Implicit–explicit Runge–Kutta methods for financial derivatives pricing models

European Journal of Operational Research 171 (2006) 991–1004

www.elsevier.com/locate/ejor

Implicit–explicit Runge–Kutta methodsfor financial derivatives pricing models

Javier de Frutos *

Departamento de Matematica Aplicada, Universidad de Valladolid, C/Prado de la Magdalena S/N, 47005 Valladolid, Spain

Available online 10 March 2005

Abstract

Implicit–explicit Runge–Kutta methods are investigated for application to financial derivatives pricing models in thepartial differential equations approach. The methods are showed to be an alternative to other existing procedures forthe numerical valuation of American type contracts. We follow the method of lines in order to have a numerical methodthat can be used with a variety of state variable discretizations including finite elements, finite differences and finite vol-ume methods. Some numerical experiments are presented.� 2005 Elsevier B.V. All rights reserved.

Keywords: Finance; American options; Implicit–explicit Runge–Kutta methods; Finite element method

1. Introduction

A wide variety of derivative securities traded in exchange markets such as call or put options on dividendpaying stocks, foreign currency options, callable bonds, among others, are American type contracts. AnAmerican contract gives the holder the right to exercise and exit the contract at any time before the expi-ration date specified in the contract. The problem of valuating such contracts is considerably more difficultthan the corresponding European contract where the holder can only exercise at expiration. In fact there isno closed form analytical formulas for valuing American contracts even in the simpler cases. Numericalmethods are then necessary for pricing such contracts.

Derivative securities can usually be defined as optimal stopping problems [18] of the form

0377-2doi:10.

* TelE-m

V ðx; tÞ ¼ sups2Tðt;T Þ

Ex½gðs;X sÞ�; x 2 X; ð1Þ

217/$ - see front matter � 2005 Elsevier B.V. All rights reserved.1016/j.ejor.2005.01.013

.: +34 983 42 31 81; fax: +34 983 42 30 13.ail address: [email protected]

mailto:[email protected]

992 J. de Frutos / European Journal of Operational Research 171 (2006) 991–1004

associated with a system of stochastic differential equations

dX t ¼ bðX t; tÞdt þ rðX t; tÞdW t; ð2Þ

and a reward function g, where X is a possibly unbounded subset of Rn and Tðt; T Þ is the set of stoppingtimes. Here, Ex denotes the mathematical expectation with respect the probability law of the processXt, starting at X0 = x. In many cases the reward function is chosen to be the discounted value of a givenpayoff w.

gðs;X sÞ ¼ exp �Z s

trðs;X sÞds

� �wðs;X sÞ:

The value function is representable deterministically as the solution of a system of time dependent var-iational inequalities [2,9,12] in terms of the characteristic differential operator associated with (2). In thisformulation the problem is similar to a parabolic, possibly degenerate, obstacle problem in which the statevariables play the role of spatial degrees of freedom.

Most practical numerical methods for the solution of the equivalent linear complementarity problem usewell known finite differences approaches, [10,23,24]. In [26,8] a penalty method coupled with a finite volumediscretization of spatial like variables and finite differences in time are proved to be a useful and very effi-cient alternative for this kind of problems. The main idea is to change the set of variational inequalities intoa set of nonlinear partial differential equations through a penalization of the restriction [2,17,20]. As it isclaimed in [8], the main advantage of the penalty method is that it is easily generalizable to multidimen-sional problems and a variety of contracts. Furthermore, it can be used coupled with any type of spatial(state) discretization, finite differences, finite volume or finite element methods.

In this paper we are concerned with implicit–explicit Runge–Kutta methods for the numerical integra-tion of the system of ordinary differential equations which arises after spatial discretization of the penaltyapproximation to the linear complementarity problem. That is, we use the method of lines to decouple thespatial like and temporal discretization allowing the use of efficient stiffly accurate time integrators othersthan simple finite differences. We remark that, in the so-called continuation region, one has to numericallysolve a system of convection–diffusion–reaction partial differential equations. This problem qualifies as stiff,see [11]. So, stiffly accurate time integrators are necessary to have efficient numerical methods. Note thatefficiency is a crucial issue in order to design numerical methods of practical use in general, possibly mul-tidimensional, financial problems. Surprisingly, it seems that there has not been any attempt to use efficient,stiffly accurate time integrators other than the first-order backward Euler method in the financial literature.The only exception is [19], where the authors propose to use a numerical differentiation formula with quasi-constant step size and variable order.

