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Reports of the Department of Mathematical Information TechnologySeries B. Scientific ComputingNo. B 12/2005
Efficient Numerical Methods for PricingAmerican Options Under Stochastic Volatility
Samuli Ikonen Jari Toivanen
University of JyvaskylaDepartment of Mathematical Information Technology
P.O. Box 35 (Agora)FI–40014 University of Jyvaskyla
FINLANDfax +358 14 260 2731
http://www.mit.jyu.fi/
Copyright c© 2005Samuli Ikonen and Jari Toivanen
and University of Jyvaskyla
ISBN 951-39-2349-5ISSN 1456-436X
Efficient Numerical Methods for PricingAmerican Options Under Stochastic Volatility
Samuli Ikonen∗ Jari Toivanen∗
Abstract
Five numerical methods for pricing American put options under Heston’sstochastic volatility model are described and compared. The option prices areobtained as the solution of a two-dimensional parabolic partial differential in-equality. A finite difference discretization on nonuniform grids leading to linearcomplementarity problems with M -matrices is proposed. The projected SOR, aprojected multigrid method, an operator splitting method, a penalty method,and a componentwise splitting method are considered. The last one is a directmethod while all other methods are iterative. The resulting systems of linearequations in the operator splitting method and in the penalty method are solvedusing a multigrid method. The projected multigrid method and the componen-twise splitting method lead to a sequence of linear complementarity problemswith one-dimensional differential operators which are solved using the Brennanand Schwartz algorithm.
The numerical experiments compare the accuracy and speed of the con-sidered methods. The accuracies of all methods appear to be similar. Thus,the additional approximations made in the operator splitting method, in thepenalty method, and in the componentwise splitting method do not increasethe error essentially. The componentwise splitting method is the fastest one.All multigrid based methods have similar rapid grid independent convergencerates. They are from two to four times slower that the componentwise splittingmethod. On the coarsest grid the speed of the projected SOR is comparable withthe multigrid methods while on finer grids it is several times slower.
Keywords: American option pricing, stochastic volatility model, linear comple-mentarity problem, finite difference method, operator splitting method, multigridmethod, penalty method
∗Department of Mathematical Information Technology, University of Jyvaskyla, PO Box35 (Agora), FI-40014 University of Jyvaskyla, Finland, [email protected],[email protected]
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1 IntroductionThe valuation of financial options leads to mathematical models which are oftenchallenging to solve. Since the seminal paper [4] by Black and Scholes in 1973, em-pirical evidence has shown their assumption on the log-normality of the value ofthe underlying asset to be oversimplifying for most of asset classes. This has ledto more sophisticated models for the value of the underlying. Examples of theseare value and time dependent volatility functions [15], jump processes for the value[11], [30], their combinations [2], stochastic volatility models [18], [20], and stochas-tic volatility models with jumps [14].
An American option can be exercised at any time during the life of the optionwhile a European option can be exercised only at the expiry date. The early exercisepossibility leads to a constraint for the value of the American option. This constraintrequires special treatment which makes usually analytical formulas intractable andalso the numerical valuation more complicated. Based the Black and Scholes par-tial differential equation (PDE) Brennan and Schwartz price American options in[6]. After a finite difference discretization they proposed a direct method for thetreatment of the early exercise constraint.
In this paper we study efficient numerical methods for pricing American putoptions with Heston’s stochastic volatility model [20]. The option pricing modelis based on a two-dimensional parabolic PDE with variable coefficients. Due tothe early exercise possiblity, the model is a time dependent linear complementarityproblem (LCP). The main purpose of this paper is to compare the computationalefficiency of five numerical solution methods for the LCP which are mentioned inthe following. In addition to this, we derive a discretization with good propertiesand we make improvements to some of the methods. The projected SOR (PSOR)method is the most well-know while the penalty method and projected multigridmethods have been applied more recently in the option pricing. The operator split-ting method and the componentwise splitting method have been proposed by theauthors in [24], [25], [26].
We propose a finite difference space discretization on a nonuniform grid result-ing an M -matrix. The cross-derivative term is approximated with a special finitedifference scheme and the first-order and the second-order partial derivatives areapproximated using usual finite differences. In order to obtain non positive codiag-onal elements, we restrict the grid step sizes and we use one-sided differences forthe convection terms in a small part of the domain. The locations of the grid pointsare computed using grid generating functions which concentrate more grid pointsnear the point where the option price is required. The Rannacher time-stepping [35]is used for the time discretization.
The projected SOR method introduced in [12] has been used widely for pricingAmerican options; see, for example, [37], [39], [43]. The convergence rate of thePSOR method deteriorates when the discretization is refined which makes it slow onfiner grids. We study the choice of the relaxation parameter, since it has a significantimpact on the convergence rate.
2
The projected full approximation scheme (PFAS) multigrid method for LCPs isintroduced in [5]. It has been applied for American option pricing by Clarke andParrott in [10] and by Oosterlee in [33]. A projected line Gauss-Seidel smoother inthe multigrid method is used in [10] and several smoothers are studied and com-pared in [33]. Based on the conclusions in [33] we use the alternating direction lineGauss-Seidel smoother in this paper. Furthermore, we use the projected multigridmethod considered by Reisinger and Wittum in [36]. It is more closely related to thebasic multigrid method for linear problems than the PFAS.
The operator splitting methods for pricing American options [24], [25], treats theearly exercise constraint in a separate fractional time step which is computationallyinexpensive. The other fractional step requires the solution of a system of linearequations. The overall speed of the operator splitting method is determined by thesolution method for the system of linear equations. Here we apply the multigridmethod described in [25] for these problems. According to our earlier studies theadditional error due to the splitting is small compared to the discretization error.The operator splitting method is particularly useful when there is a more efficientway to solve linear problems than the corresponding LCPs. This is the case for two-factor American option pricing problems.
Zvan, Forsyth, and Vetzal apply penalty methods for pricing American optionsunder stochastic volatility in [46]. One of their methods is based on the l1 exactpenalty function which we are also using in this paper. This method was studiedfor the one-dimensional Black-Scholes model in [17]. The resulting nonlinear andnonsmooth problems in [46] are solved using a semismooth Newton method. Thisleads to systems of linear equations which were solved with an ILU preconditionedBiCGSTAB method. Here we propose to use a multigrid method for the result-ing linear problems. Alternative penalty methods for the Black-Scholes model havebeen considered in [1], [27], [32], for example.
