Motivation
New Approach
Results
Barrier Option Pricing Using
Adjusted Transition Probabilities
G. Barone-Adesi
N. Fusari
J. Theal
Swiss Finance Institute, University of Lugano
Lugano, Switzerland
Milan, X Workshop on Quantitative Finance
January 29-30th, 2009
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Outline
1
Motivation
The standard Cox-Ross-Rubinstein Binomial Tree
Previous work
2
New Approach
Adjusted Binomial (and Trinomial) Tree
3
Results
Comparison
What is new?
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
The standard Cox-Ross-Rubinstein Binomial Tree
Previous work
Outline
1
Motivation
The standard Cox-Ross-Rubinstein Binomial Tree
Previous work
2
New Approach
Adjusted Binomial (and Trinomial) Tree
3
Results
Comparison
What is new?
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
The standard Cox-Ross-Rubinstein Binomial Tree
Previous work
Convergence behaviour of the Standard Tree
A barrier option is a path-dependent option whose payoff is
determined based on whether the price of the undelying asset
reaches some pre-determined level
The convergence of the standard Cox-Ross-Rubinstein
Binomial Tree displays an erratic behaviour and a persistent
bias in pricing Barrier Options
The reason for this behaviour, as pointed out by Boyle (1994),
is that “the barrier will in general lie in between two adjacent
nodes in the lattice”
If the barrier is near to the “starting node”, the standard
method generates a relevant bias
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
The standard Cox-Ross-Rubinstein Binomial Tree
Previous work
This Is the Problem
Down-out European Call
S=100, K=100, vol=25%, rf=10%, B=90, T=1
15
Option Price (unadjusted)
Option Price (adjusted)
Analytic Price
14.5
14
Option Price
13.5
13
12.5
12
11.5
11
10.5
10
1
46
91
136 181 226 271 316 361 406 451 496 541 586 631 676 721 766 811 856 901 946 991
Number of Steps in Tree
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
The standard Cox-Ross-Rubinstein Binomial Tree
Previous work
Outline
1
Motivation
The standard Cox-Ross-Rubinstein Binomial Tree
Previous work
2
New Approach
Adjusted Binomial (and Trinomial) Tree
3
Results
Comparison
What is new?
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
The standard Cox-Ross-Rubinstein Binomial Tree
Previous work
Boyle and Lau (1994) & Ritchken (1995)
Revised Binomial-Trinomial Tree
In case of a constant barrier (H), it is easy to constrain the
time partition such that the barrier lies just above a layer of
horizontal nodes, Boyle and Lau (1994)
Ritchken (1995) proposes an extension of the standard
trinomial model. As usual, the issue is to place a layer of
nodes as near as possible (from below) to the barrier
Recall that, in the trinomial model,
up
p
and down movements
are respectively up = exp λσ T /n and
p
down = exp − λσ T /n
For single and double constant barriers it is possible to select
the stretch parameter λ such that the barrier lies exactly on
the nodes
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
The standard Cox-Ross-Rubinstein Binomial Tree
Previous work
Boyle and Lau (1994) & Ritchken (1995)
Revised Binomial-Trinomial Tree
In case of a constant barrier (H), it is easy to constrain the
time partition such that the barrier lies just above a layer of
horizontal nodes, Boyle and Lau (1994)
Ritchken (1995) proposes an extension of the standard
trinomial model. As usual, the issue is to place a layer of
nodes as near as possible (from below) to the barrier
Recall that, in the trinomial model,
up
p
and down movements
are respectively up = exp λσ T /n and
p
down = exp − λσ T /n
For single and double constant barriers it is possible to select
the stretch parameter λ such that the barrier lies exactly on
the nodes
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
The standard Cox-Ross-Rubinstein Binomial Tree
Previous work
Further Extensions
Costabile (2001) proposes a discrete time algorithm for pricing
double constant barrier options
Costabile (2002) proposes an extension of the
Cox-Ross-Rubinstein algorithm for pricing options with an
exponential boundary
Figlewsky and Gao (1999) propose an adaptive mesh model
that can be used to price constant barrier options
The problem with these procedures is that they are “ad hoc”
solutions to specific cases; we look for a general algorithm that
allows to approximate simple (single constant barrier) as well as
complex (single and double time-varying) barriers options.