Implicit–explicit Runge–Kutta methods are a particular instance of additive Runge–Kutta methods [7].It has been proved in [5], see also [1,14], that being easily implementable, implicit–explicit Runge–Kuttamethods are competitive with standard stiffly accurate time integrators for convection–diffusion–reactionequations. To our knowledge this is the first time that they have been successfully applied to the numericalsolution of time dependent variational inequalities. Recently, implicit–explicit finite difference methods hasbeen successfully used for European and barrier options in jump diffusion and exponential Levy modelsin [6].

In the following, for clarity in the exposition, we restrict ourselves to a simple one dimensional modelproblem, the American put option, although the numerical methods is intended to be general enough tobe applied to more complex financial problems. In Section 2 we state the model problem and its penaltyapproximation. Implicit–explicit Runge–Kutta time integrators coupled with a finite element discretizationare presented in Section 3. The next section is devoted to show the qualities of the method proposed bymeans of some numerical experiments. We finish the paper with some conclusions and final remarks.

J. de Frutos / European Journal of Operational Research 171 (2006) 991–1004 993

2. The model problem

Consider an asset with price S which is supposed to follow a log-normal evolution process

dS ¼ lS dsþ rS dW ; ð3Þ
where l is the expected rate of return, r is the volatility, dW is the increment of a Wiener process and sdenotes time. We are concerned with the problem of determining the fair value V(S, s) of an Americanput option in which the holder can exercise at any time, receiving a payoff specified in the contract asw(S, s) = max(E � S, 0), where E is the strike. Under the usual assumptions on the market, the problemcan be formally reduced to finding V(S, s) such that
V t � LBSV P 0; 0 6 S 6 1; 0 < t 6 T ; ð4Þ

V ðS; tÞ P wðS; tÞ; 0 6 S 6 1; 0 < t 6 T ; ð5Þ

ðV t � LBSV ÞðV � wÞ ¼ 0; 0 6 S 6 1; 0 < t 6 T ; ð6Þ

V ðS; 0Þ ¼ wðSÞ; 0 6 S 6 1: ð7Þ
Here, LBS denotes the well-known Black–Scholes operator,
LBSV ¼ 1

2r2S2V SS þ rSV S � rV : ð8Þ

r is the risk free interest rate and t = T � s is the time to expiration of the contract.The problem has a unique solution which is smooth in the continuation region and is continuously dif-

ferentiable globally [12,2, Chapter 3]. Note that no boundary conditions are imposed as corresponds to asingular problem. From a practical point of view the solution of (4)–(7) can be approximated by means of asequence of problems posed over bounded increasing domains. More precisely, let us consider a sequence ofdomains ½Sk

m; SkM � � ð0;1Þ with Sk

m ! 0 and SkM ! 1. Let Vk be the solution of the localized problem

V kt � LBSV k P 0; Sk

m 6 S 6 SkM ; 0 < t 6 T ; ð9Þ

V kðS; tÞ P wðS; tÞ; Skm 6 S 6 Sk

M ; 0 < t 6 T ; ð10Þ

ðV kt � LBSV kÞðV k � wÞ ¼ 0; Sk

m 6 S 6 SkM ; 0 < t 6 T ; ð11Þ

V kðSkm; tÞ ¼ E � Sk

m; 0 < t 6 T ; ð12Þ

V kðSkM ; tÞ ¼ 0; 0 < t 6 T ; ð13Þ

V kðS; 0Þ ¼ wðSÞ; Skm 6 S 6 Sk

M : ð14Þ

Let X ¼ ½Sl; Su� � ðSkm; S

jMÞ a fixed approximation domain. Then from [16,12,13],

limk!1

maxt2½0;T �

kV ð�; tÞ � V kð�; tÞkL1ðXÞ ¼ 0: ð15Þ

We note that the boundary conditions (12) and (13) are essentially arbitrary but nevertheless financiallyreasonable. Furthermore, in [16] it is shown that every choice of the boundary conditions in an increasingexhausting sequence of bounded domains leading to well posed systems can be used to approximate V(S, t)in a given fixed bounded approximation domain. However, the choice of boundary conditions surely affectsthe quality of the approximation. The rationale for the choice (12), (13) is as follows: First it is clear that the


