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section9 (1) - Continuous Time Finance Notes Spring 2004...

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Continuous Time Finance Notes, Spring 2004 – Section 9, March 31, 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 the third segment of the course: a discussion of the volatility skew/smile – its sources and consequences. We start with a general discussion of the phenomenon, and its relation to “fat tails,” following Hull’s Chapter 15. There are three main approaches to modeling the volatility skew/smile quantitatively: (a) local vol models, (b) jump-diffusion models, and (c) stochastic vol models. We have time for just a very brief introduction to each; you’ll learn much more in the course Case Studies. Today we focus on local vol models, explaining why it is “in principle” easy but in practice quite difficult to extract a local volatility function from market data on calls. The heart of the matter is “Dupire’s equation.” ***************** The implied volatility skew/smile. Consider call options on an underlying which earns no dividends. We assume the interest rate r is constant. We write C ( S 0 , K, T ) = market price of a call option with strike K and maturity T where S 0 is the spot price and the current time is t = 0. Now define C BS ( S 0 , K, σ, T ) = Black-Scholes value of the call, using constant volatility σ . Then the implied volatility σ I ( S 0 , K, T ) is defined by the equation C ( S 0 , K, T ) = C BS ( S 0 , K, σ I ( S 0 , K, T ) , T ) . Since the Black-Scholes value of a call is a monotone function of σ , the implied volatility is well-defined. If the constant-vol Black-Scholes model were “correct,” i.e. if it gave the actual market values of call options, then σ I would be constant, independent of S 0 , K , and T . In fact however σ I is not constant. The “volatility skew/smile” refers to its dependence on K . Typically, for equities, σ I decreases as K increases. For foreign exchange the typical behavior is different: σ I is smallest when K S 0 so its graph looks like a “smile.” The definition of implied vol depends on the choice of payoff. But if we used puts rather than calls we would get the same implied vols, by put-call parity. (We use here the fact that put-call parity is model-independent!). Hull discusses at length how the skew/smile reflects “fat tails” in the risk-neutral probability distribution. No need to repeat his discussion here. But note what underlies it. Since prices are linear, the value of any option with maturity T is a linear function of its payoff at time T . We recognize from the relation option value = e - rT E RN [payoff] 1
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that (for a complete market model) this linear relation is expressed by integration against the risk-neutral probability density times e - rT . In other words, if the payoff is f ( S T ) and the risk-neutral probability density of S T at time T (given price S 0 at time 0) is p ( ξ, T ; S 0 ) then option value = e - rT -∞ f ( ξ ) p ( ξ, T ; S 0 ) dξ.
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