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
The standard Cox-Ross-Rubinstein Binomial Tree
Previous work
Further Extensions
Costabile (2001) proposes a discrete time algorithm for pricing
double constant barrier options
Costabile (2002) proposes an extension of the
Cox-Ross-Rubinstein algorithm for pricing options with an
exponential boundary
Figlewsky and Gao (1999) propose an adaptive mesh model
that can be used to price constant barrier options
The problem with these procedures is that they are “ad hoc”
solutions to specific cases; we look for a general algorithm that
allows to approximate simple (single constant barrier) as well as
complex (single and double time-varying) barriers options.
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Adjusted Binomial (and Trinomial) Tree
Outline
1
Motivation
The standard Cox-Ross-Rubinstein Binomial Tree
Previous work
2
New Approach
Adjusted Binomial (and Trinomial) Tree
3
Results
Comparison
What is new?
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Adjusted Binomial (and Trinomial) Tree
Probability Adjustment
Baldi, Caramellino and Iovino (1999) derive a series of
approximations for the exit probability of the Brownian bridge
that can be used to price multiple and time-varying barriers
options
Recall that the log of the asset price follows a Brownian
Motion
Baldi et al. use these probabilities to improve the Monte Carlo
calculations, because with the standard procedure it is
possible that the underlying asset price breaches the barrier
without being detected
Here we use these probabilities to improve the
Cox-Ross-Rubinstein binomial tree
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Adjusted Binomial (and Trinomial) Tree
The Problem: Paths Detection
There exist Brownian bridge pahts that cross the barrier. These
paths has payoff equat to 0. The standard binomial (trinomial)
approach does not detect them.
!
&'(($)'*+!,-#.!/!
$!"#!
!
"#
!
&'(($)'*+!,-#.!0!
!
%$!"#!!
!
1-22'32!
!
%!"#!
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Adjusted Binomial (and Trinomial) Tree
Procedure
A Simple Example: Down-out call (DOC) with constant barrier (1)
CTup0+δt
p up
CT0
L
pd
ow
n
The ”adjusted
tree” is similar to
the standard CRR
model; the
difference relies on
the computation
of the value of the
option in the
nodes just above
the barrier.
CTdown
0+δt
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Adjusted Binomial (and Trinomial) Tree
Procedure
A Simple Example: Down-out call (DOC) with constant barrier (2)
CTup0+δt
δt
p up
(1
− pL
)
CT0
L
pd
ow
n
CTdown
=0
0+δt
G. Barone-Adesi, N. Fusari, J. Theal
That is, we
multiply the usual
probability of
reaching the up
point by one
minus the
conditional
probability (pLδt )
that the assets
price, in the time
interval, hits the
barrier.
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Adjusted Binomial (and Trinomial) Tree
Procedure
A Simple Example: Down-out call (DOC) with constant barrier (3)
pLδt = Exit Probability. This is the probability that the asset
price, starting from ST0 and ending in STup0+δt , hits the barrier
In this case we have
"
2
pLδt (T0 , ST0 , STup0+δt , L) = exp − 2 ln
σ δt
ST0
L
!
ln
STup0+δt
!#
L
With this probability adjustment the price of the DOC near
the barrier (when STdown
< L), at time T0 , is
0+δt
CDOC = exp(−r δt)pup (1 − pLδt )C (STup0+δt )
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Adjusted Binomial (and Trinomial) Tree
Procedure
General Case
The same procedure can be extended to the pricing of more
complex options with single and multiple time-varying barriers.
Baldi at al. provide exit probability approximations for all
kinds of time-varying barriers
In the paper we focus on:
Single/double constant barriers
Single/double exponential barriers
Single/double (time) linear barriers
This simple procedure avoids the need of repositioning the
tree “on the barrier”
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Adjusted Binomial (and Trinomial) Tree
Extension
Bermudan Option
Safe paths
Knock-out paths
These are the paths that must
be considered in order to apply
the transition probability
adjustment method.
safe paths
knock-out paths
Date t discret
Knock-out Barrier
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Adjusted Binomial (and Trinomial) Tree
Extension
Bermudan Option
S0
dS0
Date t discret
Knock-out Barrier
G. Barone-Adesi, N. Fusari, J. Theal
When the barrier falls between
two successive time steps, the
probability adjustment is given
by the cumulative normal
distribution (i.e. the
probability distribution of a
Brownian Bridge):
t − t1
µ=a+
(b − a)
t2 − t
(t − t1 )(t2 − t)
σ2 =
t2 − t1
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Comparison
What is new?
