smile-lecture1 - E4718 Spring 2007 Derman Lecture...

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E4718 Spring 2007: Derman: Lecture 1:Introduction to the Smile; The Principles of Valuation Page 1 of 25 1/28/08 Lecture1.courseworks.2008.fm Lecture 1: Introduction to the Smile; The Principles of Valuation
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E4718 Spring 2007: Derman: Lecture 1:Introduction to the Smile; The Principles of Valuation Page 2 of 25 1/28/08 Lecture1.courseworks.2008.fm 1.1 Introduction According to classic theory, the Black- Scholes implied volatility of an option should be independent of its strike and expiration. Plotted as a surface, it should be flat, as shown at right. Prior to the stock market crash of October 1987, the volatility surface of index options was indeed fairly flat. Since the crash, the volatility surface of index options has become skewed. Referred to as the volatility smile, the surface changes over time. Its level at any instant is a varying function of strike and expiration, as shown at left. The smile phenomenon has spread to stock options, interest-rate options, currency options, and almost ever other volatility mar- ket. Since the Black-Scholes model cannot account for the smile, trading desks have begun to use more complex models to value and hedge their options. After 15 years, there is still no overwhelming consensus as to the correct model. Each market has its own favorite (or two). Despite initial optimism about finding the model to replace Black- Scholes, we are still in many ways searching in the dark. The volatility surface according to Black-Scholes The volatility surface according to S&P options markets
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E4718 Spring 2007: Derman: Lecture 1:Introduction to the Smile; The Principles of Valuation Page 3 of 25 1/28/08 Lecture1.courseworks.2008.fm This first class is very different in style from successive classes; it sets up the general methodology of financial modeling, mostly qualitatively. Subsequent classes will be predominantly quantitative (and less long-winded …). Aim of the Course This isn’t a course about mathematics, calculus, differential equations or sto- chastic calculus, though it does use all of them. Much of the time the approach is going to be mathematical, but not extremely rigorous. I want to develop intu- ition about models, not just methods of solution. No assumptions behind finan- cial models are genuinely true, and no financial models are really correct, so it’s very important to understand what you’re doing and why. This is a course about several themes: 1. Understanding the practical use of the Black-Scholes-Merton model. There’s more to it than just the equation and its solution. 2. The theoretical and practical limitations of the model. 3. The extensions of the model to accommodate/explain the volatility smile. 4. Understanding the consequences of these extensions. It’s easy to make up new models of increasingly greater complexity, but we want to under- stand whether they can describe the world and what they lead to. First we’ll begin with a brief look at the implied volatility smile for equity index options, a phenomenon inconsistent with the Black-Scholes model.
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