Componentwise splitting methods was proposed in [26] for pricing Americanoptions under stochastic volatility. It approximates a discrete LCP by five sets ofLCPs with tridiagonal matrices at each time step. The first three sets consist of LCPsalong grid lines in the x-direction, in the y-direction, and in the cross direction. Dueto the Strang symmetrization the two last sets are LCPs in the y-direction and in thex-direction. The early exercise regions are such that the direct Brennan and Schwartzalgorithm [6] can be used for solving LCPs with tridiagonal matrices. Options ontwo stocks were priced using an ADI method, which is somewhat similar to thecomponentwise splitting method, in [41].
This paper is organized as follows. The American option pricing model is pre-sented in Section 2. In Section 3 a finite difference scheme for the space discretiza-tion and the time discretization scheme are described. Five numerical algorithmsfor the solution of the American option pricing problem are considered in Section 4.Numerical experiments are performed in Section 5 and conclusions end the paper.
3
2 Model for American optionsIn this section we describe a model for pricing American options. The most commonway to obtain the price of an American option is to formulate a linear complemen-tarity problem and then solved it numerically; see, for example, [10], [23], [33], [34],[46]. In the following, the price of an underlying asset is modeled in a more realisticway than only applying an asset price model with a log-normal return distribution.Under Heston’s stochastic volatility model and with suitable assumptions on themarkets a linear complementarity problem with a two-dimensional parabolic par-tial differential operator can be derived for the price of American options; see [20],[43], [46].
We denote the time to expiry by t, the price of the underlying asset by x, and itsvariance by y. A generalized Black-Scholes operator is defined by
Lu =∂u
∂t− 1
2yx2 ∂2u
∂x2−ργyx
∂2u
∂x∂y− 1
2γ2y
∂2u
∂y2−rx
∂u
∂x−{α(β−y)−ϑγ
√y}∂u
∂y+ru, (1)
where the r is the risk free interest rate, β is the mean level of the variance, α isthe rate of reversion on the mean level, and γ is the volatility of the variance. Themarket price of the risk is denoted by ϑ and in the rest of the paper it is assumedto be zero as in [10], [33], [46]. The correlation between the price of the underlyingasset and its variance is ρ.
The price at the exercise moment is given by the payoff function g. Furthermore,it defines the initial condition
u(x, y, 0) = g(x, y). (2)
For a put option with the exercise price E, the payoff function is
g(x, y) = max{E − x, 0}. (3)
Due to the early exercise possibility of the American option the price u has to beat least the same as the payoff function g. This leads to the early exercise constraintu(x, y, t) ≥ g(x, y). In the region where the constraint is inactive the price u satisfies apartial differential equation Lu = 0. By combining these relations a time dependentlinear complementarity problem is obtained for the price of the American option. Itis not know a priori where the constraint is active and this makes deriving analyticalformulas for the price intractable.
The linear complementarity problem for the price of the American put option is{
Lu ≥ 0, u ≥ g,
(Lu)(u − g
)= 0,
(4)
in a domain {(x, y, t)| x ≥ 0, y ≥ 0, t ∈ [0, T ]} with the initial condition (2). On theboundaries x = 0 and y = 0, Dirichlet boundary conditions
u(0, y, t) = g(0, y) and u(x, 0, t) = g(x, 0) (5)
4
are posed. On far-field the asymptotic behavior of u satisfies the conditions
limx→∞
∂u(x, y, t)
∂x= 0 and lim
y→∞
∂u(x, y, t)
∂y= 0. (6)
These conditions are described in [10] and [33], for example.
3 Discretization on nonuniform gridsHere we consider the finite difference discretization of the generalized Black-Scholesoperator (1) and the linear complementarity problem (4). We use second-order accu-rate differences as much as possible, but when they would lead to a positive codiag-onal element we employ first-order accurate one-sides differences for the convectionterms. With certain grid step size limitations we obtain a coefficient matrix havingan M -matrix property. Such matrices have many good properties; see [44], for ex-ample. In the context of option pricing with two-factor models the importance ofnon positive codiagonal entries has been studied in [47].
For the discretization a sufficiently large computational domain is truncated fromthe infinite domain. The truncation for the one-dimensional Black-Scholes PDE isconsidered in [29]. For the computational domain the finite difference approxima-tion is constructed on a grid
(xi, yj, tk) ∈ {0 = x0, . . . , xm = X}× {0 = y0, . . . , yn = Y }× {0 = t0, . . . , tl = T}. (7)
On the truncation boundaries {X} × [0, Y ] and [0, X] × {Y }, we obtain Neumannboundary conditions
∂u(X, y, t)
∂x= 0 and
∂u(x, Y, t)
∂y= 0 (8)
as approximations of (6).In the following the finite difference approximation is introduced. Then the gen-
eration of the grid is described and finally, a time discretization scheme is given. Bycombining these a discrete linear complementarity problem is obtained in the endof this section.
3.1 Space discretizationWe use a finite difference method for the space discretization of the partial differ-ential operator (1). We employ nonuniform grids to increase the efficiency of thediscretization and also to avoid positive codiagonal elements. Here, by efficiencywe mean that the desired accuracy of the option price is obtained with a fewer gridpoints. We apply central finite differences for the diffusion terms. The convectionterms are mostly approximated using central finite differences but in those part ofthe computational domain where the convection dominates the diffusion one-sided
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finite differences are used in order to attain non positive codiagonal elements. Thecross-derivative term is approximated using a second-order accurate scheme. For ageneral description of finite difference methods see [31], [37], [39], for example.
A seven point discretization stencil is described at a reference grid point (xi, yj).We denote step sizes related to this reference point by
hl = xi − xi−1, hr = xi+1 − xi, hd = yi − yi−1, and hu = yi+1 − yi. (9)
For brevity, we use the notation (x, y) for the reference point instead of (xi, yj) in thefollowing.
