Interest Rate Swaps and Their Derivatives: A Practitioner's Guide

APPENDIX B

Mean-Reverting Processes

The process equation for the Hull-White, BDT/BK models in Chapter 8, and the Ho-Lee model in Chapter 10 can be expressed as a generalized Ornstein-Uhlenbeck process:

where for the Hull-White/Ho-Lee models x(t, ω) = r(t, ω), while for the BDT/BK models x(t,ω) = ln r (t,ω). Multiplying both sides by the integrating factor

we get

Since the increments of Brownian motion, dB(t, ω), are independent and Normal, x(s, ω) is conditionally Normal (conditioned on x(t, ω)):

Moreover,x(s, ω)ds is also conditionally normal:¹

where

where we have used the following property of Ito differentials:

NORMAL DYNAMICS

For the Vasicek/Hull-White/Ho-Lee models, r (t, ω) = ...