Markov Chain Approximation method for
Pricing Barrier Options with Stochastic
Volatility and Jump
Sumei Zhang
Department of applied mathematics
School of science, Xi’an university of Post and Telecommunications
Xi’an, China
zhanggsumei@sina.com
Abstract— The purpose of this paper is to provide an efficient
pricing method for barrier option with stochastic volatility and
jump risk. First, by constructing a nonuniform variance grid
and using local consistency arguments, this paper
approximates the stochastic volatility jump-diffusion model
with a finite and dense Markov chain; Then, the paper
computes the rate matrix of the Markov chain by solving a
system induced by local consistency conditions; And then the
paper provides the character function of the Markov chain. At
last, using Markov chain approximation method and Fourier
transform technique, the paper obtains numerical solutions for
barrier options pricing. Numerical results show that
comparing with the Monte Carlo simulation, the proposed
pricing technique is accurate, fast and easy to implement.
Keywords—Barrier option; option pricing; Markon chain;
stochastic volatility; jump diffusion
I. INTRODUCTION
Barrier options [1] are among the most popular path-
dependent derivatives that disappear or appear if the
underlying asset price crosses a given level (barrier level)
before expiration date. Such contracts form effective risk
management tools, and are liquidly traded in the foreign
exchange markets. The most frequently used standard
barrier options are knock in and knock out options. Knock
in options can be divided into two categories, up and in, and
down and in. Similarly, knocking out options can also be
used as a down and out and up and out option. This paper
focuses on standard knock out call options.
There exists currently a good deal of literature on
numerical methods for the pricing of barrier options. It is
well known that in this case a straightforward Monte Carlo
simulation algorithm will be time-consuming and yield
unstable results for the prices and especially the sensitivities.
Since option pricing can be described as the solution of a
partial differential equation (PDE) or partial integro-
differential equation (
PIDE) with boundary condition [2],
some researchers have priced barrier options through PDE
(
PIDE) method under stochastic volatility model or jump-
diffusion model [3-7]. However some empirical evidences
[8-9] show that the model that combines stochastic volatility
and jumps may be more reasonable. But a complex model
with too many stochastic factors will lead to difficulty of
obtaining the solution of the corresponding pricing equation.
A different approach, pioneered by Kushner [10], is the
Markov chain approximation method. Originally developed
for the numerical solution of stochastic optimal control
problems in continuous time, this method consists of
approximating the system of interest by a discrete time
chain that closely follows its dynamics, and solving the
problem of interest for this chain. An application to the
pricing of American options under jump diffusion model is
given in [11]. Zhang et al [12] consider lookback options
pricing in a stochastic volatility model. Mijatović [13] prices
barrier options in a local volatility jump diffusion model.
However, there is rare study for valuation of barrier option
under a stochastic volatility jump diffusion model, which is
rather challenge due to the nonlinearity and jump
discontinuity.
The rest of the paper is organized as follows. Section 2
develops the underlying pricing model. Section 3 describes
Markon chain approximation method. Section 4 presents
numerical results for barrier options pricing. Section 5
concludes the paper.
II. THE MODEL
An arbitrage-free, frictionless financial market is
considered where only riskless asset
B
and risky asset
S
are traded continuously up to a fixed horizon date
T
. Let
(
, , , )P
t
FF
be a complete probability space with a filtration
satisfying the usual conditions, i.e. the filtration is
continuous on the right. Suppose
1
()Wt
and
2
()Wt
are both
standard Brownian motion which is
t
F
-adapted, and
1
()Wt
has correlation
with
2
()Wt
.Let
()Nt
be independent
Poisson process with constant intensity
,
()Vt
and
()St
denote the volatility and price process of stock.
According to [14], the stochastic volatility jump-diffusion
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123
model can be represented by the following:
1
2
12
l
og ( ) ( ) ( ) ( ) ( )
( ) ( ( )) ( ) ( )
( ) ( )
s
d S t r dt V t dW t dN t
dV t k V t dt V t dW t
dW t W t dt
(1)
where
r
is constant interest rate,
s
is the jump size, and
l
n( 1)
s
has an normal distribution with mean
J
and
variance
2
J
.
denotes the drift term compensates for the
expected drift added by the jump component. By the
ˆ
I
to
formula,
2
(
) 1
1
exp( )
2
S J J
E
.
()Vt
is a
square root mean reverting process, first proposed by Heston
[15].
,
, k
The parameter
0k
is the mean-reversion
rate,
0
is the longterm mean,
0
is the volatility-of-
variance. In this paper, the author always assumes that
2
2k
, which is known as the Feller condition.