The treatment of the cross-derivative term in (1) is somewhat nonstandard and,hence, it is described in the following. For the discretization of this derivative we as-sume the correlation ρ to be non negative. A similar approximation to the followingone can be constructed also for a negative ρ, but we do not consider it in this paper.In order to obtain a matrix with desired properties we approximate the derivativeby a convex combination
∂2u
∂x∂y≈ w
hlhd
(u(x − hl, y − hd) − u + hl
∂u
∂x+ hd
∂u
∂y− 1
2h2
l
∂2u
∂x2− 1
2h2
d
∂2u
∂y2
)
+1 − w
hrhu
(u(x + hr, y + hu) − u − hr
∂u
∂x− hu
∂u
∂y− 1
2h2
r
∂2u
∂x2− 1
2h2
u
∂2u
∂y2
),
(10)
where w ∈ [0, 1]. We have omitted the arguments of the function u and its deriva-tives when they are evaluated at the reference grid point (x, y). With this scheme itis possible to obtain a matrix with non positive codiagonal elements while with amore usual scheme this is not the case.
The approximation (10) leads to a form which contains only the partial deriva-tives in the x-direction and in the y-direction. Before discretizing these derivativeswe give an intermediate form with compact notations
∂u
∂t+ dx
∂2u
∂x2+ dy
∂2u
∂y2+ cx
∂u
∂x+ cy
∂u
∂y+ ru
+ Aldu(x − hl, y − hd) −(Ald + Aru
)u + Aruu(x + hr, y + hu) = 0,
(11)
where the coefficients are
dx = −1
2yx2 + wργyx
hl
2hd
+ (1 − w)ργyxhr
2hu
,
dy = −1
2γ2y + wργyx
hd
2hl
+ (1 − w)ργyxhu
2hr
,
cx = −rx − wργyx1
hd
+ (1 − w)ργyx1
hu
,
cy = −α(β − y) − wργyx1
hl
+ (1 − w)ργyx1
hr
,
Ald = −wργyx1
hrhu
, and Aru = −(1 − w)ργyx1
hrhu
.
(12)
6
Both diffusion terms in (11) are approximated with central finite differences. For theconvection terms one-sided finite differences are used when central finite differenceswould lead to a coefficient matrix with positive codiagonal elements and, thus, to anon M -matrix.
Next, we describe the discretization stencils for the diffusion and convectionterms in (11). In the x-direction the central finite difference scheme for the diffu-sion term (DC) gives codiagonal elements
ADCl =
2
hl(hr + hl)dx and ADC
r =2
hr(hr + hl)dx. (13)
The convection term in the x-direction is approximated using the central, forward,or backward finite difference scheme. The central finite difference scheme for theconvection term (CC) gives codiagonal elements
ACCl = − hr
hl(hr + hl)cx and ACC
r =hl
hr(hr + hl)cx. (14)
The forward (CF) and backward (CB) schemes lead to codiagonal elements
ACFl = 0, ACF
r =1
hr
cx, ACBl = − 1
hl
cx, and ACBr = 0. (15)
Similarly, the diffusion term in the y-direction in (11) is approximated using thecentral finite difference scheme (DC) leading to codiagonal elements
ADCd =
2
hd(hu + hd)dy and ADC
u =2
hu(hu + hd)dy. (16)
The central scheme (CC) for the convection term in the y-direction gives codiagonalelements
ACCd = − hu
hd(hu + hd)cy and ACC
u =hd
hu(hu + hd)cy. (17)
The forward (CF) or backward (CB) finite difference schemes lead to codiagonalelements
ACFd = 0, ACF
u =1
hu
cy, ACBd = − 1
hd
cy, and ACBu = 0. (18)
Based on the above stencils we choose the discretization of the diffusion andconvection terms in the x-direction in (11) so that the codiagonal elements are
Al =
ADCl + ACF
l , if ADCl > ACF
l ,
ADCl + ACB
l , if ADCl < ACB
l ,
ADCl + ACC
l , otherwise,
and Ar =
ADCr + ACF
r , if ADCl > ACF
l ,
ADCr + ACB
r , if ADCl < ACB
l ,
ADCr + ACC
r , otherwise.(19)
7
Similarly, in the y-direction the codiagonal elements are
Ad =
ADCd + ACF
d , if ADCd > ACF
d ,
ADCd + ACB
d , if ADCd < ACB
d ,
ADCd + ACC
d , otherwise,and Au =
ADCu + ACF
u , if ADCd > ACF
d ,
ADCu + ACB
u , if ADCd < ACB
d ,
ADCu + ACC
u , otherwise.(20)
The diagonal element includes the sum of the codiagonal elements describedabove with a minus sign. Furthermore, the term ru in (1) adds r into the diagonalelement. Hence, the diagonal element is given by
Adiag = r − Al − Ar − Ad − Au − Ald − Aru. (21)
Now the seven point finite difference stencil at an interior grid point (x, y) is
0 Au Aru
Al Adiag Ar
Ald Ad 0
. (22)
In the discretization of the cross-derivative term we use the weight w = 0.5 in(12) at all grid points except the ones mentioned in the following. On the bound-aries {X} × [0, Y ] and [0, X] × {Y }, we apply the Neumann boundary conditions(8) together with the weight w = 1. This leads to a stencil which has the same spar-sity pattern as (22) when references outside the computational domain are removed.Furthermore, in order to have more freedom to choose the location of the grid linex = x1, we use w = 0 on it.
The finite difference space discretization of (4) leads to a semi-discrete LCP whichhas a compact matrix form
{∂u∂t
+ Au ≥ 0, u ≥ g,(
∂u∂t
+ Au)T (
u − g)
= 0,(23)
where A is a block tridiagonal (m+1)(n+1)×(m+1)(n+1) matrix and u is a vectorof length (m + 1)(n + 1). The vector g contains the values of the payoff function g atthe grid points. The initial value of u is given by g.
3.2 Bounds for grid step sizesIn the previous section we described the finite difference discretization stencils. Thestraightforward use of such a discretization can lead to oscillations although theone-sided differences are used for the convection terms. In the following we derivelower and upper bounds for the step sizes which ensure that the space discretizationgives the matrix with an M -matrix property.
The coefficients of the second-order derivatives in (11) can be positive whichleads to positive codiagonals when the central finite difference schemes are applied.This can be avoid by choosing the step sizes in a special way. In the following we
8
derive conditions leading to negative coefficients and which then allows to use thecentral finite difference schemes for the diffusion terms. The M -matrix property isachieved if the matrix is diagonally dominant with positive diagonal elements andnon positive codiagonal elements.