value of the option must be zero when the asset value tends to infinity. The boundary condition in theneighborhood of S = 0 is slightly more subtle. A wrong value will generate a boundary layer near S = 0that is, obviously, not financially acceptable. From the complementarity condition (6) one has that, atS = 0, either Vt + rV = 0 or V(0, t) = w(0, t) = E. Using (7), in the first case it is found that V(0, t) = ex-p(�rt)E < E for all time t > 0. Then, the constraint (5) forces V(0, t) = E and in consequence (12) for Sk

m

small enough.Other boundary conditions have been used in the literature. In [21] linearity type boundary conditions of

the form VSS = 0, together with some other useful alternatives, are advocated for options with piecewiselinear payoff when finite differences are used as discretization tool. However, this type of boundary condi-tions are more difficult to implement in the context of a finite element method.

2.1. The penalty problem

There are several approaches that can be used to solve the localized problem (9)–(14). Here, we shall usethe penalization technique that has been proved to be an efficient way to treat some financial problems, see[8]. To this end, let us start by choosing a smooth function u with uðSk

mÞ ¼ E � Skm, uðSk

MÞ ¼ 0, and other-wise arbitrary. For � > 0, we seek Vk,�, the unique solution of the (regularized) penalty problem

Uk;�t � LBSUk;� þ b�ðUk;�Þ ¼ f ; Sk

m 6 S 6 SkM ; 0 < t 6 T ; ð16Þ

Uk;�ðSkm; tÞ ¼ Uk;�ðSk

M ; tÞ ¼ 0; 0 < t 6 T ; ð17Þ

Uk;�ðS; 0Þ ¼ w�ðSÞ ¼ wðSÞ � uðSÞ; Skm 6 S 6 Sk

M ; ð18Þ

where f = �ut + LBSu, w*(S) = w(S) � u(S) is the modified payoff and b� is the penalty term,

b�ðuðSÞÞ ¼1

�minðuðSÞ � w�ðSÞ; 0Þ: ð19Þ

Problem (16)–(18) has a unique solution with derivatives up to second-order in LpðSkm; S

kMÞ for t > 0, see

for example, [9, Theorem 10.4.2]. Furthermore, for � small enough, there exists a constant C > 0, such that[3, Theorem 4.1]

maxt2½0;T �

kV kð�; tÞ � ðUk;�ð�; tÞ þ uð�; tÞÞkL1ðXÞ 6 C�: ð20Þ

Besides, the American restriction (10) is also satisfied to Oð�Þ terms, that is,

V k;�ðS; tÞ ¼ Uk;�ðS; tÞ þ uðS; tÞ P wðSÞ � C�; S 2 ½Skm; S

kM � ð21Þ

for some constant C > 0 independent of �.

3. The discrete problem

We follow the method of lines and discretize first the asset value and then time. One of the advantages ofthis methodology is that decoupling the discretization problem into a spatial-like part and a temporal oneallows an easier treatment of the specific characteristic of each problem. Here, we discretize the asset valueby means of the linear finite element method.

In order to simplify somewhat the exposition we take Skm ¼ 0 in the rest of the paper. For this particular

problem it can be observed that in practice all results of the previous section remain valid.


Let us consider a partition 0 ¼ S0 < S1 < � � � < SJþ1 ¼ SkM of the computational domain, and let us de-

note h = max06i6J(Si+1 � Si). We consider the space of test functions

Xh ¼ / 2 Cð½0; SkM �Þj/j½Si ;Siþ1� 2 P1;/ð0Þ ¼ /ðSk

MÞ ¼ 0n o

;

where P1 denotes the space of polynomials of degree less than or equal to one in the asset variable S.The semidiscrete problem consists in finding a function uh : [0,T] ! Xh such that
Z SkM
0

uh;t/dS þ r2

2

Z SkM

0

S2uh;S/S dS � ðr � r2ÞZ SkM

0

Suh;S/dS þ rZ SkM

0

uh/dS þZ SkM

0

b�ðuhÞ/dS

¼Z SkM

0

f/dS 8/ 2 Xh: ð22Þ

We remark that although other variational formulations are possible (see for example [9,12,16,22]), froma computational point of view this one has the merit of being numerically efficient and directly related to thefinancial problem.

