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

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

• Introduction
• Valuation of Vanillas
• Valuation of Barrier Options
• Application

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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

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o
X100
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Vanilla Option Payoffs
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Vanilla Option Value
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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 lt 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

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B110
o
o
X100
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Barrier Options Characteristics

• Cheaper than Vanillas
• Widely-traded (since the 1960s)
• Harder to value
• Flexible/Many Varieties

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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

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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 (gt800,000 vs. 39,100)

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Analytic Valuation
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Mertons 1973 Valuation
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Toward Optimality Reiner Rubinstein (91),
Rich (94), Ritchkin (94), Haug (97,99,00)
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Toward Optimality (CONTD)
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Toward Optimality (CONTD)
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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

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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

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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

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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

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CRR Models
Very Accurate Except for Barriers!
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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
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Applications and Challenges

• Hedging Application
• Option Premium Revenue Program

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Simple Hedging Application
FDX 108.75 (9/28/06) Jan'08 Put (477 Days to
expire) Vanilla Put Knock-in Put WFXMT Ja08
100 put 10.00 B90, X100 7.65 WFXMR Ja08 90
put 4.60 B90, X90 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
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Try with SPX Options

• 1,000,000 FDX 8 Standard SPX options when
  SPX1325
• 8k 1,060,000 at 1325 and 1,040,000 at 1300
• Dec07 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.

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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- Returnrf

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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-derivat
  ives.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

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References (CONTD)

• 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, 241563.1590,
  2000.

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Introduction to Barrier OptionsJohn A.
Dobelman, MBPM, PhD October 5, 2006PROS
Revenue Management
30

John A. Dobelman October 5, 2006
PROS Revenue Management