We investigate the extremal behaviour of a diiusion (X t) given by the SDE dX t = (X t)dt + (X t)dW t ; t > 0 ; X 0 = x ; where W is standard Brownian motion, is the drift term and is the diiusion coeecient. Under some appropriate conditions on (X t) we prove that the point process of "{upcrossings converges in distribution to a homogeneous Poisson process. As examples we study the ex-tremal be...

There are two issues that are of central importance in term structure analysis. One is the modeling and estimation of the current term structure of spot rates. The second is the modeling and estimation of the dynamics of the term structure. These two issues have been addressed independently in the literature. The methods that have been proposed assume a sufficiently complete price data set and ...

In this paper we calibrate the Vasicek interest rate model under the risk neutral measure by learning the model parameters using Gaussian processes for machine learning regression. The calibration is done by maximizing the likelihood of zero coupon bond log prices, using mean and covariance functions computed analytically, as well as likelihood derivatives with respect to the parameters. The ma...

These notes provide an overview of singleand multi-factor models of the short rate. We will begin with a generic single-factor model where the dynamics of rt under the physical measure, P , are given. Following the approach of Vasicek, we then derive the PDE that must be satisfied by derivative security prices. We then use the martingale approach to give an alternative (and familiar) expression...

These notes provide an overview of singleand multi-factor models of the short rate. We will begin with a generic single-factor model where the dynamics of rt under the physical measure, P , are given. Following the approach of Vasicek, we then derive the PDE that must be satisfied by derivative security prices. We then use the martingale approach to give an alternative (and familiar) expression...