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Numerical method for pricing American options under regimeswitching jump-diffusion models
Abdelmajid El hajaji
Department of Mathematics, Faculty of Science and Technology,
University Sultan Moulay Slimane, Beni-Mellal, Morocco
E-mail: a_elhajaji@yahoo.fr
Khalid Hilal
Department of Mathematics, Faculty of Science and Technology,
University Sultan Moulay Slimane, Beni-Mellal, Morocco
E-mail: hilal_khalid@yahoo.fr
Abstract
Our concern in this paper is to solve the pricing problem for American options in a Markov-modulated jumpdiffusion model, based on a cubic spline collocation method. In this respect, we solve a set of coupled partial
integro-differential equations PIDEs with the free boundary feature by using the horizontal method of lines to
discretize the temporal variable and the spatial variable by means of Crank-Nicolson scheme and a cubic spline
collocation method, respectively. This method exhibits a second order of convergence in space, in time and also
has an acceptable speed in comparison with some existing methods. We will compare our results with some
recently proposed approaches.
Keywords: American Option, Regime-Switching, Crank-Nicolson scheme, Spline collocation, Free Boundary
Value Problem.
1. Introduction
Options form a very important and useful class of financial securities in the modern financial world. They play
a very significant role in the investment, financing and risk management activities of the finance and insurance
markets around the globe. In many major international financial centers, such as New York, London, Tokyo,
Hong Kong, and others, options are traded very actively and it is not surprising to see that the trading volume of
options often exceeds that of their underlying assets. A very important issue about options is how to determine
their values. This is an important problem from both theoretical and practical perspectives
Recently, there has been a considerable interest in applications of regime switching models driven by a Markov
chain to various financial problems. For an overview of Markov Chains
The Markovian regime-switching paradigm has become one of the prevailing models in mathematical finance.
It is now widely known that under the regime-switching model, the market is incomplete and so the option
valuation problem in this framework will be a challenging task of considerable importance for market
practitioners and academia. In an incomplete market, the payoffs of options might not be replicated perfectly by
portfolios of primitive assets. This makes the option valuation problem more difficult and challenging. Among
the many researchers that have addressed the option pricing problem under the regime-switching model, we must
mention the following: [4] develop a new numerical schemes for pricing American option with regime-switching.
[20] provides a general framework for pricing of perpetual American and real options in regime-switching Levy
models. [20] investigate the pricing of both European and American-style options when the price dynamics of
the underlying risky assets are governed by a Markov-modulated constant elasticity of variance process. [17]
develop a new tree method for pricing financial derivatives in a regimes-witching mean-reverting model. [22]
develop a flexible model to value longevity bonds under stochastic interest rate and mortality with regimeswitching.
The paper is organized as follows. In Section 2, we describe briefly the problem for American options in a
Markov-modulated jump-diffusion model. Then, we discuss time semi-discretization in Section 3. Section 4 is
devoted to the spline collocation method for pricing American options under regime-switching jump-diffusion
models using a cubic spline collocation method. Next, the error bound of the spline solution is analyzed. In order
to validate the theoretical results presented in this paper, we present numerical tests on three known examples in
Section 5. The obtained numerical results are compared to the ones given in [2]. Finally, a conclusion is given in
Section 6.
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2. Regime-switching Lévy processes
Markov chains are frequently used for capturing random shifts between different states. In this section, we
review the most important definitions from continuous-time Markov chains, Lévy processes with regimeswitching (or Markov-modulated) parameters and also option pricing in this framework (see Chourdakis [11] for
a comprehensive treatment).
Let a t be a continuous-time Markov chain taking values among H different states, where H is the total
number of states considered in the economy. Each state represents a particular regime and is labeled by an
integer i between 1 and H . Hence the state space of a t is given by M = 1,..., H . Let matrix Q = (qij ) H ´H
denote the generator of a t . From Markov chain theory (see for example, Yin and Zhang [9]), the entries qij in
Q satisfy: (I) qij ³ 0 if i ¹ j ; (II) qii £ 0 and qii = -å j ¹iqij for each i = 1,..., H .
Let
Wt be a standard Brownian motion defined on a risk-neutral probability space (W, ô, P) and assume that
this process is independent of the Markov chain a t . We consider the following regime-switching exponential
Lévy model for describing the underlying asset price dynamics:
X
St = S 0 e t .
