Communications in Mathematical Finance

Valuing the Probability of Generating Negative Interest Rates under the Vasicek One-Factor Model

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  •                                             Abstract

    The generation of scenarios for interest rates is needed in many contexts, as in the valuation capital requirements (under Solvency II or Basel 3 regulation frameworks), in other risk management tasks (the application of a risk measure to a portfolio) as well as in the pricingof the financial contracts. For such purposes, a model for the term structure, as the famous Vasicek one-factor model, is needed. Though it is very often considered as a benchmark, mainly due to its tractability, unfortunately it generates negative interest rates with a non-nullprobability. Our purpose in this paper is to analyse to what extend this model can be used to the generation of yield curve scenarios, at one or more future time horizons and under both historical and risk-neutral measures given its inconsistency. In the first case, the spot rate is defined in terms of a realization of a Gaussian variable and the bounds avoiding negative yields are analysed. In the second case, the problem is described involving of a hitting time in order to value the probability to obtain negative yields during the simulation. Moreover, some numerical examples are provided in order to illustrate the computation of the probability.