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SUMMER RESEARCH PROJECT DANIEL GUETTA with PROF. PAUL GLASSERMAN Detecting Bubbles Using Option Prices

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Bubbles

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What is a Bubble? Bubbles are often associated with a large increase in the asset price followed by a collapse when the bubble bursts. In the context of financial markets, bubbles refer to asset prices that exceed the asset's fundamental, intrinsic value possibly because those that own the asset believe that they can sell the asset at a higher price in the future.

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What is a Bubble? Asset Price Bubbles in Complete Markets, Jarrow, Protter & Shimbo, 2007 Asset Price Bubbles in Incomplete Markets, Jarrow, Protter & Shimbo, 2010

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A Very (Very, Very) Short Introduction to Financial Math

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Financial Mathematics Google Stock – 1 st January 2007 to 1 st January 2011

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Financial Mathematics First Fundamental Theorem of Asset Pricing

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Price Distributions

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The Kolmogorov Forward Equation

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Detecting Bubbles

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The Bubble Test Assumption : How to Detect an Asset Bubble, Jarrow, Kchia & Protter, March 2011

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The Bubble Test Bubble exists in the asset price S t S t is a strict local martingale Assumption : How to Detect an Asset Bubble, Jarrow, Kchia & Protter, March 2011

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Using Options to Find

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What is an Option? When time T comes along, the call option gives its owner the right, but not the obligation, to buy one unit of the financial asset at price K. StrikeMaturity

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Pricing Options

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Magic!

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The Dupire Equation + = Kolmogorov Forward Equation The Dupire Equation

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Reality Option price Maturity Strike 1 st September 2006, Options on the S&P 500

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Local Least Squares Arbitrage-free Approximation of Call Price Surfaces and Input Data Risk, Glaser and Heider, March 2010

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Local Least Squares Option price Maturity Strike 1 st September 2006, calls

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Local Least Squares Option price Maturity Strike

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Local Least Squares Option price Maturity Strike

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Local Least Squares Option price Maturity Strike

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Local Least Squares Option price Maturity Strike

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Local Least Squares Option price Maturity Strike

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Local Least Squares Option price Maturity Strike 1 st September 2006, calls

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Local Least Squares Option price Maturity Strike 1 st September 2006, calls

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The Local Volatility 2 ( K, T ) K T 1 st March 2004, calls

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The Local Volatility 2 ( K, T ) K T 2 nd July 2007, calls

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The Local Volatility 2 ( K, T ) K T 2 nd July 2007, puts

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Results

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Bubble Indicator Date

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Bubble Indicator Date VIX Index Correlation coefficient : 0.15

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Bubble Indicator Date S&P 500 Correlation coefficient : 0.01

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Concluding Remarks

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Conclusions A promising approach to implementing the bubble test. The non-parametric approach we used might have been slightly too ambitious. Fitting options prices rather than volatilities might have compounded the problem.

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Other Approaches Use some sort of spline ( Reconstructing the Unknown Volatility Function, Coleman, Li and Verma, Computation of Deterministic Volatility Surfaces, 2001. Jackson, Suli and Howison, 1999. Improved Implementation of Local Volatility and Its Application to S&P 500 Index Options, 2010.) Estimate the local volatility via the implied volatility.

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Other Approaches Assume the volatility is piecewise constant, and solve the Dupire Equation to find the best constants. ( Volatility Interpolation, Andreasen and Huge, 2011). Assume some sort of parametric pricing model (such as Heston or SABR), fit to option price data and then deduce local volatility.

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Questions

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notes

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The Implied Volatility

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