The log-price process
X t will be constructed in the following manner: Consider a collection of independent
i H
t i =1
Lévy processes {Y }
indexed by i . The increments of the log-price process will switch between these
H
different Lévy processes, depending on the state at a t :
a
dX t = dYt t .
i
Each Lévy process Yt assumed to have a Lévy-Itô decomposition of the form
dYt i = mti dt + s ti dWt + ò zN i (dz, dt ), i = 1, 2,..., H ,
R
in which
m
i
is the drift and
s
i
is the diffusion coefficient of the i - th Lévy process. In this equation,
N i (.,t )
is a Poisson random measure defined on Borel subsets of R with n (.) as its associated Lévy measure,
describing the discontinuities.
We now consider the pricing of an American put option written on the underlying asset {St }t ³0 with strike
i
price K and maturity date T . To obtain an equation with constant coefficients for the price of this option in
each regime, we switch to log-prices and let x = log( St /S0 ). Then the transformed option price Vi ( x, t ) at
time 0 £ t £ T and regime at a t = i will satisfy the following equation, due to the risk-neutral pricing
principle (see for example Karatzas and Shreve [10]):
[
]
Vi ( x, t ) = sup E e- r (t -t ) ( K - e t ) + | X t = x, a t = i .
t £t £T
X
In the above equation, t is a stopping time satisfying t £ t £ T and E is the expectation operator with
respect to equivalent martingale measure P . This is the optimal stopping formulation of the American option
pricing problem. We must note here that in writing these equations and all subsequent ones, the parameters
si
mi
r is absorbed into the
constant m term to simplify the presentation of the material. One can now show that Vi ( x,t ) for
i = 1,2,..., H satisfy the following system of free boundary value problems [4]:
and
in (2.1) are taken regime-independent and constant and also the fixed interest rate
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H
ì ¶Vi
0, x > x i (t ), i = 1,2,..., H ,
- LiVi + åqijV j =
ï ¶t
j =1
ï
Vi ( x,t ) =
K - e x , x £ x i (t ), i = 1,2,..., H ,
ï
ï
Vi ( x, T ) = ( K - e x ) + ,
i = 1,2,..., H ,
ï
(1)
í
xi (t )
i = 1,2,..., H ,
lim Vi ( x,t ) = K - e ,
ï
x® xi (t )
ï
¶Vi
ï
=
i = 1,2,..., H ,
- 1,
lim
x® xi (t ) ¶x
ï
ï
x i (T ) =
K,
i = 1,2,..., H ,
î
i
in which x i (t ) for i = 1,..., H denote the optimal exercise boundaries and L is the infinitesimal generator of
the i-th Lévy process of the form
1
1
LiVi = - s 2¶ xxVi - (r - s 2 - lix )¶ xVi + (r + li )Vi - ò Vi ( x + z,t )n i (dz ),
R
2
2
with
li stands for the jump intensity at state i,
x = ò (e z - 1)dF ( z ),
R
for the function F which is the distribution of jumps sizes. This is a set of coupled partial integro-differential
equations with H free boundaries due to the regime-switching feature introduced in the underlying asset model.
The analytical solution of the above system of PIDEs is not available at hand and so the need for efficient
numerical approaches seems a necessity. In the sequel, we introduce our approach to solve this set of equations.
Remark: One should notice that if we set l = 0 and H = 1 ; (1) will become original Black-Scholes PDE.
3. Time and Spatial discretization
Our aim in this section is to use a cubic spline collocation method to find an approximate solution for the set of
Eqs. (1). By using the change of variables t = T - t and applying the Crank-Nicolson scheme in time, we can
use the collocation method in each time step to find a continuous approximation in the whole interval. It is
obvious that Vi ( x, t ) for i = 1,..., H satisfy the following set of coupled PIDEs in operator form:
H
¶Vi
+ LVi - åqijV j = 0, i = 1,..., H ,
¶t
j =1
which is valid in the space-time domain [-¥,+¥] ´ [0, T ] . In order to numerically approximate the solution, let
us truncate the x -domain into the sub domain W x = [ xmin , xmax ] .
Taking V
= [V1 ,V2 ,...,VH ]T and W = W x ´ [0, T ],
ì ¶V
¶V
¶ 2V
+ (G - Q)V
-P 2 -R
ï ¶t
¶x
¶x
ï
V ( x,0)
í
ï
V ( xmin , t )
ï
V ( xmax , t )
î
I (V ( x, t )), ( x, t ) Î W,
=
=
K .I H ,
x
= max ( K - e ,0).I H ,
=
0.I H ,
where
65
x Î Wx ,
t Î [0, T ],
t Î [0, T ],
(2)
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IH
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[1,1,...,1]T Î R H ,
ö
æ
s2
ç diag ( I j , j ) ÷
,
÷
ç
2
ø j =1,...,H
è
=
P =
ö
æ
s2
ç diag (r ,
- l jx ) ÷
÷
ç
2
ø j =1,...,H
è
diag (r + l j ) j =1,...,H ,
R =
G =
(diag (l ))
(
j
I (V ) =
)
ò V ( x + z, t ) f ( z )dz,
j =1,...,H R
with Q - G is a continuous, bounded, symmetric matrix function and each function of the matrix G - Q is
~
³ g > 0 on W and max ( K - e x ,0) is sufficiently smooth function.
