section5 (1)

# section5 (1) - Continuous Time Finance Notes Spring 2004...

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Continuous Time Finance Notes, Spring 2004 – Section 5, Feb. 25, 2004 Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. For use in connec- tion with the NYU course Continuous Time Finance. This section discusses the Hull-White model. The continuous-time analysis is not much more diﬃcult than Vasicek – everything is still quite explicit. The basic paper is Rev. Fin. Stud. 3, no. 4 (1990) 573-592, downloadable via JSTOR; my treatment is much simpler though because I keep the parameters a and σ constant rather than letting them (as well as θ ) vary with time. The real importance of Hull-White is that while it’s rich enough to match any forward curve, it’s also simple enough to be approximated by a (recombining, trinomial) tree. This topic is covered very clearly in Sections 23.11-23.12 of Hull (5th edition), so I won’t cover it separately in these notes. ***************** The Hull-White model . Also sometimes known as “extended Vasicek,” this model as- sumes that the risk-neutral process for the short rate has the form dr =( θ ( t ) - ar ) dt + σdw (1) where a and σ are constant but θ is a function of t . (Actually the 1990 paper by Hull and White also considers taking a = a ( t )and σ = σ ( t ).) We’ll show that (a) for a given choice of θ ( t ), the situation is a lot like Vasicek; (b) there is a unique choice of θ

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section5 (1) - Continuous Time Finance Notes Spring 2004...

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