Arbitrage-Free Interpolation of Forward Rates in the LIBOR Market Model
Summary
The document asks how to obtain LIBOR rates whose reset or maturity dates fall between the model’s predefined dates. It presents two approaches: use an arbitrage-free interpolation scheme designed for the displaced-diffusion LIBOR Market Model, or bootstrap a yield curve from simulated forward rates and apply a standard curve interpolation method.
The cited paper discusses interpolation alongside pricing callable range accruals, simulating coupon behavior, and calculating sensitivities. The curve-based alternative is familiar from interpolating market forward-rate agreements, but it does not account for how future forward rates evolve. The response therefore expects it to work best when the desired rate is near one of the simulated rates. The document offers pointers and a limitation rather than a detailed implementation or a quantitative comparison of interpolation methods.
Key ideas
- The LIBOR Market Model simulates rates on a fixed schedule, while some contracts need rates at intermediate dates.
- Arbitrage-free interpolation schemes can extend the model’s rates between its simulation dates.
- A bootstrapped curve with standard interpolation is another practical way to estimate intermediate rates.
- Curve interpolation ignores the evolution of future forward rates and may be most reliable near a simulated rate.
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Full text
# Rate interpolation in Libor Market Model # 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? ## Answer by TheBridge (score 6, accepted) https://quant.stackexchange.com/a/473 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 ## Answer by Mark Joshi (score 8) https://quant.stackexchange.com/a/2632 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. ## Answer by ldnquant (score 3) https://quant.stackexchange.com/a/471 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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