Introduction to Barrier Options

John A. Dobelman, MBPM, PhD

October 5, 2006

PROS Revenue Management

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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

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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

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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

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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

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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

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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

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Toward Optimality (CONT’D)

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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

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!

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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

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Applications and Challenges

• Hedging Application

• Option Premium Revenue Program

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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

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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.

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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- Return=rf

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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-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

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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