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Numerical solutions to an integro-differentialparabolic problem arising in the pricing of financialoptions in a Levy marketIonuţ Florescua, Maria Cristina Marianib & Granville Sewellb
a Department of Mathematical Sciences, Stevens Institute of Technology, Castle Point onHudson, Hoboken, NJ 07030, USA.b Department of Mathematical Sciences, The University of Texas at El Paso, Bell Hall 124,El Paso, TX 79968-0514, USA.Published online: 05 Oct 2011.
To cite this article: Ionuţ Florescu, Maria Cristina Mariani & Granville Sewell (2014) Numerical solutions to an integro-differential parabolic problem arising in the pricing of financial options in a Levy market, Quantitative Finance, 14:8,1445-1452, DOI: 10.1080/14697688.2011.618144
To link to this article: http://dx.doi.org/10.1080/14697688.2011.618144
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Numerical solutions to an integro-differential
parabolic problem arising in the pricing of
financial options in a Levy market
IONUT FLORESCUy, MARIA CRISTINA MARIANI*z and GRANVILLE SEWELLz
yDepartment of Mathematical Sciences, Stevens Institute of Technology, Castle Point on Hudson,Hoboken, NJ 07030, USA
zDepartment of Mathematical Sciences, The University of Texas at El Paso, Bell Hall 124,El Paso, TX 79968-0514, USA
(Received 22 September 2010; in final form 24 August 2011)
We study the numerical solutions for an integro-differential parabolic problem modeling aprocess with jumps and stochastic volatility in financial mathematics. We present two generalalgorithms to calculate numerical solutions. The algorithms are implemented in PDE2D,a general-purpose, partial differential equation solver.
Keywords: Applied mathematical finance; Jump-diffusion processes; Numerical methods foroption pricing; Partial differential equations
1. Introduction
In financial mathematics the old problem of finding the
price of derivatives (options, futures, etc.) leads to the
study of partial differential equations. The standard type
of equations obtained are of parabolic type. In recent
years, the complexity of the models used has increased
and in turn this has lead to more and more complicated
equations for the derivative prices. Of particular interest is
a type of differential equation containing an integral term.
These equations, aptly named partial integro-differential
equations (PIDE), are difficult to solve and numerical
methods especially constructed for them are not easy to
find. Florescu and Mariani (2010) study these types of
problems and prove the existence of the solution under
general hypotheses concerning the integral term. In the
present study we extend the work on PIDEs by providing
a completely novel algorithm which is suggested in the
proof of existence of the solution. We also present a
second algorithm—a more traditional finite-element
scheme including a discretization of the integral term at
every step. In our numerical applications the two schemes
are convergent to the same solution. We also mention
here the work of D’Halluin (2004), who presents otherfast numerical methods for soving PIDEs.
The paper is structured as follows. In sections 1 and 2we introduce the problem, as well as previous results anddefinitions. In section 3 we describe the two algorithmsthat we will use to find numerical solutions. In section 4we explain how we find numerical solutions usingPDE2D, a general-purpose partial differential equationsolver that has been used to solve many mathematicalfinance problems. In sections 4 and 5 we present anddiscuss the results obtained.
2. Problem motivation and general PIDE results
In financial mathematics, usually the Black–Scholesmodel (Black and Scholes 1973), or variants of theBlack–Scholes model (Ikeda 1989, Merton 1992, Duffie2001, Jarrow 2002, Hull 2008), have been used for pricingderivatives on the equity. By applying the fundamentaltheorem of asset pricing (Harrison and Pliska 1981,Delbaen and Schachermayer 1994), one obtains differenttypes of backward parabolic partial differential equations.In all these models, an important quantity is the volatility,which is a measure of the fluctuation (risk) in the assetprices, and corresponds to the diffusion coefficient in theBlack–Scholes equation.*Corresponding author. Email: mcmariani@utep.edu
� 2011 Taylor & Francis
http://dx.doi.org/10.1080/14697688.2011.618144
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2.1. Stochastic volatility models
In the standard Black–Scholes model, a basic assumptionis that the volatility is constant. It was soon discovered
that this assumption does not allow matching an entireoption chain (option values for different strike values).
