Derivatives NYU Lecture 6

# Derivatives NYU Lecture 6 - Dynamic Assets Option Pricing 6...

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11/05/2009 Sebastien Galy, NYU Poly Tech Institute & BNP Paribas – [email protected] 1 Dynamic Assets & Option Pricing 6 Sebastien Galy

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Overview 1. In discrete time – Review of fundamental concepts 2. In discrete time – Example of trading rule 3. In continuous time – Sebastien Galy, NYU Poly Tech Institute & BNP Paribas – [email protected]
11/05/2009 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

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Fundamental Theorem of Risk Neutral Pricing Assume that the market admits no arbitrage portfolios and that there exists a riskless lending/borrowing at rate Rf. Then, there exists a probability measure (risk neutral) defined on the set of feasible market outcomes,{1,2,…M}, such that the value of any security is equal to the expected value of its future cash flows discounted at the riskless lending rate. S. Galy
Definition: Complete Markets S. Galy M Rank : condition rank matrix the to equivalent is This . P any for solution a has ] ,..., 1 [ , ) 1 ( that implies s completnes market the us assumes.Th it M} [1,. .., j state each to associated flow cash a has which ) n , , (n securities traded of portfolio a exists there flow, cash any for if, complete be to said is states M h market wit securities A M i N 1 = N j ij i n M j S n π Thus if a portfolio can be replicated uniquely then there is one price and complete markets exist. Inversely, the uniqueness of asset prices determines a complete market

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Application: Arrow Debreu (each state has a price) framework in linear programming S. Galy [] [] [] M j S n S n M j S n M j S n S n N ij i N i i N ij i N ij i N i i ,.., 1 0 ) 1 ( and 0 ) 0 ( or ) 2 ( ,.., 1 , 0 ) 1 ( .... ,.., 1 , 0 ) 1 ( payoff expected positive weakly a create can I s), long/short (e.g. 0 ) 0 ( portfolio costless a With ) 1 ( 1 i 1 i 1 i 1 i 1 i < > = = = = = = Linear programming and optimization is used from Asset Allocation (matching factors to stock returns), to utility theory (corner solutions), to problems of allocation of capital under linear constraints…Excel has a macro solver that is particularly useful.
Application: Arrow Debreu (each state has a price) framework in linear programming S. Galy 0 ) 0 ( and M] [1,. ., j 0 ) 0 ( ) 1 ( that constraint the unde ) 0 ( Minimize 1 i 1 i 1 i 1 i ,.... 1 , = Φ = = = = = N i i N i i N ij i N i i n S n S n S n S n N i i Linear programming and optimization is used from Asset Allocation (matching factors to stock returns), to

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Microsoft Excel – Help File Solver is part of a suite of commands sometimes called what-if analysis tools. With Solver, you can find an optimal value for a formula in one cell— called the target cell— on a worksheet. Solver works with a group of cells that are related, either directly or indirectly, to the formula in the target cell. Solver adjusts the values in the changing
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Derivatives NYU Lecture 6 - Dynamic Assets Option Pricing 6...

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