Derivatives_NYU_Lecture_5

Derivatives_NYU_Lecture_5 - 06/04/10 Sebastien Galy, NYU...

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Unformatted text preview: 06/04/10 Sebastien Galy, NYU Poly Tech Institute & BNP Paribas sgaly@poly.edu 1 Dynamic Assets & Option Pricing 5 Sebastien Galy Overview 1. In continuous time Backing out the Risk Neutral Distribution 2. In continuous time example of algorithmic trading 3. In continuous time pricing knock outs Sebastien Galy, NYU Poly Tech Institute & BNP Paribas sgaly@poly.edu 06/04/10 S. Galy 3 Continuous PDE Discrete CRR Binomial Hull and White Discretise/Grid Utility Theory MRS discounting Feynman Kac BSM Analytical Convergence Monte Carlo Sim Replication Complete Markets =>No Arbitrage=> Unique state price Incomplete Markets Risk Neutral Transformation Girsanov Theorem Change distribution to move to risk free drift 1. In continuous Time: Backing out the Risk Neutral Distribution 06/04/10 4 Sebastien Galy, NYU Poly Tech Institute & BNP Paribas sgaly@poly.edu Implied Risk Neutral Distribution Q/Q 06/04/10 5 Sebastien Galy, NYU Poly Tech Institute & BNP Paribas sgaly@poly.edu Implied Risk Neutral Distribution The Martingale Approach and the Implied Risk Neutral Distribution When the underlying process is expressed in its risk neutral (or Martingale) form, calculating the price of the option is a simple matter. For example, say that we want to price a Call option with an exercise at time T and a strike of K. Further, assume that the risk neutral probability of the underlying at time is given by . In this case, the price of the option at any time t price to its exercise is: { } ( ) * ( ) * * ( ) * * ( , ) ( ( ) ,0) ( , ) ( , ) ( , ) 1 ( ) f f f R T t t t R T t K K R T t K C S t e E Max S T K e Sf S T dS K f S T dS e Sf S T dS K F K-- -- -- =- =- =-- Implied Risk Neutral Distribution If the risk neutral distribution is known (analytically or empirically) a price can be found by applying the appropriate mathematical techniques or by using Monte Carlo simulation (consisting in simulating the risk neutral distribution and calculating the average simulated values. Using this equation, we also note that, its first and second derivatives with respect to option strike is: ( ) * * * * ( , ) ( , ) 1 ( , ) ( , ) 1 ( , ) f R T t t t C e S t Kf K T F K T Kf K T F K T K- = -- + + = - + While the second derivative yields: 2 ( ) * 2 ( , ) f R T t t C e f K T K- = Simple introduction from the Brazilian Central Bank http://www.bcb.gov.br/ingles/estabilidade/2002_nov/ref200201c62i.pdf Stephen Figlewski, NYU The Risk Neutral Probability Distribution for the U.S. Stock Market Stephen Figlewski Professor of Finance New York University Stephen Figlewski, NYU The Risk Neutral Probability Distribution for the S&P 500 Index This project develops a methodology for estimating a well-behaved risk neutral probability distribution for the U.S. stock market from S&P 500 index options. We describe the procedure and provide several examples of interesting results that can...
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Derivatives_NYU_Lecture_5 - 06/04/10 Sebastien Galy, NYU...

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