X7Implied_trees

X7Implied_trees - Hedging, Pricing, and Predicting...

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Slide 12-1 Hedging, Pricing, and Predicting Volatility Supplementary Material
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Slide 12-2 Agenda The expected profit of a delta-hedged market maker depend on the volatility of the underlying asset. Today: estimating volatility (and more. ..), – Based on option prices - implied volatility – Based on historical data (if we get to it) hedging volatility, predicting volatility.
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Slide 12-3 Why estimate risk neutral densities? Violations of BS assumptions The constant volatility Black Scholes model will fail unser any of the following four violations of its assumptions: The ”local” (instantaneous) volatility is a function of S t . The local vola is a function of the path leading up to S t . The local vola is a function of a state-variable other than S t or the path leading up to S t . Market imperfections. Violations of the first two types leave arbitrage reasoning intact. Violations of the second type can lead to computational problems. Violations of the third type destroy the arbitrage foundations of BS. At the very least, additional assets must be included in the replicating portfolio. Violations of the fourth type will typically make it impossible to pin down option prices beyond specifying certain intervals within which the prices should lie.
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Slide 12-4 Implied volatility = the volatility of asset returns that is consistent with observed option prices and the Black-Scholes option pricing model. In principle, one can use the implied volatility from an option with an observable price to calculate the price of another option (on the same underlying asset) with an unobservable price. Instead of using historical volatility we can ask, “Given the option's observable market price, and assuming the market is using the Black- Scholes model to price options, what volatility number is the market using?” Checking the uniformity of implied volatilities across various options on the same underlying assets allows one to verify the validity of the pricing model. In practice implied volatilities of in, at, and out-of-the money options are generally different resulting in a volatility smile or a volatility skew .
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Slide 12-5 Finding the implied volatility The Black-Scholes option pricing model for European call options is where and The implied volatility must be found using an iterative procedure, for example the Solver routine in EXCEL. d ln S / K r T T 1 2 1 2 = + + ( ) ( ) δ σ σ d d T 2 1 = − σ C S,K, ,r,T, Se N d Ke N d - T -rT ( ) = ( ) ( ) σ δ δ 1 2
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Slide 12-6 Constructing an implied volatility index Idea: compute a weighted average of the implied volatilities of options with different strike prices. A simple way to do this: take the average of the implied volatilities of the two nearest-the-money calls.
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This note was uploaded on 10/25/2009 for the course 15 15.402 taught by Professor Bergman during the Fall '09 term at MIT.

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X7Implied_trees - Hedging, Pricing, and Predicting...

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