Let /j 2 Xh, j = 1, . . . ,J, be defined by /j(Sk) = djk. It is clear that Xh = span{/j, j = 1, . . . ,J}. There-fore, writing

uhðS; tÞ ¼XJj¼1

UjðtÞUjðSÞ;

the variational equations (22) are equivalent to the following system of ordinary differential equations:

MdUh

dtþ r2

2KUh � ðr � r2ÞCUh þ rMUh þMb�ðUhÞ ¼ MFh; ð23Þ

where Uh ¼ ½U 1; . . . ;UJ �T, Fh ¼ ½f ðS1Þ; . . . ; f ðSJ Þ�T and b�ðUhÞ ¼ ½b�ðU 1Þ; . . . ; b�ðUJ Þ�T. The mass, M, stiff-ness, K, and convection, C, matrices are defined by

M ¼Z SM

0

/i/j dS� �J

i;j¼1

;

K ¼Z SM

0

S2/i;S/j;S dS� �J

i;j¼1

;

C ¼Z SM

0

S/i;S/j dS� �J

i;j¼1

:

For the semidiscrete approximation (22) it is possible to show that for each t > 0, the following errorbound holds [4, Chapter 8]:

kUk;�ð�; tÞ � uhð�; tÞkL1ðXÞ 6 Ch2j log hj;

and in consequence, taking (20) into account,

kV kð�; tÞ � ðuhð�; tÞ þ /ð�; tÞÞkL1ðXÞ 6 Cð�þ h2j log hjÞ: ð24Þ

3.1. Implicit–explicit time discretizations

The system of ordinary differential equations (23) can be solved by means of an standard ODE time inte-grator. The main difficulty when dealing which this kind of system is that the use of explicit time integrators


is inefficient because the system becomes stiffer as the asset value mesh is refined. As a consequence the timestep must be heavily reduced in order to fulfill the severe stability restriction inherent to explicit methods.On the other hand, if a stiffly accurate, implicit time integrator is chosen, one has to solve implicit nonlinearequations with nonsmooth Jacobian. Furthermore, the convective terms introduce additional numericaldifficulties. This could be one of the reasons why very few implicit codes other than mid-point rule canbe found in the literature. To our knowledge perhaps the only exception is [19] where a backward differen-tiation type code is proposed. However, multistep based methods are reported in [25], see also [8], to per-form badly with American type financial problems as shout options. Although it is not clear in the financialliterature if this is also the case for simpler contracts, the use of one step time integrators present some prac-tical advantages. Among others, there exist easily implementable methods with variable time steps andautomatic error control that are efficient for convection–diffusion type equations that typically arise infinancial problems.

The main idea in an implicit–explicit time discretization is to use different methods for the different termsrolled into a single composite method that could treat more efficiently each one of the terms appearing in(23). To this end let us write (23) in the form

MdUh

dt¼ LðUhÞ þNðUhÞ; ð25Þ

where

LðUhÞ ¼ � r2

2KUh �Mb�ðUhÞ;

NðUhÞ ¼ ðr � r2ÞCUh � rMUh þMFh:

The partition (25) has been chosen such that L(Uh) collects the stiffer terms appearing in the equations(diffusive and penalty terms in our case) and N(Uh) gathers the less stiff terms (convective and source terms).

The simplest implicit–explicit Runge–Kutta method is a combination of the well known explicit and im-plicit Euler rules. Let us suppose that at time tn < T a numerical approximationUn

h � UhðtnÞ has been found.Then the approximation Unþ1

h to Uh(tn+1) = Uh(tn + Dt) is the solution of the system of algebraic equations

MUnþ1h ¼ MUn

h þ DtLðUnþ1h Þ þ DtNðUn

hÞ: ð26Þ
Note that as the most stiff terms, diffusion and penalty ones, are treated implicitly by means of a L-stable
method (Backward Euler method in this simple case), only a mild stability restriction Dt 6 C independentof asset grid size has to be imposed, see [5]. This is the main advantage of implicit–explicit methods overpure explicit approaches. The price to be paid is that at each time step, a nonlinear system of equationshas to be solved. However, as the convection term in (25) is treated explicitly, only a linear solver for sym-metric systems is needed when (26) is solved by means of a Newton like iteration procedure. This representsusually an advantage over fully implicit methods such as multistep BDF formulas.