Here we assume that the problem satisfies sufficient regularity and compatibility conditions which guarantee that
the problem has a unique solution V Î C (W) Ç C
(W) satisfying (see, [13, 1, and 14]):
2,1
i+ j
¶ V ( x, t )
£ k on W; 0 £ j £ 3 and 0 £ i + j £ 4,
¶x i ¶t j
where k is a constant in R
H
(3)
.
3.1. Time discretization and description of the Crank-Nicolson scheme
Discretize the time variable by setting t
m
= mDt for m = 0,1,..., M , in which Dt =
T
and then define
M
V m ( x) = V ( x, t m ), m = 0,1,..., M .
Now by applying the Crank-Nicolson scheme on (2), we arrive at the following equation
V m+1 ( x) - V m ( x) 1
1
- L(V m+1 + V m ) = I (V m+1 ) + I (V m )
2
Dt
2
m
m +1
One way is to replace V
with V in the linear terms. This leads to the following modified system:
Dt
Dt
V m+1 ( x) - LV m+1 = LV m + V m + DtI (V m ).
2
2
For m = 0,1,..., M . The final price of the American option at time level m will be of the form:
(
ì ¶ 2V m+1
¶V m+1
+ LV m+1
+R
P
ï
¶x
¶x 2
ï
ï
V 0 ( x)
í
ï
V m+1 ( xmin )
ï
ï
V m+1 ( xmax )
î
J (V m ),
"x Î W x ,
0 £ m < M,
0 £ m < M.
(4)
"x Î W x ,
= f0 ( x).I H ,
y .I H ,
=
=
0.I H ,
)
=
Where, for any m ³ 0 and for any x Î W x , we have
66
(5)
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2 ö
æ
ç Q - G - I ÷,
Dt ø
è
2 m
LV m - V - 2 I (V m ),
Dt
¶2
¶
P 2 + R - (G - Q) I ,
¶x
¶x
(K - e x )+ ,
x
K - e min ,
L =
J (V m ) =
L =
f0 ( x ) =
y =
V m+1 is solution of (5), at the (m + 1) th-time level.
The following theorem proves the order of convergence of the solution V
Theorem 3.1 Problem (5) is second order convergent .i.e.
V ( x, t m ) - V m
Proof: We introduce the notation em = V ( x, tm ) - V
em
H
m
H
m
to V ( x, t ).
£ C (Dt ) 2 .
the error at step m and
i
= sup max | em ( x) |
xÎW x 1£i £ H
By Taylor series expansion of V , we have
Dt ¶V
(Dt ) 2 ¶ 2V
V ( x, tm+1 ) = V ( x, t 1 ) +
( x, t 1 ) +
( x, t 1 ) + O((Dt )3 ).I H ,
2
m+
m+
m+
2 ¶t
8 ¶t
2
2
2
Dt ¶V
(Dt ) 2 ¶ 2V
V ( x, tm ) = V ( x, t 1 ) ( x, t 1 ) +
( x, t 1 ) + O((Dt )3 ).I H .
2
m+
m+
m+
2 ¶t
8 ¶t
2
2
2
By using these expansions, we get
V ( x, tm+1 ) - V ( x, tm ) ¶V
=
( x, t 1 ) + O((Dt ) 2 ).I H ,
(6)
m+
Dt
¶t
2
¶V
and by Taylor series expansion of
, we have
¶t
(Dt ) 2 ¶ 3V
Dt ¶ 2V
¶V
¶V
( x, t 1 ) + O((Dt )3 ).I H ,
( x, tm+1 ) =
( x, t 1 ) +
( x, t 1 ) +
m+
m+
m+
8 ¶t 3
2 ¶t 2
¶t
¶t
2
2
2
¶V
¶V
Dt ¶ 2V
(Dt ) 2 ¶ 3V
( x, tm ) =
( x, t 1 ) ( x, t 1 ) +
( x, t 1 ) + O((Dt )3 ).I H .
m+
m+
m+
¶t
¶t
2 ¶t 2
8 ¶t 3
2
2
2
By using these expansions, and
¶ 3V
£ c.I H on W (see relation (3)), we have
¶t 3
1 ¶
¶
[V ( x, tm+1 ) + V ( x, tm )] = V ( x, t 1 ) + O((Dt ) 2 ).I H .
m+
2 ¶t
¶t
2
This implies
¶V
1 ¶
[V ( x, tm+1 ) + V ( x, tm )] + O((Dt ) 2 ).I H
( x, t 1 ) =
m+
¶t
2 ¶t
2
1
= [LV ( x, tm+1 ) + I (V ( x, tm+1 )) + LV ( x, tm ) + I (V ( x, tm ))] + O((Dt ) 2 ).I H .
