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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 ...


Re 2: It's a mathematical trick. Insisting on the separability of volatility function makes LMM useless. Its power lies in its powerful calibration abilities. If you constrain the vol function to separable form, you throw that ability out of the window. You might just as well use LGM then, and it will be more intuitive and faster.


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 ...

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