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

1. Introduction to Barrier OptionsJohn A. Dobelman, MBPM, PhDOctober 5, 2006PROS Revenue Management

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

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. o X=100

5. Vanilla Option Payoffs

6. Vanilla Option Value

7. Introduction 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). • 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

8. B=110 o o X=100

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

10. Barrier Options - Varieties • 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,t2) • 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

11. Option Valuation - Vanillas Numerical - Americans and Exotics • PDE Approach (Schwartz 77) • Binomial (Sharpe 1978, CRR 1979) • Trinomial Model • Monte Carlo • Multiple Models Today 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)

12. Analytic Valuation

13. Merton’s 1973 Valuation

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

15. Toward Optimality (CONT’D)

16. Toward Optimality (CONT’D)

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

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

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: 10.00 B=90, X=100: 7.65 WFXMR Ja08 90 put: 4.60 B=90, X=90: 4.48 $1,000,000 FDX 100 Standard option contracts to hedge $100,000 vs. 75,600 Cost to insure $80,000 Loss Total $180,000 vs. $155,600 $46,000 vs. 44,800 Cost to insure $180,000 Loss Total $226,000 vs. $224,800

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 $39,200 Cost to Insure $20,000 loss total $59,200 (Much cheaper) Cheaper yet with Barriers but what if OTM? Cheapest with Self-Insurance.

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=rf

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. 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 OptionsJohn A. Dobelman, MBPM, PhDOctober 5, 2006PROS Revenue Management

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