Vasicek model

Résumé

In finance, the Vasicek model is a mathematical model describing the evolution of interest rates. It is a type of one-factor short-rate model as it describes interest rate movements as driven by only one source of market risk. The model can be used in the valuation of interest rate derivatives, and has also been adapted for credit markets. It was introduced in 1977 by Oldřich Vašíček, and can be also seen as a stochastic investment model. Details The model specifies that the instantaneous interest rate follows the stochastic differential equation: :dr_t= a(b-r_t), dt + \sigma , dW_t where Wt is a Wiener process under the risk neutral framework modelling the random market risk factor, in that it models the continuous inflow of randomness into the system. The standard deviation parameter, \sigma, determines the volatility of the interest rate and in a way characterizes the amplitude of the instantaneous randomness inflow. The typical parameters

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https://en.wikipedia.org/wiki/Vasicek_model

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On Finite-Dimensional Term Structure Models

Damir Filipovic

In this paper we provide the characterization of all finite-dimensional Heath-Jarrow-Morton models that admit arbitrary initial yield curves. It is well known that affine term structure models with time-dependent coef- ficients (such as the Hull-White extension of the Vasicek short rate model) perfectly fit any initial term structure. We find that such affine models are in fact the only finite-factor term structure models with this property. We also show that there is usually an invariant singular set of initial yield curves where the affine term structure model becomes time-homogeneous. We also argue that other than functional dependent volatility structures - such as local state dependent volatility structures - cannot lead to finite-dimensional realizations. Finally, our geometric point of view is illustrated by several examples.

Princeton University2002

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Markovian Term Structure Models in Discrete Time

Damir Filipovic

In this article we discuss Markovian term structure models in discrete time and with continuous state space. More precisely we are concerned with the structural properties of such models if one has the Markov property for a part of the forward curve. We investigate the two cases where these parts are either a true subset of the forward curve, including the short rate, or the entire forward curve. For the former case we give a sufficient condition for the term structure model to be affine. For the latter case we provide a version of the HJM [6] drift condition (see also [7]). Under a Gaussian assumption an HJM-Musiela [10] type equation is derived

2002

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We give a complete characterization of affine term structure models based on a general nonnegative Markov short rate process. This applies to the classical CIR model but includes as well short rate processes with jumps.We provide a link to the theory of branching processes and show how CBI-processes naturally enter the field of term structure modelling. Using Markov semigroup theory we exploit the full structure behind an affine term structure model and provide a deeper understanding of some well-known properties of the CIR model. Based on these fundamental results we construct a new short rate model with jumps, which extends the CIR model and still gives closed form expressions for bond options.

Springer Verlag2001

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