Several models were proposed allowing the volatility to bemodeled as a stochastic variable, the so-called stochasticvolatility models (Hull and White 1987, Scott 1987,
Wiggins 1987, Chesney and Scott 1989, Stein and Stein1991, Heston 1993, Hagan et al. 2002). To exemplify this,in the Heston (1993) model, the underlying security S
follows
dSt ¼ �St dtþ �tSt dZt,
where Z¼ {Zt}t50 is a standard Brownian motion.Unlike the classical model, the variance v(t)¼ �2(t) also
follows a stochastic process, given by
dvt ¼ �ð� � vðtÞÞdtþ �ffiffiffiffivtp
dWt,
where W¼ {Wt}t50 is another standard Brownianmotion. The correlation coefficient between W and Z isdenoted by �:
CovðdZt, dWtÞ ¼ �dt:
This leads to a generalized PDE in two state variables andone temporal variable:
1
2vS2 @
2U
@S2þ ��vS
@2U
@v @Sþ1
2v�2
@2U
@v2þ rS
@U
@S
þ ½�ð� � vÞ � �v�@U
@v� rUþ
@U
@t¼ 0:
We should mention that any stochastic volatility model
(no matter how simple) has the capability of matching anentire option chain for a fixed maturity (Fouque et al.2000, Florescu and Viens 2008).
2.2. Jump models, Levy processes
At about the same time as the stochastic volatility models
were being developed, researchers argued that the badfitting to real data was caused by the path continuity ofthe price process. Thus, the resulting model may have
difficulties fitting financial data exhibiting large fluctua-tions. The necessity of taking into account large marketmovements and a great amount of information arriving
suddenly (i.e. a jump) led researchers to propose modelswith jumps.
Merton (1992) and Andersen and Andreasen (2000)model the stock as a jump-diffusion model (a geometric
Brownian motion plus a compound Poisson processmultiplying the stock process). Under this modelingassumption, one may derive differential equations for
option prices, but these equations will contain an integralterm coming from the compensator of the Poisson
process. This has led to the study of partial integro-differential equations (PIDE). As an example, in the
above-cited work, the derivative value F(S, t) solves the
following PIDE under appropriate boundary conditions:
1
2�2S2FSS þ ðr� �kÞSFS þ Ft � rF
þ �EfFðSY, tÞ � FðS, tÞg ¼ 0: ð2:1Þ
Here r denotes the riskless rate, � the jump intensity, and
k¼E(Y� 1), where E is the expectation operator and the
random variable Y� 1 measures the percentage change in
the stock price if a jump occursThe jump-diffusion component is a particular case of
a Levy process and indeed stock evolution was soon
modeled using these more general processes (e.g.,
Barndorff-Nielsen et al. (1998), Madan et al. (1998),
Geman (2002), and Geman (2002), and Cont and Tankov
(2003). Similar to the jump-diffusion case, when using any
of these more general models, the resulting partial
differential equation for derivative prices contains an
integral term coming from the associated Levy measure.
Practical applications confirm that a Levy-like stochastic
process appears to be the best fit when modeling high-
frequency data (see Mariani et al. (2009) and references
therein).
2.3. Combining the stochastic volatility and the Levycomponent
Each of the two proposed models does something
different. One allows for varying fluctuations and the
other copes well with a large amount of information
driving the price suddenly up or down. It is only natural
to attempt to combine the two modeling assumptions.