We begin by considering the codigonal elements due to the diffusion term in thex-direction in (11). They are non positive when dx ≤ 0. We easily obtain from (12)that this is holds when
hl ≤1
ργxhd and hr ≤
1
ργxhu. (24)
The codiagonal elements due to the diffusion term the y-direction in (11) are ana-lyzed in the same way. This gives us the lower bounds
ρ
γxhd ≤ hl and
ρ
γxhu ≤ hr, (25)
for the step sizes hl and hr. Assuming hd < hu and combining the inequalities (24)and (25), we obtain the restrcitions
ρ
γxhu ≤ hl ≤
1
ργxhd and
ρ
γxhu ≤ hr ≤
1
ργxhd, (26)
for the step sizes. In order to simplify (26) we consider inequalities for hr. At thenode xi, these inequalities read
ρ
γxihu ≤ hr,i ≤
1
ργxihd, (27)
where the subscript i in hr refers to the node xi. In the same way we can apply theinequalities (26) at the node xi+1. By noticing xi+1 = xi + hr,i and hl,i+1 = hr,i, weobtain
ρ
γxihu
(1 − ρ
γhu
)−1
≤ hr,i ≤1
ργxihd
(1 − 1
ργhd
)−1
, (28)
where it is assumed that 1 − (ρhu)/γ > 0.Finally, combining (27) and (28), the lower and upper bounds for the step size hr
at the node xi are of the form
ρ
γxihu
(1 − ρ
γhu
)−1
≤ hr,i ≤1
ργxihd. (29)
It should be noted that in the previous inequalities the lower bound is smaller thanthe upper bound only when the inequality
hu
hd
<1
ρ2
(1 − ρ
γhu
), (30)
is satisfied. This sets one more restriction for the choice of the nonuniform gridin the y-direction. The M -matrix property is achieved when the partial differentialequation is discretized as described in Section 3.1 using step sizes satisfying thelower and upper bounds at each node.
9
3.3 Space discretization gridIn this section we consider the generation of nonuniform grids for the space dis-cretization. First, we descibe the generation of grid points in the x-direction and inthe y-direction and then we describe a way to take the lower and upper bounds forthe step sizes into account in the x-direction.
An advantage of nonuniform grids is that the grid can be made more densewhere it increases the accuracy of the desired price the most. Thus, the same ac-curacy can be obtained with a fewer grid points with nonuniform grids than withuniform grids. When the correlation ρ between the price of the asset and its vari-ance is non zero it is generally necessary to use a nonuniform grid in the x-directionin order to satisfy the lower and upper bounds for the grid step sizes so that anM -matrx can be obtained. Particularly with larger values of ρ, uniform grids leadnon M -matrices and unstable numerical solutions with the componentwise splittingmethods.
3.3.1 Grid generating function in the x-direction
We use a grid generating function
hr(x) = ax2 + bx + c, x ∈ [0, X], (31)
to define step sizes in the x-direction. With the parabola function (31) we can gener-ate more dense grid near the exercise price E. Conditions
∫ X
0
1
hr(x)dx = mt, h′
r(E) = 0, and κx =hr(X)
hr(E), (32)
are used to determine the coefficients a, b, and c. Our aim is to have approximatelymt+1 grid points in the [0, X] interval and for this reason we have the first conditionin (32). The second condition defines the grid generating function in a way that thesmallest step size is located at E. The last condition says that the ratio between thegrid step sizes at X and at E is κx. These three conditions defines the coefficients a,b, and c uniquely.
3.3.2 Grid generating function in the y-direction
In the y-direction we apply a linear grid generating function
hu(y) = dy + e, y ∈ [0, Y ], (33)
to define step sizes. The coefficients in (33) are chosen to be
d =log κy
nand e =
d
(d + 1)n − 1Y. (34)
With the choice (34) there will be exactly n + 1 grid points in the y-direction and theratio of the last step size and the first step size is approximately κy. In the followingand in (34), we assume that κy > 1.
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3.3.3 Nonuniform grid
We describe how the nonuniform grid is generated. At the first place, the nonuni-form grids are generated using the grid generation functions hr and hu. In order tohave stable discretization we have to take into account the lower and upper boundsfor the step sizes in the x-direction. Next, we describe how we implement the dis-cretization grid.
First, the function hu is used to generate the nodes in the y-direction and then thefunction hr is applied in the x-direction. The function hr(x) should be applied withthe lower and upper bounds defined in Section 3.2. In order to be well within thebounds we add 10 % margin to them. The modified grid generating function
hr(x) = min
{max
{hr(x), 1.1
(1 − ρ
γhu(Y )
)−1 ρ
γhu(Y )x
}, 0.9
1
ργhu(0)x
}(35)
is then used in our numerical experiments. Moreover, the limitation for the stepsizes in the y-direction was given in (30) is verified numerically.
3.4 Time discretizationThe option pricing model has a nonsmooth initial value. That is because the payofffunction has discontinuous first derivative. Although the Crank-Nicolson methodis popular it does not have good stability properties. In many cases the Crank-Nicolson method can lead to a numerical solution with oscillation because it is notL-stable. In order to obtain numerical solution without undesired oscillations weuse the Rannacher time-stepping method in our numerical experiments [35]. Thistype of method is also applied in the option pricing in [17], [34], for example.
The Rannacher time-stepping scheme performs a few first time steps with theimplicit Euler method and after that it uses the Crank-Nicolson method. This waythe scheme has good damping properties and second-order accuracy. For the semi-discrete LCP in (23) the Rannacher time-stepping scheme reads
(I + 1
2∆tA
)u(k+1) ≥ u(k), u(k+1) ≥ g,
[(I + 1
2∆tA
)u(k+1) − u(k)
]T (u(k+1) − g
)= 0,
k = 0, 1, 2, 3,
(I + 1
2∆tA
)u(k+1) ≥
(I − 1
2∆tA
)u(k), u(k+1) ≥ g,
[(I + 1
2∆tA
)u(k+1) −
(I − 1
2∆tA
)u(k)
]T (u(k+1) − g
)= 0,
k = 4, . . . , l − 1,
(36)where the constant time step is ∆t = T/(l−2) and the initial value u(0) is given by g.The implicit Euler method is applied with step size ∆t/2 in the first four time stepsand the time-stepping is continued with the Crank-Nicolson method using step size∆t. The advantage of this specific choice is that only one coefficient matrix needs tobe formed.
11
We remark that the efficiency of the time stepping could be improved by varyingthe time steps adaptively. For option pricing this approach has been considered in[1], [10], [17], for example.
