Forward-Measure Expectations of Future LIBOR Rates

Summary

The document asks how to compute the expectation of a LIBOR rate over an accrual period under a later forward measure. The answer says the calculation requires a model for rate dynamics between the accrual end date and the forward-measure date; without specifying those dynamics, the expectation is underdetermined.

Under lognormal rate dynamics, as in the Brace–Gatarek–Musiela or LIBOR Market Model framework, a measure change introduces a state-dependent drift. The response states that the expectation generally has no closed-form expression. It points toward further mathematical finance material but does not provide a derivation, approximation, or numerical example. The result therefore conveys the modeling dependency and qualitative form of the drift, while leaving practical computation and assumptions beyond lognormal dynamics unspecified.

Key ideas

• The expectation depends on specifying rate dynamics between the accrual end date and the later measure date.
• Under lognormal rate dynamics, a change to the forward measure leads to a state-dependent drift.
• The expectation generally lacks a closed-form solution in the stated setting.
• The document provides no derivation or computational method, so practical use requires further model details.

Tags

# How to calculate $E^{T_N}(L(T_i, T_{i+1}))$?


# How to calculate $E^{T_N}(L(T_i, T_{i+1}))$?












suppose $L(T_i, T_{i+1})$ is the LIBOR rate between $T_i$ and $T_{i+1}$, and $T_N$ is some time later than $T_{i+1}$. $E^{T_N}$ is the $T_N$-forward measure.

I tried to work this out using John Hull's timing adjustment methods (ch 29.2 of "Options, futures and other derivatives"), but to no avail.

Could you pls throw some lights here?

## Answer by Mark Joshi (score 1)

https://quant.stackexchange.com/a/20690

well you need to specify dynamics for the rates between $$T_{i+1}$$ and $T_N.$ If you make them log-normal then the standard BGM/LMM drift computation applies and you get a state dependent drift.

The expectation does not exist in closed form however.

(See eg More Mathematical Finance for detailed discussion.)

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