2
By using this relation in (6) we get
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Dt
Dt
Dt
L)V ( x, tm+1 ) = (1 + L)V ( x, tm ) + [ I (V ( x, tm+1 )) + I (V ( x, tm ))] + O((Dt )3 ).I H ,
2
2
2
by (4). Then, we obtain
Dt
Dt
Dt
L)em+1 = (1 + L)em + [I (em+1 ) + I (em )] + O((Dt )3 ).I H .
2
2
2
3
3
We may bound the term O((Dt ) ) by c(Dt ) for some c > 0, and this upper bound is valid uniformly
throughout [0,T ] . Therefore, it follows from the triangle inequality that
Dt
Dt
Dt
( I - L)em+1 £ ( I + L)em + ( I (em ) H + I (em+1 ) H ) + c(Dt )3 .
2
2
2
H
H
(1 -
We use the cross-correlation function (see [3]) defined by
R
i
fem
i
i
( x) = f ( x) * *em ( x) = ò f ( z )em ( x + z )dz, for i = 1,..., H ,
R
we have
R fe
m H
(I -
Dt
L)em+1
2
£ (I +
H
£ (I +
Dt
L)em
2
Dt
L)
2
em
H
H
+
H
+
£ f
Dt
I (em )
2
(
Dt
l
2
H
f
em
H
H
H
H
.
+ I (em+1 )
(e
m H
)+ c(Dt ) ,
3
H
+ em+1
)+ c(Dt ) .
3
H
Dt ö
æ
ç I H ± L ÷ satisfies a maximum principle (see, [7, 5]) and consequently
2 ø
è
æ
ö
ç
÷
-1
Dt ö
æ
ç 1 ÷
£ç
ç IH ± L÷
÷.
2 ø
è
Dt ~ ÷
H
ç1+ g ÷
ç
2 ø
è
Since we are ultimately interested in letting Dt ® 0 , there is no harm in assuming that Dt. < 2 , with
h
Clearly, the operator
h = (L H + l
H
f
H
). We can thus deduce that
em+1
H
1
æ
ö
ç 1 + Dt.h ÷
2
÷ em
£ç
ç 1 - 1 Dt.h ÷
ç
÷
2
è
ø
H
æ
ö
ç
÷
c
÷(Dt ) 3 .
+ç
ç 1 - 1 Dt.h ÷
ç
÷
2
è
ø
(7)
We now claim that
m
éæ
ù
1
ö
1 + Dt.h ÷
êç
ú
c
2
÷ - 1ú (Dt ) 2 .
em H £ êç
h êç 1 - 1 Dt.h ÷
ú
÷
êç
ú
2
ø
ëè
û
The proof is by induction on m . When m = 0 we need to prove that e0
(8)
H
£ 0 and hence that e0 = 0 . This
t0 = 0 the numerical solution matches the initial condition and the error is zero.
For general m ³ 0 , we assume that (8) is true up to m and use (7) to argue that
is certainly true, since at
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m
ù
1
1
æ
ö éæ
æ
ö
ö
1 + Dt.h ÷ êç 1 + Dt.h ÷
ç
ç
÷
ú
c
c
2
2
2
÷ êç
÷ - 1ú (Dt ) + ç
÷(Dt ) 3
em+1 H £ ç
h ç 1 - 1 Dt.h ÷ êç 1 - 1 Dt.h ÷
ç 1 - 1 Dt.h ÷
ú
ç
÷ êç
ç
÷
÷
ú
2
2
2
è
ø ëè
è
ø
ø
û
m +1
éæ 1
ù
ö
1 + Dt.h ÷
êç
ú
c
÷ - 1ú (Dt ) 2 .
£ êç 2
h êç 1 - 1 Dt.h ÷
ú
÷
êç 2
ú
ø
è
ë
û
This advances the inductive argument from m to m + 1 and proves that (8) is true. Since 0 < Dt. < 2, it is
h
true that
l
æ 1
æ
æ
ö
ö
ö
æ
ö
ç 1 + Dt.h ÷
ç Dt.h ÷ ¥ 1 ç Dt.h ÷
ç Dt.h ÷
ç 2
÷ = 1+ ç
÷£å ç
÷ = exp ç
÷.