3.5 Discrete linear complementarity problemsThe LCPs appearing in the Rannacher time-stepping (36) have the general form
Bu(k+1) ≥ Cu(k), u(k+1) ≥ g,(Bu(k+1) − Cu(k)
)T (u(k+1) − g
)= 0,
(37)
for k = 0, . . . , l − 1. The matrix B is the same for all k while C changes after foursteps. LCPs like (37) appear repeatedly in the following and in order to simplify thenotations we denote the problem (37) for a given k by
LCP(B,u(k+1),Cu(k), g). (38)
4 Solution methods for time dependent complementar-ity problems
Here we describe five methods for time dependent LCPs. The speed and accuracy ofthese methods are compared in the numerical experiments in Section 5. The two firstones are the projected SOR method and the projected multigrid method. They bothsolve iteratively the LCPs in (37) without making additional approximations. Thethree other considered methods, namely the operator splitting method, the penaltymethod, and the componentwise splitting method, approximate the LCPs and, thus,make them easier to solve. First four methods are iterative while the componentwisesplitting method is a direct method.
4.1 Projected SOR methodThe projected SOR method for linear complementarity problems was proposed byCryer in [12]. The method is well-known and widely applied for pricing Americanoptions; see, for example, [37], [39]. For completeness we present the followingalgorithm which performs one PSOR iteration for the problem LCP(B,u,f , g). Alsothe stopping criterion used in the numerical experiments is based on the vector r
generated by the algorithm.
Algorithm PSOR(B,u,f ,g)Do i = 1, dim B
ri = f i −∑
j Bi,juj
ui = max {ui + ωri/Bi,i, gi}End Do
12
The convergence rate of the (P)SOR method depends strongly on the choice of therelaxation parameter ω. Let ρG be the spectral radius of the Jacobi iteration matrixG = D−1(B − D), where D is the diagonal of B. Then a classical result states thatthe relaxation parameter
ω =2
1 +√
1 − ρ2G
(39)
is optimal for the SOR method when the eigenvalues of G are real; see [3], [45], forexample. In our case, some of the eigenvalues of G have relatively small imaginaryparts. However the numerical experiments in Section 5 show that (39) suggest rea-sonable values for the relaxation parameter. Using the Gershgorin circle theoremwe can conveniently approximate the spectral radius
ρG ≈ maxi
1
Bi,i
∑
j 6=i
|Bi,j|. (40)
We propose to use the approximation of ω in (39) computed using (40) with thePSOR method.
4.2 Projected multigrid methodMultigrid methods are scalable iterative methods in the sense that the convergencerate of a well designed method does not deteriorate when the grid is refined. Herewe describe the problem specific parts of our multigrid methods while for generaldetails we refer to the extensive scientific literature on the topic; see, for example, thebooks [7], [19], [40], [42]. The considered projected multigrid (PMG) method solveslinear complementarity problems (38). In this section we also present a fairly similarmultigrid method for systems of linear equations which is used with the operatorsplitting method and the penalty method.
Brandt and Cryer introduced a projected full approximation scheme (PFAS) mul-tigrid method for solving LCPs in [5]. Reisinger and Wittum described a PMGmethod for LCPs which resembles more closely to a classical multigrid method forlinear problems in [36]. Partly due to this reason we employ the PMG approachin this paper. For pricing American options under stochastic volatility the PFASmethod has been used by Clarke, Parrott in [9], [10] and Oosterlee in [33].
The following recursive algorithms describe one V cycle of the PMG method forLCPs and the multigrid (MG) method for linear problems. The restriction operatorsare denoted by R, R and R while P is the prolongation (interpolation) operator. Thesmoothers for LCPs and linear problems are denoted by PS and S, respectively. Thechoices of all these operators are described in the following. The superscript c refersto vectors and matrices for one level coarser grid. For coarser grids we constructcoefficient matrices by discretizing the generalized Black-Scholes operator on them.
13
Algorithm PMG(B,u,f ,g) Algorithm MG(B,u,f )If Bc does not exist Then If Bc does not exist Then
LCP(B,u,f ,g) u = B−1f
Else ElsePS(B,u,f ,g) S(B,u,f )uc = 0 uc = 0
rc = R(f − Bu) rc = R(f − Bu)
gc = R(g − u)PMG(Bc,uc,rc,gc) MG(Bc,uc,rc)u = u + Puc u = u + Puc
PS(B,u,f ,g) S(B,u,f )End If End If
The operators R and R are the full weighting restriction for a uniform grid andthe injection, respectively. The one-sided restriction R introduced in [22] is definedby
(Rr)i =
{(RrA)i, if ui = gi
(RrI)i, otherwise,
where the vectors rA and rI are the restrictions of the residual into the active setand inactive set, respectively, given by
(rA)i =
{ri, if ui = gi
0, otherwise,and (rI)i =
{0, if ui = gi
ri, otherwise.(41)
We use the bilinear interpolation operator for a uniform grid as P in the algorithms.The generalized Black-Scholes operator requires a more involved smoother than
one based on the (point) Gauss-Seidel or SOR method. Clarke and Parrott used ax-line Gauss-Seidel type smoother for LCPs in their multigrid [9]. Oosterlee per-formed Fourier analysis for smoothers for the generalized Black-Scholes operator[33]. According to the analysis the x-line Gauss-Seidel smoother is efficient whenthe grid is sufficiently fine in the y-direction and an alternating direction line Gauss-Seidel method leads to a good convergence rate on all grids. Thus, we employsmoothers based on the alternating direction line Gauss-Seidel method.
Our alternating direction line Gauss-Seidel smoother solves first one-dimensionalproblems in the x-direction and then in the y-direction. Due to the discretizationstencil (22) all one-dimensional problems lead tridiagonal coefficient matrices. Forlinear problems these can be solved efficiently using LU decomposition. For LCPsthe smoother has to respect the complementarity condition, that is, the early exer-cise constraint. Fortunately, along any line in the x-direction or in the y-directionthe grid points corresponding the the active and inactive sets form two continuous(nonoverlapping) segments. This can be seen well in Figure 2. We remark that wehave not proved this, but it has been the case for all problems we have encountered.For such LCPs the Brennan and Schwartz algorithm described in Section 4.2.1 offersa very efficient method and we employ it in our line smoother for LCPs.
