1. Introduction to Barrier Options John A. Dobelman, MBPM, PhD October 5, 2006 PROS Revenue Management

2. 2 Overview Introduction Valuation of Vanillas Valuation of Barrier Options Application

3. 3 Introduction What is an option? –Contingent Claim on cash or underlying asset –Long Option – Rights –Short Option – Obligation –CALL: Right to buy underlying at price X –PUT: Right to sell underlying at price X – –ITM/OTM: Moneyness

4. 4 X=100

5. 5 Vanilla Option Payoffs

6. 6 Vanilla Option Value

7. 7 Introduction What is a Barrier Option? Barrier Options – 8 Types Knock-in-up and in down and in Knock-Out-up and out down and out A barrier option is an option whose payoff depends on whether the price of the underlying object reaches a certain barrier during a certain period of time. One barrier options specify a level of the underlying price at which the option comes into existence (“knocks in”) or ceases to exist (“knocks out”) depending on whether the level L is attained from below (“up”) or above (“down”). There are thus four possibile combinations: up-and-out, up-and-in, down-and-out and down-and-in. To be specific consider a down-and-out call on the stock with exercise time T, strike price K and a barrier at L < S0. This option is a regular call option that ceases to exist if the stock price reaches the level L (it is thus a knock-out option).

8. 8 X=100 B=110

9. 9 Barrier Options Characteristics Cheaper than Vanillas Widely-traded (since the 1960’s) Harder to value Flexible/Many Varieties

10. 10 Barrier Options - Varieties Time-varying barriers Rebates. Upon KO, not KI Double Barriers Look Barriers. St/end; if not hit, fixed strike lookback initiated Partial Time Barriers. Monitored only during windows Delayed Barrier Options. Total length time beyond barrier Reverse Barriers. KO or KI while ITM Soft/Fluffy Barriers. U/L Barrier. Knocked in/out proportionally Multi-asset Rainbow Barriers 2-factor/Outside Barrier Protected Barrier. Barrier not active [0,t 2 )

11. 11 Option Valuation - Vanillas Analytic – First Cut Black-Scholes-Merton (1973) Modified B-S European/American Black Model Quadratic Approximation (Whaley) Transformations/Parity Multiple Models Today (>800,000 vs. 39,100) Numerical - Americans and Exotics PDE Approach (Schwartz 77) Binomial (Sharpe 1978, CRR 1979) Trinomial Model Monte Carlo Multiple Models Today

12. 12 Analytic Valuation

13. 13 Merton’s 1973 Valuation

14. 14 Toward Optimality: Reiner & Rubinstein (91), Rich (94), Ritchkin (94), Haug (97,99,00)

15. 15 Toward Optimality (CONT’D)

16. 16 Toward Optimality (CONT’D)

17. 17 BSOPM Assumptions European exercise terms are used Markets are efficient (Markov, no arbitrage) No transaction costs (commission/fee) charged (no friction) Buy/Sell prices are the same (no friction) Returns are lognormally distributed (GBM) Trading in the stock is continuous, with shorting instantaneous Stock is divisible (1/100 share possible) The stock pays no dividends during the option's life Interest rates remain constant and known Volatility is constant and estimatable

18. 18 Numerical Valuation Finite Difference Methods (PDE) Monte Carlo Methods Easy to incorporate unique path-dependencies of actual options Modeling Challenges: –Price Quantization Error –Option Specification Error

19. 19 Finite Difference Methods Explicit: –Binomial and Trinomial Tree Methods –Forward solution Implicit: –Specific solutions to BSOPM PDE and other formulations –Improve convergence time and stability

20. 20 Binomial and Trinomial Tree Methods Cox, Ross, Rubinstein 1979 Wildly Successful Finance vs. Physics Approach Hedged Replicating Portfolio Arbitrary Stock Up/Dn moves Equate means to derive the lognormal Limits to the exact BSOPM Solution

21. 21 CRR Models Very Accurate – Except for Barriers!

