Julien Duguet

We address in this paper the issue of a bond valuation when the underlying shadow rate is assumed to stay inside a given tunnel, but not only above a lower bound as is considered by various authors when dealing with the zero-lower bound for the interest rate or as with the negative interest rates situation arisen after the policy adopted by the European Central Bank from 2014. The model considered here is referred to as LUB (Lower and Upper Bound) model, and for the simplicity, the underlying shadow rate is assumed to follow the one-factor Vasicek model for the interest rate. Our consideration of the LUB model is not only done under the willing to deal with a model representation consistent with market situations observed both in developed and emerging countries, but it is also performed with the intention to provide a helping tool when structuring bespoke financial products linked to interest rates. Indeed, discarding the ranges of interest rate levels not attainable under the market regime may mildly/drastically lower or rise zero-coupon prices. As is well-known for the Black's approach in the context of interest rate zero-lower bound, the LUB restriction on the shadow rate level leads to technical complications in the bond valuation such that getting tractable zero-coupon bond prices is challenging. We first provide a Monte-Carlo explicit based expression for the zero-coupon price, in the sense that this last is given as a deterministic function of the bond characteristic(s), the model parameters, the underlying state variable and independent realizations of the standard normal Gaussian random variable. Not only the obtained price is helping from the pricing audit point of view, but it has also the advantage to provide a starting point for the derivation of the zero-coupon price sensitivities. Next, using a cubature approach, we derive an approximate closed form for the zero-coupon price which has the advantage to be free of any standard Gaussian random variable realizations and may serve as a quick tool for the LUB model parameter calibration and the underlying shadow rate estimation.