Introduction to Barrier Options John A. Dobelman, MBPM, PhD October 5, 2006 PROS Revenue Management.
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Transcript of Introduction to Barrier Options John A. Dobelman, MBPM, PhD October 5, 2006 PROS Revenue Management.
Introduction to Barrier Options
John A. Dobelman, MBPM, PhD
October 5, 2006
PROS 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
( , , , , )V V S X T r
4
X=100
5
Vanilla Option Payoffs
6
Vanilla Option Value
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
( , , , , , )V V S X B T r
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
X=100
B=110
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Barrier Options Characteristics
• Cheaper than Vanillas
• Widely-traded (since the 1960’s)
• Harder to value
• Flexible/Many Varieties
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,t2)
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Option Valuation - VanillasAnalytic – 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
Analytic Valuation
1 2( ) ( )rtC S d Xe d
2 2
1 2 1
1 1log( ) ( ) log( ) ( )
2 2;
S SX Xr t r t
d d d tt t
2 1( ) ( )rtP Xe d S d
13
Merton’s 1973 Valuation
1 2 3 4
212
( ) ( ) ( ) ( ) / 2
( , ) ( , )
D O rt rt
BS BS
S SC S d Xe d B d Xe dB B
SV S t V B S tB
2 2123
2124
2ln ln ( ) / 1 2
2ln ln ( ) /
SBd r t t rX X
SBd r t tX X
KO KIV V V
14
Toward Optimality: Reiner & Rubinstein (91), Rich (94), Ritchkin (94), Haug (97,99,00)
1 1
2 2
2( 1) 2
1 1
2( 1) 2
2 2
2
2 2
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( )
t rt
t rt
m mt rt
m mt rt
mrt
m
A Se x Xe x t
B Se x Xe x t
B BC Se y Xe y tS S
B BD Se y Xe y tS S
BE Xe x t y tS
BF X zS
( 2 )m
B z tS
15
Toward Optimality (CONT’D)
2122
2 2
1 2
2
1 2
2( )
ln( ) ln( )(1 ) (1 )
ln( ) ln( )(1 ) (1 )
r rm m
S SX Bx m t x m tt t
B BSX Sy m t y m tt t
ln( )BSz tt
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
0
!(1 ) max 0, /
!( )!
nj n j j n j n
j
nC p p u d S X r
j n j
Very Accurate – Except for Barriers!
22
Other MethodsOscillation 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 ApplicationFDX 108.75 (9/28/06)
Jan'08 Put (477 Days to expire)
Vanilla Put Knock-in PutWFXMT Ja08 100 put: 10.00 B=90, X=100: 7.65WFXMR 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 LossTotal $180,000 vs. $155,600
$46,000 vs. 44,800 Cost to insure $180,000 LossTotal $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 losstotal $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.
Introduction to Barrier Options
John A. Dobelman, MBPM, PhD October 5, 2006
PROS Revenue Management
John A. Dobelman October 5, 2006 PROS Revenue Management