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?
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3 Answers
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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
answered Feb 12, 2011 at 17:46
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$\begingroup$ What about papers.ssrn.com/sol3/papers.cfm?abstract_id=1729828 ? $\endgroup$ Feb 12, 2011 at 18:25
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$\begingroup$ 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 $\endgroup$ Feb 12, 2011 at 20:21
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$\begingroup$ 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. $\endgroup$ Feb 12, 2011 at 22:03
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$\begingroup$ 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. $\endgroup$ Feb 14, 2011 at 12:31
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$\begingroup$ 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. $\endgroup$ Feb 15, 2011 at 10:01
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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.
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$\begingroup$ 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. $\endgroup$ Dec 20, 2011 at 20:17
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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.
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$\begingroup$ There is the question of the "short stub", say the first simulated rate starts accruing 1 week from the filtration date. $\endgroup$ Feb 12, 2011 at 18:25