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section8 (1)

# section8 (1) - Continuous Time Finance Notes Spring 2004...

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Continuous Time Finance Notes, Spring 2004 – Section 8, March 24, 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 by wrapping up our discussion of HJM – discussing its strengths and weaknesses. Then we turn to the “Libor Market Model” – which though relatively new is rapidly becoming the method of choice for many purposes. Our discussion of the Libor Market Model follows Hull’s treatment (Section 24.3 of the 5th edition). ***************** Some comments on the HJM approach to interest rate modelling: HJM is a “framework” not a “model” . When modeling equity-based derivatives, our usual framework is to assume the underlying solves a diﬀusion process ds = μ ( s, t ) dt + σ ( s, t ) dw . The market is complete for any choice of the functions μ ( s, t ) and σ ( s, t ) (with some minor restrictions, for example σ ( s, t ) > 0). But what to choose for μ and σ ? The choice of μ is of course irrelevant for option pricing, because under the risk-neutral measure the underlying satisﬁes ds = r dt + σ ( s, t ) dw . But the choice of σ is crucial. We commonly choose σ ( s, t ) = σ 0 s with σ 0 constant, i.e. we commonly assume the underlying has lognor- mal dynamics. (This is not the only possibility; in a week or two we’ll discuss “local vol” models, i.e. the idea of using the skew/smile of implied volatilities to infer an appropriate choice of σ ( s, t ).) The situation with HJM is analogous. For one-factor HJM, the framework is d t f ( t, T ) = α ( t, T ) dt + σ ( t, T ) dw . Working under the risk-neutral measure (i.e. assuming w is a Brownian motion in the risk-neutral measure) the drift α is completely speciﬁed by σ . The choice of σ is again crucial. It is no longer so obvious how to choose it; we saw how to choose σ ( t, T ) to get either Ho-Lee or Hull-White. But many other choices are possible. Calibration to data is not easy, because when σ is not deterministic (e.g. if it depends explicitly on f ( t, T )) we have no exact pricing formulas. It is natural to suppose σ is a function of f as well as t and T . Indeed, if σ is a deterministic function of t and T then f ( t, T ) is Gaussian, so there is a positive probability that it is negative. We tolerated this in Hull-White, and one often tolerates it in HJM for the same reason – namely analytical tractability of the model. However a model that permits negative interest rates cannot be the last word. Amongst short-rate models the simplest way to keep the short rate positive is to use a state-dependent volatility: for example the Cox-Ingersoll-Ross model assumes dr = ( θ ( t ) - ar ) dt + σ 0 r dw (compare this to Hull-White: dr = ( θ ( t ) - ar ) dt + σ 0 dw ). Our discussion of HJM (in particular: our explanation how the volatility determines the drift) did not assume σ was deterministic. It applies with no change if, for example, σ has the form

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section8 (1) - Continuous Time Finance Notes Spring 2004...

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