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

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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:
453
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
454
we get
455
Since the increments of Brownian motion, dB(t, ω), are independent and Normal, x(s, ω) is conditionally Normal (conditioned on x(t, ω)):
456
Moreover,457x(s, ω)ds is also conditionally normal:1
458
where
459
where we have used the following property of Ito differentials:
460

NORMAL DYNAMICS

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

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