# Rate interpolation in Libor Market Model

Libor Market Model (LMM) models the interest rate market by simulating a set of simply compounded, non-overlapping Libor rates which reset and mature on predefined dates. How do I obtain from them a Libor rate which resets and/or mature between these fixed dates?

## 3 Answers

The subject is interesting and not so easy if you want to interpolate in an arbitrage-free way, to my knowledge a good paper on the subject is this one

• – quant_dev Feb 12 '11 at 18:25
• As this article is quite recent, I haven't read it. I'll take a look and tell you what I think about it. Anyway, there is also some interesting material about this topic in books of Brigo and Mercurio and in the book(s) by Andersen and Piterbarg. Regards – TheBridge Feb 12 '11 at 20:21
• I used the scheme described in the paper I linked to in my implementation of LMM, but I'm interested if there are better solutions. Thanks. – quant_dev Feb 12 '11 at 22:03
• Well I haven't implemented the paper but argument seems fine even if the solution to make the fixed Libor is a little "articial" in my opinion and the author could have reformulated it in a more natural way within the framework of Schlögl with "a model" for short rates. Regards. – TheBridge Feb 14 '11 at 12:31
• Thanks. Btw, I think it's somewhat simpler in the "artificial" setup -- you won't have any market data for the short rate vol anyway. – quant_dev Feb 15 '11 at 10:01

The following paper, Interpolation Schemes in the Displaced-Diffusion LIBOR Market Model and the Efficient Pricing and Greeks for Callable Range Accruals, addresses this issue:

We introduce a new arbitrage-free interpolation scheme for the displaced-diffusion LIBOR market model. Using this new extension, and the Piterbarg interpolation scheme, we study the simulation of range accrual coupons when valuing callable range accruals in the displaced-diffusion LIBOR market model. We introduce a number of new improvements that lead to significant efficiency improvements, and explain how to apply the adjoint-improved pathwise method to calculate deltas and vegas under the new improvements, which was not previously possible for callable range accruals. One new improvement is based on using a Brownian-bridge-type approach to simulating the range accrual coupons. We consider a variety of examples, including when the reference rate is a LIBOR rate, when it is a spread between swap rates, and when the multiplier for the range accrual coupon is stochastic.

• Hi Mark, welcome to the site! We're very glad to have you contribute. We generally discourage posting of links alone, so I copied and pasted the title and abstract from your paper into the answer. – Tal Fishman Dec 20 '11 at 20:17

You could bootstrap a curve based on the forward rates you get, plus your standard interpolation scheme.

That's certainly what you'd do if the rates were presented to you as a set of market quotes for FRAs. It does ignore the evolution of the future forward rates though, so I'd expect it to work best if the intetpolated rate is close to one of your simulated rates.

• There is the question of the "short stub", say the first simulated rate starts accruing 1 week from the filtration date. – quant_dev Feb 12 '11 at 18:25