14
We use one presmooth iteration before moving to a coarsed grid and one post-smooth iteration after adding the correction from the coarser grid. Computationallyone MG cycle is roughly eight times more expensive than one (P)SOR iteration. Inthe PMG method there are some additional computations due to the injection of thelower bound g and due to a more involved restriction operator for the residual.
4.2.1 Brennan and Schwartz algorithm
The direct Brennan and Schwartz algorithm [6] was introduced for pricing a putoption using on the one-dimensional Black-Scholes PDE discretized with finite dif-ferences. It solves the linear complemetarity problem with a tridiagonal matrix T
using a UL decomposition with a projection in the backsubstitution step, where U
and L are bidiagonal upper and lower triangular matrices, respectively. Under theassumption that T is an M -matrix the algorithm was analyzed in [28]. A similarmethod for obstacle problems was described and analyzed in [16]. The followingdescribes the Brennan and Schwartz algorithm for LCP(T ,x, b, g); see also [1].
Algorithm BS(T ,x,b,g)Form decomposition UL = T
Solve Uy = b
x1 = max{y1/L1,1, g1}Do i = 2, dim T
xi = max{(yi − Li,i−1yi−1)/Li,i, gi}End Do
Sufficient conditions for the algorithm to yield the correct solution are: T is an M -matrix and there exists an integer k such that for the solution x it holds that xi = gi
for i ≤ k and xi > gi for i > k.
4.3 Operator splitting method with multigrid methodOperator splitting methods have been applied to American options pricing prob-lems in [24], [25]. These methods are also well-suited for linear complementarityproblems with a coefficient matrix arising from the discretization of a higher di-mensional partial differential operator. In order to introduce the operator splittingmethod we give the following formulation with a Lagrange multiplier λ(k+1). Thelinear complementarity problem (37) has an equivalent form
Bu(k+1) = Cu(k) + ∆tλ(k+1),
λ(k+1) ≥ 0, u(k+1) ≥ g,(λ(k+1)
)T (u(k+1) − g
)= 0,
(42)
for k = 0, . . . , l − 1.
15
The basic form of the operator splitting method has two fractional steps at eachtime step. At the first step a system of linear equations is solved and at the secondfractional step an intermediate solution u(k+1) and the Lagrange multiplier λ(k+1)
are updated. The operator splitting method reads
Bu(k+1) = Cu(k) + ∆tλ(k), (43)
u(k+1) − u(k+1) = ∆t(λ(k+1) − λ(k)
),
λ(k+1) ≥ 0, u(k+1) ≥ g,(λ(k+1)
)T (u(k+1) − g
)= 0,
(44)
for k = 0, . . . , l − 1. The initial value u(0) is the payoff vector g and the λ(0) equals tozero. A more detailed description of the operator splitting method is given in [25].
The update step (44) can be performed componentwise and it is computationallyinexpensive. Thus, the efficiency of the operator splitting method depends on theefficiency of the solution method for the systems of linear equations. We apply themultigrid method for linear problems, described in Section 4.2, for the solution of(43).
4.4 Penalty method with multigrid methodPenalty methods force the solution towards feasible one by penalizing violations ofearly exercise constraint. For American option pricing under stochastic volatilitythey have been considered by Zvan, Forsyth, and Vetzal in [46]. Particularly, weuse their l1 exact penalty function formulation which was also studied in [17]. Thepenalty approximation of the LCPs in (37) reads
Bu(k+1) = Cu(k) +1
εmax
{g − u(k+1), 0
}, (45)
where the minimum is taken componentwise.The problems (45) are nonlinear and nonsmooth. Following [46] we employ a
semismooth Newton method to solve these problems; see, for example, [21] for anal-ysis of this case. In order to present the method we denote x = u(k+1) and f = Cu(k).Furthermore, the initial guess is chosen to be x0 = u(k). Then the (p + 1)th iterant isgiven by
xp+1 = xp + dp, (46)where the vector dp is the solution of the system of linear equations
J(xp)dp = f +1
εmax {g − xp, 0} − Bxp = rp. (47)
The matrix J(xp) in (47) belongs to the generalized Jacobian [8] at xp and it is chosento be
[J(xp)]i,j = Bi,j +
{1ε, if i = j and x
pi < gi,
0, otherwise.(48)
16
We solve the systems of linear eqautions in (47) using the multigrid method forlinear problems described in Section 4.2. In [46], an ILU preconditioned BiCGSTABmethods was employed to solve these linear problems.
4.5 Componentwise splitting methodThe option pricing problem (37) has been solved using componentwise splittingmethods in [26]. These splitting methods are based on a decomposition of the matrixresulting from the space discretization. This large and sparse matrix is decomposedinto three matrices as
A = Ax + Axy + Ay, (49)
where the matrices in the right-hand side have simpler structure than A. The ma-trices Ax, Axy, and Ay contain the couplings of the finite difference stencil in thex-direction, in the xy-direction, and in the y-direction, respectively. Due to the termru in (11) one third of r is added to the diagonals of these three matrices. After asuitable permutation of rows and columns each of these matrices are block diagonalwith tridiagonal diagonal blocks.
In the basic componentwise splitting method each linear complementarity prob-lem in (37) is splitted into three subproblems [26]. This type of componentwisesplitting method is only first-order accurate. The splitting becomes more accuratewhen the Strang symmetrization is used [26], [38]. This modified componentwisesplitting method has five subproblems at each time step and it is of the form
LCP(Bx/2,u(k+1/5),Cx/2u
(k), g),
LCP(By/2,u(k+2/5),Cy/2u
(k+1/5), g),
LCP(Bxy,u(k+3/5),Cxyu(k+2/5), g),
LCP(By/2,u(k+4/5),Cy/2u
(k+3/5), g),
LCP(Bx/2,u(k+1),Cx/2u
(k+4/5), g),
(50)
for k = 0, . . . , l−1. The matrices in the Strang symmetrized componentwise splittingmethod (50) depend on the time discretization scheme. For example, in the case ofthe Crank-Nicolson method matrices are
Bx/2 = I +1
4∆tAx, Bxy = I +
1
2∆tAxy, By/2 = I +
1
4∆tAy,
Cx/2 = I − 1
4∆tAx, Cxy = I − 1
2∆tAxy, and Cy/2 = I − 1
4∆tAy.