III. OPTION PRICING BASE D ON MARKON CHAIN
APPROXIMATION METHOD
According to [16], the volatility process
()Vt
can be
approximated arbitrarily well using a carefully selected
Markov chain which satisfies the local consistency
requirements.
A. The “local consistency” concept
Suppose
01
{
, , , }
h
h h h h
N
V V V V
is a variance grid where
0h
denotes the spacing between discrete points. Assume
that
(
) ( ) 0
hh
V t V t
,
h
N
and the grid can cover
the domain of
()Vt
as
0h
. Suppose
()
h
V
t V
denote
the approximating chain of the process
()Vt
, and the
corresponding rate matrix is
[]
hh
ij
Qq
.
Assume that the values of the variance process and the
approximating chain coincid,
(
) ( ).
h
V t V t
Suppose
h
t
E
denotes the expectation. For some
0
,
()
h
Vt
meets
the following local consistency conditions [10]:
{
( ) ( )} ( ( )) ( )
h h h
t
E V t V t k V t o
(2)
22
{
( ) ( )} ( ) ( )
h h h
t
E V t V t V t o
(3)
(
) ( ) ( )
hh
V t V t o h
(4)
B. Construction of the approximating Markov chain of the
model
1) The grid
A suitable choice of the grid is essential for the
effectiveness of the pricing algorithm. One of the features of a
good grid is that it has sufficient resolution in regions of
interest, such as the current spot value and the barrier levels,
which is a necessary condition for constructing a Markov
chain market model that approximates well the dynamics of
the given price process. Another desirable feature is that the
grid “covers” a sufficiently large part of the state-space,
which is needed to control the truncation error that arises
when approximating an infinite state-space by a finite state-
space. To employ a uniform grid that satisfies these
conditions would be computationally expensive. Here the
author employs the following procedure for generating a
suitable nonuniform grid G, based on an algorithm from [13]:
a) Pick
i
NN
and
(
0, ),
i
d
1
,2,3i
and the
smallest and largest values
1
,
N
VV
of the grid
V
, such that
1
2 3
.N N N N
b) Define the subgrid
(
, , , , , ), 1,2,3.
i i i i i i i
G G a s b N d d i
1
1 1 2 0
, , ,where a V s l s S
2
3 3
3 1 2
. , , ,
N
a b Vs u b a b
The subgrid
i
G
is generated by the following procedure:
● Compute
12
12
a
r
csin ( ), arcsin ( ).
a s b s
c h c h
gg
● Define the lower part of the grid by the formula
11
s
inh( (1 ( 1) / ( / 2 1))),
k
x s g c k M
where
{
1, , / 2}kM
.
● Define the upper part of the grid by the formula
/
2 2 2
sinh( 2 / ),
kM
x s g c k M
where
{
1, , / 2}kM
.
c)
1
2 3
GG
GG
.
2) Computation of the rate matrix
Assume that at time t, the variance is equal to
h
j
V
. Over a
time interval
, there are three possibilities: remain at
h
j
V
,
move up by
U
d
to
1
h
j
j U
V V d
, or move down by
D
d
to
1
h
j
j D
V V d
. Local consistency condition (2) and (3) can
therefore be restated as
,
1 , 1
( ) ( ),
hh
j j D j j U
q d q d r o
(5)
22
,
1 , 1
( ) ( ).
h h h
j j D j j U
q d q d V t o
(6)
Solving the above system, it can be obtained
22
2
,1
e
xp( ) exp( ) 1 ,
2
2
hh
jj
h
jj
k
VV
hk
h
q
2
2
,
e
xp( ),
h
j
h
jj
V
h
q
22
2
,1
e
xp( ) exp( ) 1 .
2
2
hh
jj
h
jj
k
VV
hk
h
q
3) The characteristic function
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Using the decompositions
1 2 2
1,W W Z
where
Z
is a standard Brownian motion, independent of all other
processes, Model (1) can be written as
22
log ( ) ( ) ( ) ( ) ( ) 1
( )
s
d S t r dt V t dW t V t dZ
dN t
(7)
2
( ) ( ( )) ( ) ( )dV t k V t dt V t dW t
(8)
Equation (8) implies that
2
,
( ) ( ( ))
()
dW
dV t k V t dt
Vt
which
can be substituted into the first expression, and substitute
()Vt
by
()
h
Vt
, it can be obtained
2
( ( ) ( ( )) )
( ) 1
log ( ) ( )
( ).
hh
s
dV t k V t dt
V t dZ
d S t r dt
dN t
As [16] shows, the approximating characteristic function of
()
log
(0)
St
S
is given by
( ) exp{ B(u)}
T
ii
u l t e
where
l
is an (n☓1) vector of ones, and
i
e
is the i-th (n☓1)
unit vector. The matrix function
B(u)
has elements of the
form:
,
,,
( ),
( ), 1
0, .
hh
j j j
hh
j i j i j
q u i j
q u i j
otherwise
where
22
1
( ) ( ) (1 ) ( ) ( 1),
2
hh
jJ
u i r u V t u
2
1
exp (1 ) exp ( 1) 1 ,
2
iu
JJ
iut t iu iu
( ) exp .
h
j
u i hu
C. Barrier option pricing
Let
log Sx
and denote with
( , )U x t
the value of the
barrier option at time
t
.