Outline
1
Motivation
The standard Cox-Ross-Rubinstein Binomial Tree
Previous work
2
New Approach
Adjusted Binomial (and Trinomial) Tree
3
Results
Comparison
What is new?
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Comparison
What is new?
Down-out Call Price Approximation
S0
94.0
93.0
92.0
91.0
90.1
90.01
Number of Time
500
1000
4.849/4.723 4.814/4.839
(4.863)
(4.864)
3.667/3.610 3.601/3.634
(3.700)
(3.701)
2.450/2.526 2.509/2.398
(2.504)
(2.506)
1.222/1.333 1176/1.155
—
(1.274)
0.127/0.122 0.129/0.122
—
—
0.012/0.012 0.012/0.012
—
—
Steps in the Tree
3000
5000
4.840/4.880 4.820/4.833
(4.864)
(4.864)
3.641/3.708 3.654/3.703
(3.701)
(3.702)
2.507/2.497 2.454/2.464
(2.506)
(2.506)
1.221/1.248 1.224/1.272
(1.275)
(1.274)
0.129/0.124 0.137/0.127
—
—
0.012/0.012 0.012/0.012
—
—
Alyt. P.
4.864
3.702
2.506
1.274
0.129
—
0.013
Table: K=100, Vol=25%, rf=10%, T=1, L=90, ()=Ritchken’96.
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Comparison
What is new?
Double Knock-out Call with Constant Barriers
Short Maturity (1 Month - Trinomial Tree): Pellser(2000)
Vol
σ = 0.20
σ = 0.30
σ = 0.40
U
1500
1200
1050
1500
1200
1050
1500
1200
1050
L
500
800
950
500
800
950
500
800
950
KI
25.12
24.16
2.15
36.58
29.45
0.27
47.58
25.84
0.02
FD
24.47
24.69
2.15
36.04
29.40
0.27
47.31
25.82
0.01
Approx1
25.12
24.77
2.15
36.58
29.40
0.26
47.85
25.88
0.01
Approx2
25.12
24.76
2.15
36.58
29.46
0.28
47.84
25.86
0.01
Table: U = Upper Barrier, L = Lower Barrier, KI = Kunitomo and Ikeda
method, FD = Finite Difference method, Approx1 = 1000 steps, Approx2
= 2000 steps. Parameters: S0 =1000, K=1000, rf=5%, T=0.5
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Comparison
What is new?
Double Knock-out Call with Constant Barriers
Geman and Yor Comparison
Tree Lvls.
1000
2000
3000
4000
5000
Analyt. Value
Option Value
Case 1
0.0412
0.0412
0.0411
0.0411
0.0410
0.0411
Option Value
Case 2
0.0178
0.0178
0.0176
0.0179
0.0178
0.0178
Option Value
Case 3
0.0759
0.0762
0.0755
0.0762
0.0762
0.0761
Table: Comparison of results between the adjusted trinomial tree and the
analytical values calculated by Geman and Yor (1996).
Case1: S0 = 2, K = 2, σ = 20%, r = 2%, T = 1y , L = 1.5, U = 2.5
Case2: S0 = 2, K = 2, σ = 50%, r = 5%, T = 1y , L = 1.5, U = 3
Case3: S0 = 2, K = 1.75, σ = 50%, r = 5%, T = 1y , L = 1.5, U = 3
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Comparison
What is new?
Down-out Call with Exponential Barrier
Tree Lvls.
17
77
181
327
515
2100
4754
Analytic
Slope=-0.1
Cost.
Adj.
7.002 6.494/6.631
6.958 6.580/6.753
6.920 6.710/6.787
6.910 6.761/6.786
6.912 6.796/6.802
6.902 6.854/6.868
6.900 6.872/6.889
6.896
Tree Lvls
24
92
203
356
552
2174
4865
Analytic
Slope=0.1
Cost.
