A flexible spot multiple-curve model

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Acknowledgements

We are grateful to Fabio Mercurio, Wolfgang Runggaldier and the participants of the XV Workshop on Quantitative Finance (Florence, January 2014) for helpful comments.

Notes

No potential conflict of interest was reported by the authors.

1 Notice that if we assume that admissible trading strategies involve finitely many assets, we should require the NFLVR (No Free Lunch With Vanishing Risk) condition in order to grant the existence of an equivalent martingale measure in our framework.

2 In a recent version of their paper the authors included a short-rate approach with deterministic shifts in the spirit of our present paper. However, we emphasize that our model is not a particular case of their framework since spreads and shifts are defined in a completely different way.

3 The model is inspired by the paper of Kijima et al. (2009) not yet written in a modern multiple curve perspective.

4 We mention that in the version of the paper published in Morini and Bianchetti (2013), Kenyon (2013) addresses this issue and some amendment is performed.

5 The extension is in principle straightforward because neither the spot LIBOR processes , nor the forward LIBOR curves for t fixed have to satisfy any particular no arbitrage requirement, as noted already in Mercurio (2010a), section 5. Note, however, that the presence of multiple tenors implicitly introduces a problem in the emerging correlation structure as well as in the ordering and positivity of spreads that is far from being trivial. We leave for further research these issues that play a crucial role from an empirical point of view.

6 For the rest of the paper, we will follow the common practice of referring to such interbank rates simply as ‘LIBOR’, but everything we will say applies as well to EURIBOR or any similar rate.

7 Notice that here the dimension of the vector space E is , so that, for example, the diffusion matrix a(x) should be represented as a symmetric matrix and b(x) is a vector in .