For the numerical solution of (26), following [8], we propose to use the following fixed point Newton-likeiteration: Given an initial iterate Y0, for example, Y0 ¼ Un

h, for k = 0,1, . . . ,

Mþ DtKþ DtMBk� �

Ykþ1 ¼ MUnh þ DtNðUn

hÞ þ DtMBkw�; ð27Þ

where w� ¼ ½w�ðS1Þ; . . . ;w�ðSJÞ�T and Bk is the diagonal matrix defined by

Bkjj ¼

1=� if ðYki Þj < w�ðSjÞ;

0 if ðYki Þj P w�ðSjÞ:

(

The iteration is stopped when the difference between two consecutive iterates is small enough. We refer toSection 4 for the practical details concerning the implementation of the nonlinear iteration (27).


We note that (27) is, formally, the Newton method applied to the nonlinear system (26) in which matrixB plays the role of the Jacobian of the (nonsmooth) nonlinear penalty term. In [8] it is proved that if alumped (diagonal) mass matrix [15] is used in (27), the iterates converge monotonically to the unique solu-tion of (26). In our experiments we have found that with consistent mass matrix, (27) still converges in asmall number of iterations.

3.2. Implicit–explicit Runge–Kutta methods

The implicit–explicit Euler method is a first-order method, that is, the error decreases proportionally totime step Dt and in consequence is somewhat inefficient. However, high order one step methods sharing thesame ideas that the implicit–explicit Euler methods are available in the literature.

In an implicit–explicit Runge–Kutta method, the equations that describe the step tn # tn+1 when appliedto (25) take the form

MY1 ¼ MUnh; ð28Þ

MY i ¼ MUnh þ DtaiiLðY iÞ þ Dt

Xi�1

j¼2

aijLðY jÞ þXi�1

j¼1

ai;jNðY jÞ !

; 2 6 i 6 sþ 1; ð29Þ

MUnþ1h ¼ MUn

h þ DtXsþ1

i¼2

biLðY iÞ þXsþ1

i¼1

biNðY iÞ !

; ð30Þ

where Yi denotes the internal stages. The coefficients of the method, that is, the matrices A = (aij), bA ¼ ðaijÞand the vectors bT = (bi) and b

T ¼ ðbiÞ are computed as to satisfy the so-called order conditions (to a cer-tain attainable order depending on the number of stages s + 1) and some stability restrictions. Usually, thecoefficients are chosen such as the following simplifying assumptions are satisfied:

ci ¼Xi

j¼2

aij ¼Xi�1

j¼2

aij; bi ¼ bi:

We refer to [5], see also [11], for a review of the order conditions and a study of the stability properties ofimplicit–explicit Runge–Kutta methods. Several efficient implicit–explicit Runge–Kutta methods exist inthe literature. We refer for example to [1,14], where an exhaustive comparison among different possibilitiesare reported.

Each stage of the method is an implicit–explicit Euler like step, see Eq. (29), that is treated as has beenpreviously described by means of an iteration similar to (27). Note that only an efficient procedure to solvesymmetric positive definite linear systems is needed. Furthermore, the method is easily generalizable toother models were a special stabilization procedure of the convective terms could be needed. We do notpursue further this issue in this paper as this is not the case for the simple model of the American putoption.

We remark that the method (28)–(30) can be seen as pair of Runge–Kutta methods specified by thecoefficients A, bT, c and bA, b

T, c, respectively. The first method is implicit and has to be applied to the stiff

terms, in this case the diffusive and penalty terms; the second one is explicit and is applied only to the con-vective and source terms. Moreover, if the implicit method is L-stable [11] it is possible to construct an ex-plicit companion method in such a way that the full implicit–explicit procedure needs only a mild stabilityrestriction of the type Dt 6 C with C independent of the mesh size, see [5]. This is a necessary condition inorder to have an efficient numerical procedures for partial differential equations. Besides, it is also a veryconvenient property when one is dealing with nonsmooth initial data as (18), avoiding the need of special


starting procedures required when one uses a non strongly stable procedure as for example the implicit mid-point rule (Crank–Nicolson method) [8].

