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section4 (1)

# section4 (1) - Continuous Time Finance Notes Spring 2004...

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Continuous Time Finance Notes, Spring 2004 – Section 4, Feb. 18, 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 begins our discussion of interest rate models. You should already have some familiarity with basic terminology (e.g. bond prices, instantaneous forward rates), instru- ments (e.g. caps and captions, swaps and swaptions), and their valuation using Black’s formula. If you need to review these topics, Hull is excellent; see also my Derivative Securi- ties lecture notes, sections 10 and 11 (warning: the notation there is slightly diﬀerent from the present notes). Today we’ll get started by discussing relatively simple one-factor models of the short rate (Vasicek, Cox-Ingersoll-Ross). Next time we’ll discuss the Hull-White (also known as modiﬁed Vasicek) model, a richer one-factor short rate model that can be cali- brated to an arbitrary initial yield curve. After that we’ll turn to the Heath-Jarrow-Morton theory. I’ve adopted Baxter & Rennie’s notation. However my pedagogical strategy is diﬀerent from theirs. I’m starting with short-rate models, because they’re easier. Baxter & Rennie start with HJM, specializing afterward to short-rate models, because it’s more eﬃcient. The Vasicek model is by now quite standard; treatments can be found (with slightly diﬀerent organization and viewpoints) in Brigo & Mercurio, Lamberton & Lapeyre, and Avellaneda & Laurence. For a good survey of the “big picture,” I recommend reading chapter 1 of R. Rebonato, Modern pricing of interest-rate derivatives: the Libor market model and beyond (2nd edition, 2002, on reserve in the CIMS library). ***************** Basic terminology. The time-value of money is expressed by the discount factor P ( t, T )=va lueatt ime t of a dollar received at time T. This is, by its very deﬁnition, the price at time t of a zero-coupon (risk-free) bond which pays one dollar at time T . If interest rates are stochastic then P ( t, T )w i l lnotbeknown until time t ; its evolution in t is random, and can be described by an SDE. Note however that P ( t, T ) is a function of two variables, the initiation time t and the maturity time T . The dependence on T reﬂects the term structure of interest rates; we expect P ( t, T ) to be relatively smooth as a function of T , since interest is being averaged over the time interval [ t, T ]. We usually take the convention that the present time is t =0–thuswhatis observable now is P (0 ,T ) for all T> 0. There are several equivalent ways to represent the time-value of money. The yield R ( t, T ) is deﬁned by P ( t, T )= e - R ( t,T )( T - t ) ; it is the unique constant interest rate that would have the same eﬀect as P ( t, T ) under continuous compounding. Evidently R ( t, T - log P ( t, T ) T - t .

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section4 (1) - Continuous Time Finance Notes Spring 2004...

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