ç 1 - 1 Dt.h ÷
ç 1 - 1 Dt.h ÷ l = 0 l! ç 1 - 1 Dt.h ÷
ç 1 - 1 Dt.h ÷
ç
ç
ç
÷
÷
÷
ç
÷
è 2
è 2
è 2
ø
ø
ø
è 2
ø
Consequently, relation (8) yields
m
1
ö
æ
ö
æ
1 + Dt.h ÷
ç mDt.h ÷
c(Dt ) 2 ç
c(Dt ) 2
2
÷ £
ç
÷.
em H £
exp ç
h ç 1 - 1 Dt.h ÷
h
ç 1 - 1 Dt.h ÷
÷
ç
÷
ç
2
2
ø
è
ø
è
This bound is true for every nonnegative integer m such that mDt < T . Therefore
æ
ö
ç T .h ÷
c(Dt ) 2
÷.
em H £
exp ç
h
ç 1 - 1 Dt.h ÷
ç
÷
è 2
ø
We deduce that
V ( x, t m ) - V m
H
£ C (Dt ) 2 .
In other words, problem (5) is second order convergent.
For any m ³ 0 , problem (5) has a unique solution and can be written on the following form:
ì
ï PV ¢¢( x) + RV ¢( x) + LV ( x) =
ï
V ( xmin ) =
í
ï
V ( xmax ) =
ï
î
~
f ( x) Î R H , "x Î W x ,
y .I H ,
0,
(9)
In the sequel of this paper, we will focus on the solution of problem (9).
3.2. Spatial discretization and cubic spline collocation method
Let Ä denotes the notation of Kronecker product, . the Euclidean norm on
Rn+1+ H and S (k ) the k th
derivative of a function S .
In this section we construct a cubic spline which approximates the solution V of problem (9), in the
interval W x
Let
ÌR.
Q = {xmin = x-3 = x-2 = x-1 = x0 < x1 < L < xn-1 < xn = xn+1 = xn+2 = xn+3 = xmax}
subdivision of the interval
be
a
W x . Without loss of generality, we put xi = a + ih , where 0 £ i £ n
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xmax - xmin
2
. Denote by S4 (W x , Q) = P3 (W x , Q) the space of piecewise polynomials of degree less
n
2
than or equal to 3 over the subdivision Q and of class C everywhere on W x . Let Bi , i = -3,L, n - 1 , be the
and h =
B-splines of degree 3 associated with Q. These B-splines are positives and form a basis of the
space S4 (W x , Q).
Consider the local linear operator
Q3 which maps the function V onto a cubic spline space S4 (W x , Q) and
2
which has an optimal approximation order. This operator is the discrete C cubic quasi-interpolant (see [15])
defined by
n -1
Q3V = åmi (V ) Bi ,
i = -3
where the coefficients
m j (V )
are determined by solving a linear system of equations given by the exactness of
Q3 on the space of cubic polynomial functions P3 (W x ). Precisely, these coefficients are defined as follows:
ìm -3 (V ) = V ( x0 ) = V ( xmin ),
ï
1
ïm -2 (V ) = (7V ( x0 ) + 18V ( x1 ) - 9V ( x2 ) + 2V ( x3 )),
18
ï
ï
1
ím j (V ) = (-V ( x j +1 ) + 8V ( x j +2 ) - V ( x j +3 )), for j = -1,..., n - 3,
6
ï
ïm (V ) = 1 (2V ( x ) - 9V ( x ) + 18V ( x ) + 7V ( x )),
n-3
n-2
n -1
n
ï n-2
18
ïm n-1 (V ) = V ( xn ) = V ( xmax ).
î
It is well known (see e.g. [16], chapter 5) that there exist constants C k , k = 0,1,2,3, such that, for any
function V Î C (W x ) ,
4
V ( k ) - Q3V ( k )
H
£ Ck h 4-k V (4-k )
By using the boundary conditions of problem (9), we obtain
H
, k = 0,1,2,3,
(10)
m-3 (V ) = Q3V ( xmin ) = V ( xmin ) = y .I H
and mn-1 (V ) = Q3V ( xmax ) = V ( xmax ) = 0.I H . Hence
Q3V = z1 + S ,
where
T
n -2
é n -2
ù
z1 = yB-3 I H and S = ê å m j (V1 ) B j ,L, å m j (VH ) B j ú .
j = -2
ë j =-2
û
From equation: (10), we can easily see that the spline S satisfies the following equation
PS (2) ( x j ) + RS (1) ( x j ) + LS (0) ( x j ) = g ( x j ) + O(h 2 ).I H , j = 0,..., n
(11)
with
~
g ( x j ) = f ( x j ) - ( Pz1(2) ( x j ) + Rz1(1) ( x j ) + Lz1(0) ( x j )) Î R H , j = 0,..., n.