Finally we remark that since the asset and time discretizations are decoupled, the same time integratorcan be used in conjunction with other spatial like discretization such as finite differences, finite volumemethods or even high order finite elements.

4. Numerical experiments

In this section we consider the third-order method (LIRK3) with s + 1 = 4 stages constructed in [5]. Theimplicit part of the method is based in the well known L-stable, three stage, third-order SDIRK [11]. Thecoefficients of the explicit part of the method are computed in such a way that the method have quasi-optimal stability properties when applied to convection–diffusion equations, see [5, Section 4], like theBlack–Scholes equation.

The method is implemented in variable-step mode, so that the time step is automatically chosen by thecode depending on an estimation of the local errors. To this end, we have considered the second-orderformula:

M eU nþ1

h ¼ MUnh þ Dt b2ðLðY2Þ þNðY2ÞÞ þ b3ðLðY3Þ þNðY3ÞÞ þ b4ðLð eU nþ1

h Þ þNðY4ÞÞ� �

;

where Y2, Y3 and Y4 are the internal stages of the third-order scheme. The second-order approximationeU nþ1

h can be interpreted as the solution generated with a second-order implicit–explicit Runge–Kutta

method with an additional stage Y5 ¼ eU nþ1

h defined by the coefficients a5;i ¼ a5;i ¼ bi, 1 6 i 6 3, a5;4 ¼a5;5 ¼ 0, a5;5 ¼ a5;4 ¼ b4.

Each time a new approximation Unþ1h is computed, the local error is estimated by

ERR ¼ kUnþ1h � eU nþ1

h k1:
The step is accepted if ERR 6 TOL, where TOL is the precision required by the user. If ERR > TOL the
step is rejected and the computation is restarted from the previously accepted approximation using a smal-ler time step. In both cases the new time step size is selected according to the rule

Dtnew ¼ min hlim;Dtold min FACMAX;FACTOL

ERR

� �1=3 ! !

; ð31Þ

where ‘‘hlim’’ is the maximum admissible step size (in our code hlim = T/5) and FAC and FACMAX aresafety factors that limit the variation of step size in order to make the code more robust. In our code wechoose

FAC ¼ 0:9; FACMAX ¼ 1:5

if the step has been accepted (EST 6 TOL), and

FAC ¼ 0:7; FACMAX ¼ 1

after a rejection (ERR > TOL).Each stage, a nonlinear problem of the form (26) has to be solved. To this end, we have used the Newton

like iteration (27), stopping the iteration when

maxjY kþ1

j � Y kj j

maxð1; jY kj jÞ

6 10�3 TOL;

where Y kj , j = 1, . . . , denote the components of the iterant Yk in (27).


In order to control the cost of the iterations in the solution of the nonlinear systems, we have alsodecided to adopt a conservative policy and limit the maximum number of iterations in each stage. If, inthe computation of one stage, the nonlinear iteration has not converged after four iterations, the full timestep is rejected and the computation restarted with time step halved.

We have tested the method for several values of the parameters. Here, we present the results obtained forthe particular values r = 0.2, r = 0.1 and E = 100. The conclusions for other values were similar to theones reported here. In all the numerical experiments we have taken Sk

M ¼ 200. For this computational do-main the error caused by the arbitrary boundary conditions was found to be small enough compared withthe errors made in the discretization process.

Fig. 1 shows the time step history for a single run using a grid with 160 nodes and setting TOL = 10�3

and � = 10�4. In this experiment time to expiry was T = 1. As we can see the code selects small time steps atthe beginning of the computation and smoothly increments the time step as the computation proceeds. Weremark that the initial condition (18) has only piecewise smooth derivatives although is smoother for t > 0due to the parabolic character of the Black–Scholes equation. The small steps are the response of the codeto the lack of regularity at t = 0. Once the solution has been smoothed, the code increments the time stepuntil a default in the convergence of the nonlinear iterations (27) is detected. Then, the code automaticallyreduces the time step, see Fig. 1 near T = 0.4, and continues the time integration increasing again the timestep according to the rule (31). Note that no special procedure is needed to deal with nonsmooth initial con-ditions as is the case with other time integrators of common use as, for example, mid-point rule, see [8]. Thetime step history was found to depend only on the time integrator tolerance and is essentially independentof the asset grid size, see below.

