1. 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

2. Outline Introduction and MotivationIntroduction and Motivation BackgroundBackground Specific Problem StatementSpecific Problem Statement Solution Strategy and ImplementationSolution Strategy and Implementation Results and DiscussionResults and Discussion Conclusion and Future WorkConclusion 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 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.Barrier options cannot be priced analytically except in some special cases. A barrier option is very similar to standard call and put options.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 priceBarrier 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.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 modelBinomial model S0S0 p 1-p S 0 u S 0 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: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 :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 European "In" Barrier Options –Down-and-in Call: max [0, S T – K] if for some t  T, S t  H –Up-and-in Call: max [0, S T – K] if for some t  T, S t  H –Down-and-in Put: max [0, K – S T ] if for some t  T, S t  H –Up-and-in Put:max [0, K – S T ] if for some t  T, S t  H

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

9. Results and Discussion In this project, the whole class of European barrier options was examined. 1.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

12. 2.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.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 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).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.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.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.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.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.