( , )U x t
can be computed by
( , ) [ ( ( ), )]
BB
U x t E C S
where
( ( ), )
BB
CS
is a discounted payoff function and
B
is the first hitting time of the given barrier level
B
by the
underlying asset process
()St
. For down-and-out call barrier
options the payoff
( ( ), )
BB
CS
is defined by
max( ( ) ,0), ,
( ( ), )
0, .
rT
B
BB
B
e S T K T
CS
T
where
K
is given exercise price at expiration date
T
.
Using Markon chain approximation method,
( , )U x t
can
be obtain by the recursive relationship
1
( , ) exp( ) ( ( , , ) , )
exp( ) ( , , )
P( , )P( )
t t t t t t
N
R
j
t t t t t t t t
U x t r t E U X t t s X x s i
r t U y t t j
X x dy s i s j s j s i
If denote with
()fy
the log-return density over a time
interval of
t
, then the above relationship can be written as
1
( , , ) exp( ) ( , , ) ( , , ).
N
R
j
U x t i r t U y t t j f y x i j
( , , )f y i j
can be retrieved by taking the inverse Fourier
transform of the characteristic function
()
i
u
.
IV. NUMERICAL EXAMPLE
This section uses the method from Section 3 to price
barrier options. This paper first evaluates barrier options
using Markov chain approximation method. Then a
comparison of the speed and accuracy between the Markov
chain approximation and the exact Monte Carlo simulation
proposed by [17] is provided. For comparison the default
parameters are used in [17] and listed in Table Ⅰ.
TABLE I. DEFAULT PARAMETERS FOR BARRIER OPTIONS PRICING
Parameter
Value
Initial asset price
(0) 100S
Intensity of the Poisson process
0.11
Volatility of volatility
0.27
Interest rate
0.0319r
Long-run variance
0.014
Initial variance
(0) 0.008836V
Mean reversion
3.99k
expectation of the jump size
0.12
Correlation between returns and
volatility
0.79
Maturity date
5T
Table Ⅱ lists the comparison of the two methods. Except
for barrier level
90B
, other parameters are the same as
the ones in Table 1. For Monte Carlo, number of simulation
is 100000. For Markov chain approximation, number of grid
is 200.
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TABLE II. COMPARISON OF DOWN AND OUT CALL OPTIONS PRICES
BETWEEN MARKOV CHAIN APPROXIMATION AND MONTE CARLO
SIMULATION
Exercise
price
Exact Monte Carlo
simulation
Markov chain
approximation
80
32.3423(0.0284)
32.3417
85
28.9055(0.0271)
28.9060
90
25.4688(0.0259)
25.4679
95
22.1107(0.0248)
22.1099
100
18.9709(0.0235)
18.9701
105
16.0827(0.0220)
16.0833
110
13.4641(0.0204)
13.4633
115
11.1232(0.0188)
11.1225
120
9.0646(0.0171)
9.0651
Note: Numbers in parentheses are standard errors for the estimates of options prices.
The numerical experiment shows that Markov chain
approximation is considerably faster than the Monte Carlo
simulation. For the pricing of down and out call options
Markov chain approximation takes about 0.03 seconds,
while Monte Carlo simulation takes about 9.1 seconds.
Moreover, Table Ⅱ suggests the accuracy of the Markov
chain approximation method. If the Monte Carlo is
considered to be the benchmark, the relative percentage
pricing differences of Markov chain approximation are all
less than 0.09%.
V. CONCLUSION
This paper combines stock price jumps and stochastic
volatility and considers a general jump-diffusion model for
pricing barrier options. By Markov chain approximation
method and Fourier transform technique, the paper obtains
numerical solutions for barrier options pricing. Numerical
results show that the proposed pricing technique is accurate,
fast and easy to implement. The paper presents an efficient
pricing method for barrier option with stochastic volatility
and jump risk.
ACKNOWLEDGMENT
This work is supported by the National Natural Science
Foundation of China under Grant No.11426176, the Science
Plan Foundation of the Education Bureau of Shaanxi
Province under Grant No.14JK1672 and the Youth
Foundation of Xi’an University of Post and
Telecommunications under Grant No.ZL2013-33.
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