In the next test we have checked the behaviour of the method with respect to the penalty parameter. Wehave monitored the maximum relative error in enforcing the American constraint by the penalty method bymeans of the quantity

AMERR ¼ maxS2½0;SkM �

maxðw�ðSÞ � uhðSÞ; 0Þmaxð1;w�ðSÞÞ : ð32Þ

Table 1 shows the results obtained for several values of the discretization parameters, number of nodesand tolerance in the step control procedure of the time integrator, varying the size of the penalizationparameter �. We observe that the size of the relative American error is proportional to �, in good agreement

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.02

0.04

0.06

0.08

0.1

0.12

Fig. 1. Time step selection history. r = 0.2, r = 0.1, K = 100, T = 1, grid with 160 nodes, TOL = 10�3, � = 10�4.

Table 1American put option

� Nodes TOL American error Ratio Steps

10�2 160 10�2 7.56 · 10�3 www 1710�3 160 10�2 9.45 · 10�4 8.00 1710�4 160 10�2 1.24 · 10�4 7.62 1710�5 160 10�2 1.12 · 10�5 11.07 27

10�2 320 10�3 7.57 · 10�3 www 2510�3 320 10�3 8.70 · 10�4 8.70 2510�4 320 10�4 1.04 · 10�4 8.36 3010�5 320 10�5 9.64 · 10�6 10.79 32

10�2 640 10�4 7.56 · 10�3 www 4210�3 640 10�4 8.63 · 10�4 8.76 4610�4 640 10�4 9.42 · 10�5 9.16 5010�5 640 10�4 1.25 · 10�5 7.54 52

Computational domain [0, 200]. T = 0.25, r = 0.1, r = 0.2, E = 100, 640 nodes, � = 10�6.


with bound (21). Furthermore, the error is nearly independent of the discretization parameters. It is inter-esting to see that the number of time steps and, as a consequence, the cost of time integration is nearly inde-pendent of the penalization parameter. This is a consequence of the strong stability properties of the timeintegrator. The slight increment in the number of time steps observed with decreasing values of � is due tothe conservative policy adopted for time step control. Each time a slow convergence of the iterates in (27) isdetected, the time step is halved. The number of time step increases because of this somewhat drastic reduc-tion in the time step.

In order to check the accuracy of the implicit–explicit Runge–Kutta method in this application we per-formed several runs for different values of the parameter TOL. The number of nodes was set to be 640 in allthe reported experiments so that the numerical solutions are nearly free of spatial discretization error. Theresults of the experiment are shown in Table 2. The most precise result corresponds to TOL = 10�6 and has


TOL Steps V (1 0 0) Difference Ratio

10�2 18 3.04907 2.13 · 10�2 www

10�3 37 3.06794 2.40 · 10�3 8.8810�4 50 3.07003 3.01 · 10�4 7.9710�5 91 3.07032 2.00 · 10�5 15.0510�6 153 3.07034 www www

Computational domain [0, 200]. T = 0.25, r = 0.1, r = 0.2, E = 100, 640 nodes, � = 10�6.


Nodes TOL Steps V (1 0 0) Difference Ratio

40 10�2 17 3.10001 2.965 · 10�2 www

80 10�2 17 3.07736 7.019 · 10�3 4.2160 10�3 30 3.07262 2.274 · 10�3 3.1320 10�3 30 3.07097 5.267 · 10�4 4.3

Computational domain [0, 200]. T = 0.25, r = 0.1, r = 0.2, E = 100, TOL = 10�6, � = 10�6.

50 60 70 80 90 100 110 120 130 140 1500

5

10

15

20

25

30

35

40

45

50

50 60 70 80 90 100 110 120 130 140 150-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

50 60 70 80 90 100 110 120 130 140 150-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Fig. 2. Top: Option value (V), middle: delta (VS), bottom: gamma (VSS). Grid with 160 nodes, TOL = 10�3, � = 10�4, T = 0.25.


been taken as a reference solution in order to measure the efficiency of the numerical method. For eachvalue of the tolerance we show in Table 2 the number of steps, the computed price of the option at


S = 100 and the difference between computed prices and the reference solution. This difference is an indi-cator of the error due to the temporal discretization. Last column shows the ratio of consecutive differences.We observe that the error is of the same order as the prescribed tolerance.