22. 22 Other Methods Oscillation Problems when Underlying near the barrier price Trinomial and Enhanced Trees – Very Successful Adaptive Mesh New PDE Methods Monte Carlo Methods – For Integral equations

23. 23 Applications and Challenges Hedging Application Option Premium Revenue Program

24. 24 Simple Hedging Application FDX 108.75 (9/28/06) Jan'08 Put (477 Days to expire) Vanilla PutKnock-in Put WFXMT Ja08 100 put: 10.00B=90, X=100: 7.65 WFXMR Ja08 90 put: 4.60B=90, X=90: 4.48 $1,000,000 FDX 100 Standard option contracts to hedge $100,000 vs. 75,600Cost to insure $80,000 Loss Total $180,000 vs. $155,600 $46,000 vs. 44,800Cost to insure $180,000 Loss Total $226,000 vs. $224,800

25. 25 Try with SPX Options $1,000,000 FDX ~ 8 Standard SPX options when SPX=1325 8k: $1,060,000 at 1325 and $1,040,000 at 1300 Dec’07 SPX 1300 Put: $49.00 $4,900/k * 8 Contracts $39,200Cost to Insure $20,000 loss total $59,200 (Much cheaper) Cheaper yet with Barriers but what if OTM? Cheapest with Self-Insurance.

26. 26 Option Premium Revenue Program Risk of Ruin vs. Risk-Free Rate Sell Covered or Uncovered vanilla calls and puts each month to collect premium; buy back if needed at expiration. Cp. With barriers. Pr(Ruin)=1 -or- Return=r f

27. 27 References Michael J. Brennan; Eduardo S. Schwartz (1977) "The Valuation of American Put Options," The Journal of Finance, Vol. 32, No. 2 Mark Broadie, Jerome Detemple (1996) "American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods," The Review of Financial Studies, Vol. 9, No. 4. (Winter, 1996), pp. 1211-1250. Peter W. Buchen, 1996. "Pricing European Barrier Options," School of Mathematics and Statistics Research Report 96-25, Univeristy of Sydney, 13 June 1996 Cheng, Kevin, 2003. "An Overivew of Barrier Options," Global Derivatives Working Paper, Global Derivatives Inc. http://www.global-derivatives.com/options/o-types.php John C. Cox; Stephen A. Ross; Mark Rubinstein 1979. "Option pricing: A simplified approach," Journal of Financial Economics Volume 7, Issue 3, Pages 229-263 (September 1979) Derman, Emanuel; Kani, Iraj; Ergener, Deniz; Bardhan, Indrajit (1995) "Enhanced numerical methods for options with barriers," Financial Analysts Journal; Nov/Dec 1995; 51, 6; pg. 65-74

28. 28 References (CONT’D) M. Barry Goldman; Howard B. Sosin; Mary Ann Gatto. Path Dependent Options: "Buy at the Low, Sell at the High," The Journal of Finance, Vol. 34, No. 5. (Dec., 1979), pp. 1111-1127. Haug, E.G. (1999) Barrier Put-Call Transformations. Preprint available on the web at http://home.online.no/ espehaug. J.C. Hull, Options, Futures and Other Derivatives (fifth ed.), FT Prentice-Hall, Englewood Cliffs, NJ (2002) ISBN 0-13-046592-5. Shaun Levitan (2001) "Lattice Methods for Barrier Options," University of the Witwatersran Honours Project. Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring. Antoon Pelsser, 1997. "Pricing Double Barrier Options: An Analytical Approach," Tinbergen Institute Discussion Papers 97-015/2, Tinbergen Institute. L. Xua, M. Dixona, c,,, B.A. Ealesb, F.F. Caia, B.J. Reada and J.V. Healy, "Barrier option pricing: modelling with neural nets," Physica A: Statistical Mechanics and its Applications Volume 344, Issues 1-2, 1 December 2004, Pages 289-293 R. Zvan, K. R. Vetzal, and P. A. Forsyth. PDE methods for pricing barrier options. Journal of Economic Dynamics and Control, 24:1563.1590, 2000.

29. Introduction to Barrier Options John A. Dobelman, MBPM, PhD October 5, 2006 PROS Revenue Management

30. John A. Dobelman October 5, 2006 PROS Revenue Management