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Bounding Option Prices Using Semidefinite Programming S ACHIN J AYASWAL Department of Management Sciences University of Waterloo, Canada Project work for MSCI 700 Fall 2007 Semidefinite Programming: Models, Algorithms & Computation Course Instructor DR. M. F. ANJOS

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Introduction 2 Call Option: An agreement that gives the holder the right to buy the underlying by a certain date for a certain price. European vs. American call option

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Definitions 3 T: Specific time when the underlying can be purchased (Maturity) K: Specific price at which the underlying can be purchased (Strike Price) S t : Price of the underlying (stock) at time t r: Risk-free interest rate α: Expected return on the underlying σ: Volatility in the price of the underlying C: Price of call option

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Call Option Payoff 4

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Call Option Pricing 5 Call Option Pricing – An interesting and a challenging problem in finance Black-Scholes (1973) – Asset price follows geometric Brownian motion Stock prices observed in the market often do not satisfy this assumption Can we price the option without assuming any specific distribution for stock price?

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Bounds on Option Price 6

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Dual Problems 7

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Propositions 8

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SDP Formulation: Upper Bound 9

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SDP Formulation: Lower Bound 10

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Comparison with Black-Scholes 11 Black-Scholes assume stock prices follow geometric Brownian motion where N(·) is the cumulative distribution of a normally distributed random variable

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Computational Experiments 12

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Computational Experiments 13

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Computational Experiments 14

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Computational Experiments 15

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Computational Experiments 16

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Cutting Plane Method for Solving[UB_SDP] 17 Let (1) Observations: where represents the polyhedral set corresponding to the linear constraints of Adding constraint (1) tightens the bound. Relaxing SDP constraint on X, Z makes the problems LP

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Cutting Plane Algorithm 18

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Performance of the Cutting Plane Algorithm 19 Number of cuts required by the algorithm

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Conclusions 20 The SDPs produce good bounds on the option price in absence of the known distribution of the stock price. The approach may be used in pricing complex financial derivatives for which closed-form formula is not possible (Boyle and Lin, 1997).

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