~
~
n -1
The goal of this section is to compute a cubic spline collocation Sp i = å j = -3 c j ,i B j , i = 1,..., H which
satisfies the equation (9) at the points
t n+1 = xn-1
and
t j , j = 0,..., n + 2
t n+ 2 = xn .
Then, it is easy to see that
70
with
t 0 = x0 , t j =
x j -1 + x j
2
, j = 1,L, n ,
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~
c -3,i = y
and
~
c n -1,i = 0, for i = 1,..., H .
Hence
~
~
Sp i = z1 + S i ,
~
where S i =
n -2
~
åc
j ,i
B j , for i = 1,..., H
j = -2
~
and the coefficients c j ,i , j = -2,..., n - 2 and i = 1,..., H satisfy the following collocation conditions:
(2)
(1)
(0)
~
~
~
P S (t j ) + R S (t j ) + L S (t j ) = g (t j ), j = 1,..., n + 1,
(12)
where
~ ~
~
S = [ S 1 ,..., S H ]T ,
~
g (t j ) = f (t j ) - ( Pz1(2) (t j ) + Rz1(1) (t j ) + Lz1(0) (t j )) Î R H , j = 1,..., n + 1.
Taking
C = [m -2 (V1 ),..., m n-2 (V1 ),..., m -2 (VH ),..., m n-2 (VH )] Î R n+1+ H ,
T
~ é~
~
~
~
ù
C = ê c -2,1,..., c n-2,1,..., c -2,H ,..., c n-2,H ú Î R n+1+ H ,
ê
ú
ë
û
T
and using equations (11) and (12), we get:
((P Ä A
(2)
h
)
(1)
(0)
) + ( R Ä Ah ) + ( L Ä Ah ) C = F + E
(13)
~
(1)
(0)
) + ( R Ä Ah ) + ( L Ä Ah ) C = F ,
(14)
and
((P Ä A
(2)
h
)
with
F = [ g1 ,..., g n+1 ]T and g j =
1
g (t j ) Î R H ,
Dt
h2
h2
),..., O( )]T Î R n+1+ H ,
Dt
Dt
(k )
= ( B-3+ p (t j ))1£ j , p£n+1 , k = 0,1,2,
E = [O(
(
Ahk )
It is well known that
(
Ahk ) =
1
Ak for k = 0,1,2 where matrices A0 , A1 and A2 are independent of h , with
hk
the matrix A2 is invertible [8].
Then, relations (13) and (14) can be written in the following form
( P Ä A2 )(I + U + V )C = h2 F + h2 E,
(15)
~
( P Ä A2 )(I + U + V )C = h 2 F~ ,
(16)
U = h( P Ä A2 ) -1 ( R Ä A1 ),
(17)
C
with
-1
V = h ( P Ä A2 ) ( L Ä A0 ).
2
~
In order to determine the bounded of || C - C || ¥ , we need the following Lemma.
71
(18)
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Lemma 3.1 If
h2 r <
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Dt
-1
, then I + U + V is invertible, where r =|| ( P Ä A2 ) ||¥ .
4
Proof: From the relation (17), we have
|| U ||¥ £ h || ( P Ä A2 )-1 ||¥ || ( R Ä A1 ) ||¥
£ hr || ( R Ä A1 ) ||¥ .
For h sufficiently small, we conclude
1
|| U ||¥ < .
4
From the relation (18) and ||
(19)
A0 ||¥ £ 1, we have
|| V ||¥ £ h2 || ( P Ä A2 ) -1 ||¥ || L Ä A0 ||¥
£ h2 || ( P Ä A2 ) -1 ||¥ || L ||¥
2ö
æ
£ h 2 r ç || Q - G ||¥ + ÷
Dt ø
è
2h 2 r
.
£ h 2 r || Q - G || ¥ +
Dt
For h sufficiently small, we conclude that h
2
1
. Then
4
1 2h 2 r
|| Q - G ||¥ < +
.
4
Dt
r || Q - G ||¥ <
(20)
2h 2 r 1
< . So, || U + V || £ || U || + || V || < 1, and therefore I + U + V is invertible.
Dt
2
Dt
2
Proposition 3.1 If h £
, then there exists a constant cte which depends only on the functions p, q , l
4r
and g such that
~
|| C - C || £ cte.h 2 .
(21)
As
Proof: Assume that h
Since E
= O(
2
£
Dt
. From (15) and (16), we have
4r
~
C - C = h 2 ( I + U + V ) -1 ( P Ä A2 ) -1 E.
h2
h2
), then there exists a constant K1 such that || E || £ K1 . This implies that
Dt
Dt
~
|| C - C || £ h 2 || ( I + U + V ) -1 ||¥ || ( P Ä A2 ) -1 ||¥ || E ||
h2 r
|| ( I + U + V ) -1 ||¥ K1h 2
Dt
1
£ || ( I + U + V ) -1 ||¥ K1h 2 .
