Volatility Separability and Markovian Structure in the LIBOR Market Model
Summary
The document considers when the LIBOR market model (LMM) can be represented as Markovian in its driving Brownian motions. It notes that state-dependent drifts prevent strict Markovianity in those motions. Predictor-corrector schemes can approximate the drifts step by step, while, when drifts are ignored, a sufficient condition is separable volatility: each rate’s volatility is a rate-specific scalar times a common deterministic time function.
The answers add that a matrix-style separability condition can offer more flexibility, with rates Markovian in combinations of Brownian motions. They also caution that imposing simple separability can undermine the LMM’s calibration flexibility, and suggest a simpler lognormal Gaussian model as an alternative in that constrained setting. The post raises whether separability is necessary and whether weaker conditions exist, but does not resolve those questions or provide derivations; it points to further mathematical finance material.
Key ideas
- State-dependent drifts prevent the LMM from being strictly Markovian in its underlying Brownian motions.
- Predictor-corrector schemes can approximate drift behavior over a single time step.
- With drifts ignored, separable volatility is sufficient for Markovianity in the Brownian motions.
- A matrix separability condition can provide more flexibility and yield Markovian rate combinations.
- Restricting volatility separability can reduce the LMM’s calibration flexibility.
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Full text
# When is the LIBOR market model Markovian? # When is the LIBOR market model Markovian? The question is inspired by a short passage on the LMM in Mark Joshi's book. The LMM cannot be truly Markovian in the underlying Brownian motions due to the presence of state-dependent drifts. Nevertheless, the drifts can be approximated 'in a Markovian way' by using predictor-corrector schemes to make the rates functions of the underlying increments across a single step. Ignoring the drifts, the LMM would be Markovian in the underlying Brownian motions if the volatility function is separable. The volatility function $\sigma_i(t)$ is called separable if it can be factored as follows $$\sigma_i(t)=v_i\times\nu(t),$$ where $v_i$ is a LIBOR specific scalar and $\nu(t)$ is a deterministic function which is the same across all LIBOR rates. Questions. - The separability condition above is sufficient for the LMM to be Markovian in the Brownian motion. How far is it from being a necessary one? - What is the intuition behind the separability condition? - Are there any weaker sufficient conditions? ## Answer by Mark Joshi (score 8, accepted) https://quant.stackexchange.com/a/2631 You can use a matrix type seperability condition as well. This is similar but the equation has more flexibiliity. The rates are then markovian in some combinations of the Brownian motion. See More Mathematical Finance for details. ## Answer by quant_dev (score 5) https://quant.stackexchange.com/a/1021 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.
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