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Bounding Option Prices Using Semidefinite Programming

Bounding Option Prices Using Semidefinite Programming. Sachin Jayaswal Department of Management Sciences University of Waterloo, Canada Project w ork for MSCI 700 Fall 2007 Semidefinite Programming: Models, Algorithms & Computation Course Instructor DR. M. F. ANJOS. Introduction.

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Bounding Option Prices Using Semidefinite Programming

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  1. Bounding Option Prices Using Semidefinite Programming Sachin Jayaswal 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

  2. Introduction • 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

  3. Definitions • T: Specific time when the underlying can be purchased (Maturity) • K: Specific price at which the underlying can be purchased (Strike Price) • St: 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

  4. Call Option Payoff

  5. Call Option Pricing • 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?

  6. Bounds on Option Price

  7. Dual Problems

  8. Propositions

  9. SDP Formulation: Upper Bound

  10. SDP Formulation: Lower Bound

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

  12. Computational Experiments

  13. Computational Experiments

  14. Computational Experiments

  15. Computational Experiments

  16. Computational Experiments

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

  18. Cutting Plane Algorithm

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

  20. Conclusions • 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).

  21. Thank You!

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