# 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?

• The technique is now standard. See the interest rate book by Brigo. – Gordon Sep 9 '15 at 19:43
• @Gordon are you referring to Ch13.8 The Convexity Adjustment and Applications to CMS . . . . . . . pp 559? – athos Sep 9 '15 at 23:31
• See Proposition 6.3.1 on Page 258 of the second edition. For implementation, certain approximation is needed, for example, to frozen the drift. – Gordon Sep 10 '15 at 13:10
• @Gordon thank you. After read Ch 6 till $\S 6.3$, I think the third case ($i>k, t \le T_{k-1}$) of this proposition can solve the problem. May I ask what's the next step? Assume the calibration is done (I guess it's discussed in $\S 6.4$), is the next step Monte Carlo as pp 261 said?-- "discretize equations (6.14) between 0 and t with a sufficiently (but not too) small time step Δt, and generate the distributionally-known Gaussian shocks Zt+Δt − Zt." – athos Sep 11 '15 at 15:22
• See Section 6.13 for clues, in particular, Page 272. – Gordon Sep 11 '15 at 15:50

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.)

• @Joshi thanks for the recommendation. May I ask which chapter is the result in? – athos Sep 10 '15 at 4:13