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CS757 Computational Finance Project No. CS757.2003Win03-25

CS757 Computational Finance Project No. CS757.2003Win03-25. Pricing Barrier Options using Binomial Trees Gong Chen Department of Computer Science University of Manitoba 1 Instructor: Dr.Ruppa K. Thulasiram. 1 gongchen@cs.umanitoba.ca. Outline. Introduction and Motivation Background

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CS757 Computational Finance Project No. CS757.2003Win03-25

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  1. CS757 Computational FinanceProject No. CS757.2003Win03-25 Pricing Barrier Options using Binomial Trees Gong Chen Department of Computer Science University of Manitoba1 Instructor: Dr.Ruppa K. Thulasiram 1gongchen@cs.umanitoba.ca

  2. Outline • Introduction and Motivation • Background • Specific Problem Statement • Solution Strategy and Implementation • Results and Discussion • Conclusion and Future Work

  3. Introduction and Motivation • Barrier options are cheaper than regular options. They have become very popular in recent years in over-the-counter market. • Barrier options cannot be priced analytically except in some special cases. • A barrier option is very similar to standard call and put options.

  4. Background • Barrier options are similar to standard options except that they are extinguished or activated when the underlying asset price reaches a predetermined barrier or boundary price • A knock-out option will expire early if the barrier price is reached whereas a knock-in option will come into existence if the barrier price is reached. • Binomial model S0 u p S0 1-p S0 d

  5. Background ctn’d Binomial Tree

  6. Specific Problem Statement • The Binomial model makes certain assumptions, the most important of which for our purposes are: (1) The option is European, i.e., exercisable only at expiration; (2) The stock does not pay dividends; (3) The option expires without rebate; (4) The asset price is continuously monitored; (5) The expected return from all traded securities is the risk-free interest rate; (6) One barrier.

  7. Solution Strategy and Implementation • We investigate the value boundary conditions of the European Barrier Options. As usual, S is the asset price, K is strike price, and H is the barrier price. Based on the assumption mentioned above, we have the following : • European "In" Barrier Options • Down-and-in Call: max [0, ST – K] if for some t  T, St H • Up-and-in Call: max [0, ST – K] if for some t  T, St H • Down-and-in Put: max [0, K – ST] if for some t  T, St H • Up-and-in Put: max [0, K – ST] if for some t  T, St H

  8. Solution Strategy and Implementation ctn’d • European “Out" Barrier Options • Down-and-out call: max[0,ST – K] if for all t  T, St > H • Up-and-out call: max[0,ST – K] if for all t  T, St < H • Down-and-out put: max[0, K – ST] if for all t  T,St>H • Up-and-out put: max[0, K – ST] if for all t  T, St < H

  9. Results and Discussion In this project, the whole class of European barrier options was examined . • Down-and-In European Call Binomial Tree K=20.00 S= 25.00 H= 35.00 T=1 (years) n=5 sigma =0.2500 r= 0.1200 u=1.1183 d=0.8942 p=0.5805

  10. Results and Discussion ctn’d

  11. Results and Discussion ctn’d

  12. Results and Discussion ctn’d • Up-and-out European put Binomial Tree K=40.00 S= 30.00 H= 35.00 T=1 (years) n=5 sigma =0.2500 r= 0.1200 u=1.1183 d=0.8942 p=0.5805

  13. Results and Discussion ctn’d

  14. Conclusions and Future Work • This project focuses exclusively on one-dimensional, single European barrier options, which include eight possible types: up(down)-and-in (out) call (put) options. • when the time step increases, the execution time increases, Meanwhile, the problem with convergence arises. • When the barrier increases, the option value of the down-and-in call, down-and-in put, up-and-out call and up-and-out-put increase, after the barrier greater than one certain number, the option value converges to a constant ( > 0).

  15. Conclusions and Future Work ctn’d • When the barrier increases, the option value of down-and-out call, down-and-out put, up-and-in call and up-and-in put decreases. After the barrier increases to a certain number, the option value for these four types converges to zero. • This project can be extended to value American barrier options. • Options with more than one barriers such as 'double barriers' are similar to the single barrier options except that they have one more barrier level. • In many cases, the barrier option does not become worthless when the assert price hits the barrier and pays a fixed rebate at that point. The project can be modified to value those barrier options.

  16. Reference [1] M. Broadie, P. Glasserman, and S. Kou. A continuity correction for discrete barrier options. Mathematical Finance, October 1997. [2] Bin Gao Dong-Hyun, Stephen Figlewski. Pricing discrete barrier options with an adaptive mesh. The Journal of Derivatives, 6:33-44, Summer 1999. [3] S. Figlewski and B. Gao. The adaptive mesh model:a new approach to efficient option pricing. Journal of Financial Economics, 1999. [4] John C. Hull. Options, Futures and Other Derivatives. Prentice Hall.

  17. Reference ctn’d [5] Boyle P.P. and S.H.Lau. Bumping up against the barrier with the binomial method. Journal of Derivatives, (2):6-14, 1994. [6] Mark Rubinstein and Eric Reiner. Breaking down the barriers. RISK 4, pages 28-35, September 1991. [7] G. Thompson. Bounds on the value of barrier options with curved boundaries. Working paper.

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