(51)
Here, the coefficient 1/4 signifies that the step size ∆t/2 is applied instead of thestep size ∆t. Again after suitable permutations of rows and columns the matricesin (50) are block diagonal with tridiagonal diagonal blocks which are M -matrices.This together with suitable early exercise regions enable to the use of the Brennan
17
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 5 10 15 20
h
x
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0 0.2 0.4 0.6 0.8 1
h
y
Figure 1: The grid generating functions hr(x) together with its bounds for m = 160(left) and hu(y) for n = 64 (right).
and Schwartz algorithm described in Section 4.2.1. The applicability of this algo-rithm was discussed with the projected multigrid method in Section 4.2. Here thesolutions have to have suitable structure also along the diagonal grid lines, since theLCPs with the matrix By/2 are problems along these lines.
5 Numerical experimentsHere we study numerically the efficiency of the methods described above. TheAmerican option pricing problem under stochastic volatility (4) is solved with theparameter values
E = 10, T = 0.25, r = 0.1, α = 5, β = 0.16, γ = 0.9, and ρ = 0.1.
The same values of the parameters have been used also in [9], [33], and [46]. Simi-larly to [33], we truncate domain so that X = 20 and Y = 1.
Based on numerical tests we have chosen the grid refinement ratios to be κx = 2in (32) and κy = 5 in (34). Thus, in the x-direction the ratio between the longestand shortest grid step sizes is about two and similarly in the y-direction it is five.The grid generating function hu(y) in (33) constructs a grid in the y-direction withn + 1 points. In the x-direction we choose the target number mt of grid steps so thatthe use of the modified grid generating function hr(x) in (35) leads to a grid with msteps. We remark that care must be taken when choosing m, n, κx, and κy so thata feasible grid exists. Particularly, the ratio m/n has to be large enough. Figure 1shows the grid generating function in the x-direction for m = 160 together with thebounds for the grid step sizes and the grid generating function in the y-direction forn = 64.
In Table 1, we report the prices for the four values of x when the variance is y =0.0625. We have computed these prices using all described methods on four grids.In the iterative methods we have used a very strict stopping criterion. The PSORand PMG methods use the same discretizations and, thus, they lead to the same
18
Table 1: The prices of the put option when the variance is y = 0.0625.
x
method grid (m,n, l) 9 10 11 12PSOR/ (160, 64, 34) 1.10683 0.51430 0.21031 0.08129PMG (320, 128, 66) 1.10699 0.51735 0.21219 0.08169
(640, 256, 130) 1.10731 0.51864 0.21291 0.08184(1280, 512, 258) 1.10749 0.51939 0.21332 0.08196
operator (160, 64, 34) 1.10716 0.51444 0.21035 0.08130splitting (320, 128, 66) 1.10715 0.51744 0.21222 0.08170
(640, 256, 130) 1.10738 0.51868 0.21293 0.08185(1280, 512, 258) 1.10752 0.51941 0.21333 0.08197
penalty (160, 64, 34) 1.10683 0.51430 0.21031 0.08129method (320, 128, 66) 1.10699 0.51735 0.21219 0.08169
(640, 256, 130) 1.10731 0.51864 0.21291 0.08184(1280, 512, 258) 1.10749 0.51939 0.21332 0.08196
comp.wise (160, 64, 34) 1.10756 0.51475 0.21053 0.08137splitting (320, 128, 66) 1.10737 0.51759 0.21231 0.08174
(640, 256, 130) 1.10750 0.51876 0.21297 0.08187(1280, 512, 258) 1.10759 0.51946 0.21335 0.08198
0
0.2
0.4
0.6
0.8
1
5 6 7 8 9 10
y
x
Figure 2: The locations of the free boundaries at times 0.25k/16, k = 0, 1, . . . , 16,computed using the PSOR/PMG method and (640, 256, 130) grid.
prices. For all numerical results with the penalty method we have used the penaltyparameter ε = 10−4. With this choice the prices computed with the penalty methodare the same as with the PSOR method in Table 1. Moreover, the prices computedusing the operator splitting method and the componentwise splitting method differonly slightly from the other prices. The difference is especially small on the finestgrid. Figure 2 shows the behavior of the free boundary with respect to time.
We computed reference prices at 10 points using the componentwise splittingmethod with a very fine grid defined by m = 5120, n = 2048, and l = 2050. Theseprices are given in Table 2 together with the prices published in [9], [46], and [33].Particularly our reference prices and the ones computed by Zvan, Forsyth, and Vet-zal in [46] are in good agreement. They have at least four same decimals after round-ing our prices to four decimals.
For our next experiment we chose the stopping criterion for the iterative methods
19
Table 2: The reference prices computed using the componentwise splitting methodon a (5120,2048,2050) grid and the prices published in scientific literature.
x
reference y 8 9 10 11 12comp.wise 0.0625 2.000000 1.107641 0.520030 0.213668 0.082036splitting 0.25 2.078381 1.333647 0.795982 0.448278 0.242815[9] 0.0625 2.0000 1.1080 0.5316 0.2261 0.0907
0.25 2.0733 1.3290 0.7992 0.4536 0.2502[46] 0.0625 2.0000 1.1076 0.5202 0.2138 0.0821
0.25 2.0784 1.3337 0.7961 0.4483 0.2428[33] 0.0625 2.00 1.107 0.517 0.212 0.0815
0.25 2.079 1.334 0.796 0.449 0.243
to be‖r‖ ≤ 1
mn‖f‖, (52)
where f is the right-hand side vector. For linear problems r in (52) is the residualvector and for the projected multigrid it is the residual rI in (41) reduced on theinactive set, and for the PSOR method it is the vector r in Algorithm PSOR reducedon the inactive set using (41). We have used also the condition (52) for the semis-mooth Newton method. For all methods this choice leads to additional errors dueto the termination of the iteration which are between 2% and 8% of the discretiza-tion errors measured by the l2 norm using the 10 reference prices given in Table 2.The PSOR results were computed using the value of the relaxation parameter givenby (39) with the approximation (40). The resulting average number of iterations,the errors at the 10 reference points in the l2 norm, the ratios of consecutive errors,and the CPU times in seconds are reported in Table 3. We performed all numericalexperiments on a 3.40 GHz Intel Xeon PC. Our Fortran 90 implementations of themethods were compiled using the NAGWare f95 compiler.