We remark that in a step size control procedure, the objective is to have an error in the numericalsolution as close as possible to the user prescribed tolerance. In this way, the user is able to optimize,for a given precision, the number of time steps and in consequence the CPU time required by the com-putation. The high values in the last column of Table 2 nicely show a very rapid convergence as the tol-erance is reduced. Note also that each time the time integrator error is reduced by an order ofmagnitude, the number of steps is increased by a factor less than two. So, in spite of the some timesalleged lack of regularity of the American put problem, by means of a stiffly accurate, third-order timeintegrator, the error can be drastically reduced with only a moderate increment in the cost of thecomputation.

It is clear that, in a practical computation, an equilibration of the errors coming from asset and temporaldiscretizations is needed. We present in Table 3 the results obtained by minimizing the cost of the compu-tation for a given error. The value of the time integrator tolerance showed in each row is the best value inthe sense that a smaller value does not reduce the error while it increases the number of steps and in con-sequence the cost of the computation. In all runs the value of the penalization parameter was taken to be� = TOL/10. As we have done before we take as an indicator of the error in the computation the differencewith respect to the numerical solution computed with 640 nodes, TOL = 10�6 and � = 10�7. As we can see,the ratio between two consecutive differences is very near 4 as corresponds to a second-order method.Furthermore, thanks to the high order temporal discretization, the error in the asset discretization canbe reduced while maintaining a small number of time steps.

To sum up, the numerical experiments show that the error in the proposed method decouples in threecomponents, the error due to the penalty term, the temporal discretization error and the asset discretizationerror which can be controlled independently. The high order strongly stable method used for the time inte-gration makes the global procedure very efficient providing rapid convergence with respect to the time inte-gration tolerance and quadratic convergence with respect to the number of degrees of freedom used in theasset discretization.

The sensitivity of the option value to variations in asset price and parameters of the model is measuredby the so called Greeks [21,23]. In practice, besides the option price, accurate approximations to the Greeksare needed for edging purposes [23]. Fig. 2 shows value, delta (oV/oS) and gamma (o2V/oS2) of the optionfor the same values of the parameters as before and time to expiry T = 0.25. We can observe that thenumerically calculated Greeks are free of spurious oscillations which are typically present in the numericalsolution of American type contracts, specially in the computed gamma, due to the discontinuity of the sec-ond derivative in the optimal exercise price. This unwanted behaviour of the gamma has been reported in [8]where a second-order mid-point rule (Crank–Nicolson method) is used as time integrator. There, theauthors claim that the oscillations disappear if the first-order, fully implicit, Backward Euler method is usedinstead. Note that the implicit–explicit Runge–Kutta method is a strongly stable method that shares thedissipative properties of implicit Euler method while it retains the efficiency derived from the third-orderaccuracy and variable step implementation.

5. Conclusions

We have presented a third-order variable step implicit–explicit Runge–Kutta method which is showed tobe an efficient alternative to other existing procedures for the numerical valuation of American put options.The method is coupled with a linear finite element discretization for the asset variable and treats the Amer-ican restriction by the penalty method. The computed Greeks are free of the spurious oscillations which are


typically present when no strongly stable time integrators, such as Crank–Nicolson method, are used, whilethe efficiency inherent to high order methods is retained.

Using the method of lines to state the full discretization it is possible to completely decouple asset dis-cretization from time integration. This allows the time integrator to be used with other asset discretizationssuch as the finite difference or finite volume method. We remark that, for the same reason, the extension tomultifactor problems is also straightforward.

Acknowledgments

Research supported by DGI-MCYT under grant BFM2001-2138 (cofinanced by FEDER funds) and byJCYL under project VA044/03. This research was completed while the author was visiting professor atGERAD and HEC Montreal partially supported by grant SEEU PR2002-0223.

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Implicit–explicit Runge–Kutta methods for financial derivatives pricing models

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