spot rate. The question arises while interpreting a formula for pricing constant-maturity-swap par rates, where the formula uses forward rates observed at times that coincide with the start of their accrual periods. The author frames the issue in the setting of Monte Carlo pricing with a forward-rate model such as the Libor Market Model.

No answer or derivation is included, so the post does not settle the terminology or explain the pricing formula. It does identify a useful distinction for interpreting rate notation: the observation time, accrual start, and accrual end are separate arguments, and matching the first two does not by itself explain how the cited model defines the rate. Readers need the paper’s convention and the market model’s rate definitions to resolve the question.

Key ideas

  • The question concerns interpreting a forward rate whose observation time equals its accrual start.
  • The rate notation distinguishes observation time from the accrual period endpoints.
  • The context is Monte Carlo pricing of constant-maturity swaps with a forward-rate model.
  • The document contains no answer, so the rate convention must be checked in the cited framework.

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Fixed incomeDerivatives pricingStatistics
Full text
# Are forward rates starting at observation date spot rates?


# Are forward rates starting at observation date spot rates?












In part 3.2 of Lu and Neftci (2003) "Convexity Adjustments and Forward Libor Model: Case of Constant Maturity Swaps", the authors propose a new way of pricing CMS swaps, with Monte Carlo simulations. They show that the par CMS swap rate is a function of forward rates. So we just need a model of forward rate to do the Monte Carlo pricing (such as the Libor Market Model).

Specifically, the formula they find for the par CMS swap rate (equation 22 and 23 in their paper) is a function of these forward rates: $F(t_1,t_1,t_2)$, $F(t_2,t_2,t_3)$ and $F(t_3,t_3,t_4)$, with the following notation: $F(a,b,c)$ = forward rate viewed from time $a$ between time $b$ and time $c$.

These forward rates are then of the form : $F(a,a,c)$. I don't understand why they are said to be forward then. It seems to me that if a forward rate start at the date of observation, it is not a forward rate but a spot rate. Isn't it? Why would they say that then?

Many thanks for your help!

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