The convergence rates of multigrid iterations for the systems of linear equationsand for the LCPs are similar. The projected multigrid method and the penaltymethod require some additional computational work when compared to the op-erator splitting method and this shows up as a slightly increased CPU times. Thenumber of multigrid iterations is increasing on finer grids due to the tighteningstopping criterion. If the stopping criterion would have been grid independent thenthe number of multigrid iterations would be decreasing on the finer grids due toimproving initial guesses. Regardless the optimized relaxation parameter the num-ber of PSOR iterations increases quite a bit when grids are refined. Hence, the PSORmethod cannot compete in efficiency with the other methods on finer grids.
The errors in the prices are fairly similar for all methods. The errors with thecomponentwise splitting method are slightly smaller partly because of the iterativemethods have up to 8 % additional error due to the termination of the iteration. Forthe multigrid based methods this additional error is larger on coarser grids while onthe finest grid it is less than 3%. For the PSOR method the behavior is the opposite:for the coarsest grid the additional error is less than 3%, but for the finest one it is
20
Table 3: The average number of iterations, the errors at the 10 reference points, theratios of consecutive errors, and the CPU times in seconds for all methods on fourgrids.
Newtonmethod grid (m,n, l) iter. iter. error ratio CPUPSOR (160, 64, 34) 8.8 0.00697 0.27
(320, 128, 66) 13.7 0.00327 2.13 3.09(640, 256, 130) 20.4 0.00174 1.88 35.39(1280, 512, 258) 29.6 0.00086 2.02 401.10
PMG (160, 64, 34) 1.0 0.00703 0.19(320, 128, 66) 1.0 0.00336 2.09 1.59(640, 256, 130) 1.2 0.00174 1.93 15.18(1280, 512, 258) 1.4 0.00079 2.21 141.39
operator (160, 64, 34) 1.0 0.00680 0.15splitting (320, 128, 66) 1.0 0.00321 2.12 1.29
(640, 256, 130) 1.2 0.00166 1.93 11.80(1280, 512, 258) 1.4 0.00075 2.21 108.43
penalty (160, 64, 34) 1.1 1.0 0.00731 0.28method (320, 128, 66) 1.2 1.0 0.00335 2.18 2.66
(640, 256, 130) 1.4 1.1 0.00173 1.93 24.39(1280, 512, 258) 1.4 1.3 0.00079 2.20 211.59
comp.wise (160, 64, 34) 0.00622 0.08splitting (320, 128, 66) 0.00284 2.19 0.81
(640, 256, 130) 0.00147 1.93 6.86(1280, 512, 258) 0.00067 2.21 56.82
almost 8%. We also note that the vector r used in the stopping criterion of the PSORmethod had up to 10 times larger norm than the reduced residual. The ratios inTable 3 suggest the discretizations to be first-order accurate when both space andtime steps are reduced at the same rate.
In the last numerical experiment we study the choice of the relaxation parameterω of the PSOR method. We compare the relaxation parameter (39) computed withthe approximation (40) and with the exact spectral radius. We also computed thebest possible ω by minimizing the number of iterations. These three values of ω aregiven in Table 4 along the number of the PSOR iterations. We have also reported thel2 errors computed at the 10 reference points, since the termination of the iterationis influenced by the relaxation parameter and, thus, the additional error due to thetermination varies as well.
Table 4 reveals that the relaxation parameter given by (39) is not optimal for theconsidered LCPs. Furthermore, it is better to use the approximation (40) in (39)than the exact spectral radius. Also, the computation of the exact spectral radiusof the Jacobi iteration matrix is expensive while the approximation (40) is easy-to-implement and computationally cheap. On the finer grids the best possible ω leadsto slightly more than 20 % reduction in the number of iterations. Also, a reduction inthe error is clearly visible with the best ω. The tuning of ω by minimizing the numberof iterations requires several runs with the PSOR solver and, thus, it is feasible way
21
Table 4: The values of the relaxation parameter ω of the PSOR method computedin three different ways, the average number of iterations, and the errors at the 10reference points.
appr. (39) exact (39) best possiblegrid (m,n, l) ω iter. error ω iter. error ω iter. error(160, 64, 34) 1.562 8.8 0.00697 1.520 9.6 0.00703 1.645 7.3 0.00686(320, 128, 66) 1.632 13.7 0.00326 1.606 14.7 0.00328 1.722 10.5 0.00322(640, 256, 130) 1.689 20.4 0.00174 1.673 21.6 0.00175 1.771 15.5 0.00170(1280, 512, 258) 1.734 29.6 0.00086 1.725 30.6 0.00087 1.825 22.9 0.00078
to choose ω only when many options are priced using the same discretization.
6 ConclusionsWe proposed a finite difference space discretization on nonuniform grids leadingto an M -matrix for the operator derived from Heston’s stochastic volatility model.Based on such discretizations and the Rannacher time-stepping we priced Amer-ican options using five different methods: the projected SOR method, a projectedmultigrid method, an operator splitting method, a penalty method, and a compo-nentwise splitting method. All these methods are iterative except the last one. Withthe operator splitting method and the penalty method we used a multigrid methodto solve the resulting systems of linear equations.
The M -matrix property guarantees many good qualities for the approximationand for the solution methods. In our earlier experiments when the matrix did nothave this property the solutions computed by the componentwise splitting methodoften blew up while non M -matrices did not affect the other four methods. Theoperator splitting method, the penalty method, and the componentwise method in-troduce additional approximations in order to solve the time dependent linear com-plementarity problem. Nevertheless in our numerical experiments the accuracies ofall studied methods were similar.
The componentwise splitting method was about twice faster than the opera-tor splitting method which was the second fastest among the methods. The pro-jected multigrid method and the penalty method have some additional computa-tions when compared to the operator splitting method and, thus, they were about1.3 and 1.9 times slower, respectively. On the coarsest grid the projected SOR methodwas about three times slower than the componentwise splitting method while on thefinest grid it was seven times slower.
The projected SOR method is much easier to implement than the multigrid basedmethods. Our effort to implement the componentwise splitting method was com-parable to the multigrid methods. As a direct method it does not require to choose astopping criterion and, thus, it is the easiest one to use. The penalty method and theoperator splitting method seem to be the most well suited for generalizations likeadding a jump diffusion to the model. The use of the Brennan and Schwartz algo-
22
rithm in the projected multigrid and in the componentwise splitting method requirea specific form for the early exercise region. By replacing this algorithm with a moregeneral method, for example, the one in [13], they can be made more general, butthis would probably hamper the efficiency of the methods.
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