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Unformatted text preview: Derivative Securities Fall 2007 Section 6 Notes by Robert V. Kohn, extended and improved by Steve Allen. Courant Institute of Mathematical Sciences. Stochastic differential equations and the Black-Scholes PDE . We derived the Black- Scholes formula by using no-arbitrage-based (risk-neutral) valuation in a discrete-time, bino- mial tree setting, then passing to a continuum limit. We started that way because binomial trees are very explicit and transparent. However the power of the discrete framework as a conceptual tool is rather limited. Therefore we now begin developing the more powerful continuous-time framework, via the Ito calculus and the Black-Scholes differential equation. This material is discussed in many places. Baxter & Rennie emphasize risk-neutral expecta- tion, avoiding almost completely the discussion of PDEs. The student guide by Wilmott, Howison, & Dewynne takes almost the opposite approach: it emphasizes PDEs, avoiding almost completely the discussion of risk-neutral expectation. Neftcis book provides a good introduction to Brownian motion, the Ito calculus, stochastic differential equations, and their relation to option pricing, at a level that should be accessible to students in this class. (Students taking Stochastic Calculus will learn this material and much more over the course of the semester.) A brief introduction to stochastic calculus (similar in spirit to whats here, but going somewhat deeper, with exercises and many more examples) can be found at the top of my Spring 2003 PDE for Finance course notes (on my web page). ******************** Why work in continuous time? . Our discrete-time approach has the advantage of being very clear and explicit. However there is a different approach, based on Taylor expansion, the Ito calculus, and the Black-Scholes differential equation. It has its own advantages: Passing to the continuous time limit is clearly legitimate for describing the stock price process. But is it legitimate for describing the value of the option, as determined by arbitrage? This is less clear, since a continuous-time hedging strategy is unattainable in practice. In what sense can we approximately replicate the option by trading at discrete times? The Black-Scholes differential equation will help us answer these questions. We assumed the stock price process (or the forward price process) was lognormal, then approximated it using a binomial tree, then applied arbitrage pricing to this tree. But what if we had approximated the stock price process by a trinomial tree? Then arbitrage pricing wouldnt have worked, since the one-period trinomial market isnt complete. It should make you nervous that we relied so heavily on the use of a particular approximation....
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- Fall '11