Adj.
5.020 5.581/5.400
4.949 4.993/4.985
4.934 4.926/4.876
4.934 4.907/4.877
4.932 4.902/4.912
4.929 4.905/4.915
4.929 4.929/4.918
4.928
Table: Comparison of results between the adjusted-probability method
and the extended Cox-Ross-Rubinstein method of Costabile(2002)
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Comparison
What is new?
Down-out Call with Linear Barrier
Convergence Behaviour
Douw-out European Call with Linear Barrier
S=100, K=100, vol=25%, rf=10%, Intercept=95, slope=10
10
Trinomial Tree (unadjusted)
Trinomial Tree (adjusted)
9
Option Price
8
7
6
5
4
1
45
89
133 177 221 265 309 353 397 441 485 529 573 617 661 705 749 793 837 881 925 969
Number of Steps in Tree
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Comparison
What is new?
Down-out Call with Linear Barrier
Convergence Behaviour
Douw-out European Call with Linear Barrier
S=100, K=100, vol=25%, rf=10%, Intercept=95, slope=10
10
Trinomial Tree (unadjusted)
Trinomial Tree (adjusted)
9
Option Price
8
7
6
5
4
1
45
89
133 177 221 265 309 353 397 441 485 529 573 617 661 705 749 793 837 881 925 969
Number of Steps in Tree
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Comparison
What is new?
Double Knock-out Call with Linear Barriers
Convergence Behaviour
Down-out European Call with Double Linear Barriers
S=100, K=100, vol=25%, rf=10%, L=92, L1=-22, U0=105, U1=105
1.5
Trinomial Tree (unadjusted)
Trinomial Tree (adjusted)
1.3
Option Price
1.1
0.9
0.7
0.5
0.3
1
46
91
136
181 226
271
316
361
406 451
496
541
586
631 676
721
766
811
856 901
946
991
Number of Steps in Tree
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Comparison
What is new?
Double Knock-out Call with Linear Barriers
Convergence Behaviour
Down-out European Call with Double Linear Barriers
S=100, K=100, vol=25%, rf=10%, L=92, L1=-22, U0=105, U1=105
1.5
Trinomial Tree (unadjusted)
Trinomial Tree (adjusted)
1.3
Option Price
1.1
0.9
0.7
0.5
0.3
1
46
91
136
181 226
271
316
361
406 451
496
541
586
631 676
721
766
811
856 901
946
991
Number of Steps in Tree
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Comparison
What is new?
Bermuda Options
Broadie, Glasserman and Kou (1997) Comparison
Barrier
85
87
89
91
93
95
97
99
Corrected
Continous
6.322
6.281
6.184
5.977
5.585
4.907
3.836
2.271
Probability
Adjusted
6.322
6.281
6.184
5.976
5.583
4.905
3.841
2.337
True
Price
6.322
6.281
6.184
5.977
5.584
4.907
3.834
2.337
Broadie Rel.
Error (%)
0
0
0
0
0.02
0
0.05
-2.82
Adj. Rel.
Error (%)
0
0
0
-0.02
-0.02
-0.04
0.18
0
Table: Parameters: S0 = 100, K = 100, σ = 30%, rf = 10%, T = 0.2.
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Comparison
What is new?
Outline
1
Motivation
The standard Cox-Ross-Rubinstein Binomial Tree
Previous work
2
New Approach
Adjusted Binomial (and Trinomial) Tree
3
Results
Comparison
What is new?
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities
Motivation
New Approach
Results
Comparison
What is new?
Goals of the Paper and Possible Extensions
The “Adjusted Tree”:
does not require to reposition the tree “on the barrier”
demonstrates good convergence properties towards the
analytical price for both single and double constant barriers
options
produces an accurate approximation when the stock price is
very close to the barrier
can produce price approximation for time-varying barrier
options including exponential, single linear and double linear
barriers
A possible extension concerns the approximation convergence
behaviour: is it possible to obtain monotone convergence?
G. Barone-Adesi, N. Fusari, J. Theal
Barrier Option Pricing Using Adjusted Transition Probabilities