4
On the other hand, from (19) and (20), we get || U +V ||¥ < 1. Thus
£
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1
= cte.
1- || U + V ||¥
|| ( I + U + V ) -1 ||¥ £
Finally, we deduce that
~
|| C - C || £ cte.h 2 .
Now, we are in position to prove the main theorem of our work.
Proposition 3.2 The spline approximation
i.e.,
~
V - Sp
~
Sp converges quadratically to the exact solution V of problem (2),
= O(h 2 ) .
H
Proof: From the relation (10), we have
V - Q3 (V )
H
= O(h 4 ) , so V - Q3 (V )
H
£ Kh 4 , where K is a positive constant. On the other
hand we have
~
Q3 (Vi ( x)) - Sp i ( x) =
~
n-2
å (m (V ) - c
j
i
j ,i
) B j ( x), for i = 1,..., H .
j = -2
Therefore, by using (21) and
n-2
å
B j ( x) £ 1, we get
j = -2
~
~
| Q3 (Vi ( x)) - Sp i ( x) |£|| C - C ||
~
n-2
å B ( x) £|| C - C || £ K h ,
2
1
j
for i = 1,..., H .
j = -2
~
Since || Q3 (V ) - Sp || H £
~
|| V - Q3 (V ) || H + || Q3 (V ) - Sp || H , we deduce the stated result.
4. Numerical examples
In this section we verify experimentally theoretical results obtained in the previous section. If the exact
solution is known, then at time t £ T the maximum error
E max =
max
E max can be calculated as:
| SiM , N ( x, t ) - Vi ( x, t ) | .
xÎ[ xmin , xmax ], tÎ[0,T ],1£i £ H
Otherwise it can be estimated by the following double mesh principle:
max
EM , N =
where S
M ,N
i
max
xÎ[ xmin , xmax ], tÎ[0,T ],1£i £ H
| SiM , N ( x, t ) - Si2 M ,2 N ( x, t ) |,
( x, t ) is the numerical solution on the M + 1 grids in space and N + 1 grids in time, and
S
( x, t ) is the numerical solution on the 2M + 1 grids in space and 2 N + 1 grids in time, for
1£ i £ H .
2 M ,2 N
i
We need to estimate the integral
òV
m
i
R
formula in a bounded interval of the form
( x + z )n i dz and for this purpose we use a Gaussian quadrature
[ zmin , zmax ] to arrive at
p
z
m
i
i max m
i
m
ò Vi ( x + z)n dz » l ò Vi ( x + z) f ( z)dz » l åwkVi ( x + zk ) f ( zk ),
R
zmin
(22)
k =1
for i = 1,.., H in which the wk ’s are the Gaussian quadrature coefficients; cf. [21, 6] for details.
We present two examples to better illustrate the use of the switching Lévy approach and the proposed pricing
methodology in concrete situations. These examples are concerned with American put options in three and five
world states respectively. In the first example, we assume that the stock price follows a Merton jump-diffusion
process with an intensity parameter governed by a three-state hidden Markov chain. In the second one, we
consider the Kou jump diffusion model with jump intensities having a discrete five-state Markov dynamics.
4.1. Example 1
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In this example, we assume a three-regime economy in which the dynamics of the underlying stock price in the
i-th regime obeys a Merton jump-diffusion process with the Lévy measure
(z - m j )
li
exp{n ( z) =
},
2s 2
2p s j
j
2
i
where the intensity vector is given by:
the generator matrix is defined by
l = [0.3,0.5,0.7]T ,
é- 0.8 0.6 0.2 ù
Q = ê 0.2 - 1 0.8 ú.
ê
ú
ê 0.1 0.3 - 0.4ú
ë
û
T
and the a priori state probabilities are given to be p = [0.2,0.3,0.5] .
Other useful data are provided in the following table:
Table 1. Data used to value American options under regime-switching jump-diffusion models.
Parameter
S
values
100
K
100
T
1
s
r
sj
mj
0.15
0.05
0.45
- 0.5
For this problem, we use a uniform distribution of points in the interval
[ xmin , xmax ] = [-6,6] for the
collocation process and truncate the integration domain in (22) according to
zmax = (-2s 2 log(es
j
j
2p /2)) + m j ,
zmin = - zmax ,
-12
with e = 10e . We must note here that using these two bounds forces the total truncation error to be
uniformly bounded by e and the derivation of them is described in full detail in [12] and [19].
