This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 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 Hulls Chapter 15. There are three main approaches to modeling the volatility skew/smile quantitatively: (a) local vol models, (b) jumpdiffusion models, and (c) stochastic vol models. We have time for just a very brief introduction to each; youll 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 Dupires 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 , K, T ) = market price of a call option with strike K and maturity T where S is the spot price and the current time is t = 0. Now define C BS ( S , K, , T ) = BlackScholes value of the call, using constant volatility . Then the implied volatility I ( S , K, T ) is defined by the equation C ( S , K, T ) = C BS ( S , K, I ( S , K, T ) , T ) . Since the BlackScholes value of a call is a monotone function of , the implied volatility is welldefined. If the constantvol BlackScholes model were correct, i.e. if it gave the actual market values of call options, then I would be constant, independent of S , K , and T . In fact however I is not constant. The volatility skew/smile refers to its dependence on K . Typically, for equities, I decreases as K increases. For foreign exchange the typical behavior is different: I is smallest when K S 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 putcall parity. (We use here the fact that putcall parity is modelindependent!). Hull discusses at length how the skew/smile reflects fat tails in the riskneutral 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 that (for a complete market model) this linear relation is expressed by integration against the riskneutral probability density times e rT . In other words, if the payoff is f ( S T ) and the riskneutral probability density of S T at time T (given price S at time 0) is p ( , T ; S ) then option value = e rT Z  f ( ) p ( , T...
View
Full
Document
This note was uploaded on 01/02/2012 for the course FINANCE 347 taught by Professor Bayou during the Fall '11 term at NYU.
 Fall '11
 Bayou
 Finance

Click to edit the document details