Fixed Income Securities, 4th Edition

CHAPTER 9The Vasicek and Gauss+ Models

This chapter, the last of the three on term structure models, presents the well‐known Vasicek and Gauss+ models. The Vasicek model started the literature on short‐term rate models;¹ remains an extremely good starting point for learning about these models; and can still be used in some applied contexts. The Gauss+ model has proven very popular for proprietary trading, for both relative value and macro‐style trading. The presentation of this model here is directed toward determined readers who would like to implement a term structure model for their own trading purposes.

9.1 THE VASICEK MODEL

The Vasicek model assumes mean reversion to set the expected path of the short‐term rate. When below its long‐term value, the short‐term rate is expected to increase; when above its long‐term value, the short‐term rate is expected to decrease. Mathematically, the risk‐neutral dynamics for the short‐term rate, , are given by,

(9.1)

In words, the instantaneous change in the short‐term rate, , is determined by a trend or drift plus a random fluctuation or shock. The drift is equal to the parameter of mean reversion, , times the distance between the long‐run ...