The comparison of the maximum error values between the method developed in this paper with the one
developed in [2] will be taken at five different values of the number of space steps
N = 256, 512, 1024, 2048, and time steps M = 128, 256, 512, 1024, .
We conduct experiments on different values of N , M and s . Table 2 show values of the maximum error
(max_error) obtained in our numerical experiments and the one obtained in [2]. We see that the values of
maximum error obtained by our method improve the ones in [2].
Table 2. Numerical results for three world states
M
Our max_error
max_error in [2]
N
256
128
0.83 ´10-3
2.86 ´10-3
512
256
0.20 ´10-3
1.78 ´10-3
1024
512
0.52 ´10-4
0.88 ´10-3
2048
1024
0.13 ´10-4
0.36 ´10-3
4.2. Example 2
In this example, we assume that the stock price process follows the Kou jump-diffusion model where the
jumps arrive at Poisson times and are distributed according to the law
n i ( z ) = li ( ph1e
-h1z
h z
1z ³0 + (1 - p)h2e 2 1z <0 ).
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We assume that our five-state Markov chain has a generator of the form:
é -1
ê0.25
ê
Q = ê0.25
ê
ê0.25
ê0.25
ë
and
that
the
economy
0.25
-1
0.25
0.25
0.25
between
different
switches
0.25
0.25
-1
0.25
0.25
0.25
0.25
0.25
-1
0.25
jump
0.25ù
0.25ú
ú
0.25ú,
ú
0.25ú
-1 ú
û
intensities
described
by
the
vector
l = [0.1,0.3,0.5,0.7,0.9]T .
In this case, we suppose that the market could be in any of the five regimes with equal probability. Other
corresponding information is given in the following table:
Table 3. Data used to value American options under regime-switching jump-diffusion models.
Parameter
S
values
100
K
100
T
0.25
s
r
p
h1
h2
0.5
0.05
0.5
3
2
We use a uniform distribution of points in the interval
[ xmin , xmax ] = [-6,6] as collocation points and use the
following bounds for the truncation process in (22):
zmax = log(e/p)/(1 -h1 ),
zmin = - log(e/(1 - p))/(1 -h2 ) ,
-12
where we use the value of e = 10e . We refer the reader to [19] to see a full derivation of these bounds in
order to obtain uniform truncation error bounds. Table 3 contains the option prices corresponding to each
intensity regime reported for different values of N and M .
The comparison of the maximum error values between the method developed in this paper with the one
developed in [2] will be taken at five different values of the number of space steps N = 512, 1024, 8198 and
= 0.1, l2 = 0.3, l3 = 0.5, l4 = 0.7 and l5 = 0.9 .
We conduct experiments on different values of N , M and l . Table 4 show values of the maximum error
time steps M = 256, 512, 512 for l1
(max_error) obtained in our numerical experiments and the one obtained in [2]. We see that the values of
maximum error obtained by our method improve the ones in [2].
5. Conclusion
In this paper, a cubic spline collocation approach is introduced to price American options in a regime-switching
Lévy context. After a brief review of the process of deriving the set of coupled PIDEs describing the prices in
different regimes, we present the details of our methodology which consists of first discretizing in time (by
Crank-Nicolson scheme) and then collocating in space (by a cubic spline ollocation method). Then, we have
shown provided an error estimate of order
O(h 2 ) with respect to the maximum norm
H
. In our paper we
consider a cubic spline space defined by multiple knots in the boundary and we propose a simple and efficient
new collocation method by considering as collocation points the mid-points of the knots of the cubic spline space.
It is observed that the collocation method developed in this paper, when applied to some examples, can improve
the results obtained by the collocation methods given in the literature. The two test problems which are studied
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in this paper demonstrate that this approach has an efficient alternative to the one proposed in [2].
Table 4. Numerical results for different intensity regimes and discretization parameters.
Our max_error
max_error in [2]
M
N
For
l1 = 0.1
512
256
0.00150
0.0356
1024
512
0.00037
0.0132
8198
512
0.00030
0.0060
For
l2 = 0.3
512
256
0.00146
0.0348
1024
512
0.00036
0.0136
8198
512
0.00029
0.0066
For
l3 = 0.5
512
256
0.00143
0.0341
1024
512
0.00035
0.0138
8198
512
0.00028
0.0070
For
l4 = 0.7
512
256
0.00139
0.0339
1024
512
0.00034
0.0140
8198
512
0.00027
0.0071
For
l5 = 0.9
512
256
0.00133
0.0338
1024
512
0.00033
0.0142
8198
512
0.00026
0.0072
Acknowledgment
We are grateful to the reviewers for their constructive comments that helped to improve the paper.
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