As we shall see, this complicates the resulting equations
for the stock price, but they do remain of the same
integro-differential type.As a simple example we consider a jump-diffusion
process with volatility replaced by a geometric Brownian
motion:
dSt ¼ Stð� dtþ �t dZt þ Yð gÞ dNtÞ,
d�t ¼ �tð dtþ dWtÞ,
where Z and W are two standard Brownian motions with
correlation coefficient �, Nt is a Poisson process with
intensity �, and Y(g) is the jump amplitude random
variable with density g. The jump part may be written in a
perhaps more traditional way as a compound Poisson
process:PNt
i¼1 Yi.To obtain the price of a derivative in the above model
we follow Merton (1992) and obtain the following PIDE:
@F
@tþ1
2�2S2 @
2F
@S2þ1
2�22
@2F
@�2þ ��2S
@2F
@S @�
þ ðr� �kÞS@F
@S�1
2��2
@F
@�
þ �
ZR
½FðSY, �, tÞ � FðS, �, tÞ� gðYÞ dY� rF ¼ 0:
ð2:2Þ
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Here, r denotes the riskless rate, and k¼E(Y� 1), whereE is the expectation operator, and g is the density of the Yrandom variable. We mention that the quantity Y� 1measures the percentage change in the stock price if ajump occurs (for further discussion, see Merton (1992)).
We note that the derivative price still solves a PIDE andthis is a general characteristic of these types of processes.
2.4. A general existence result
The previous discussion motivates us to consider moregeneral integro-differential parabolic problems. In recentwork, Florescu and Mariani (2010) prove the existence ofsolutions to a general partial integro-differential equationin an unbounded smooth domain (2.3) using the methodof upper and lower solutions. Since we shall use some ofthe parts of the proof in the algorithm we state theproblem and the essential results here as well.
Let ��Rd be an unbounded smooth domain, and L a
second-order elliptic operator in non-divergence form, i.e.
Lu :¼Xdi,j¼1
aijðx, tÞuxixj þXdi¼1
biðx, tÞuxi þ cðx, tÞu,
with coefficients of L in the Holder SpaceC�,�=2ð�� ½0,T�Þ and satisfying the following conditions:
�jvj2 �Xdi,j¼1
aijðx, tÞvivj � �jvj2 ð05 � � �Þ,
jbiðx, tÞj � C, cðx, tÞ � 0:
These are classical conditions to ensure the strict elliptic-ity of the operator.
The problem we are solving is
Lu� ut ¼ Gðt, uÞ, in �� ð0,TÞ,
uðx, 0Þ ¼ u0ðxÞ, on �� f0g,
uðx, tÞ ¼ hðx, tÞ, on @�� ð0,TÞ:
8><>: ð2:3Þ
The operator G is a completely continuous integraloperator as those defined in (2.1) and (2.2), comingfrom the jump distribution. More precisely, we assumethat G(t, u)¼
R� g(x, t, u)dx, where g is any continuous
function.y The proof, however, is not constrained tointegral operators, indeed any operator with the sameproperties will work.
The existence result of Florescu and Mariani (2010)reads as follows.
Theorem 2.1: Let L and G be the operators defined above.Assume that either
. G is non-increasing with respect to u, or
. there exists some continuous, and increasing one-dimensional function f such that G(t, u)� f(u) isnon-increasing with respect to u.
Furthermore, assume there exist and , a lower and uppersolution of the problem with � in �� (0,T).
Then, problem (2.3) admits a solution u such that� u� in �� (0,T).
Remark 1: The above assumptions state that the result isapplicable in particular for any G that is dominated by apolynomial.
Remark 2: We remind the interested reader that asmooth function u is called an upper (lower) solution ofproblem (2.3) if
Lu� ut � ð�Þ Gðt, uÞ, in �� ð0,TÞ,
uðx, 0Þ � ð�Þ u0ðxÞ, on �� f0g,
uðx, tÞ � ð�Þ hðx, tÞ, on @�� ð0,TÞ:
8><>:
The new algorithm we propose is inspired by the proofof the theorem and we outline here the main steps of theproof.
We construct a series of regular PDE discretizations(for n2 {1, 2, . . . ,K}) for problem (2.3),
Lunþ1 � unþ1t ¼ Gðt, unÞ, in U� ð0, ~TÞ,
unþ1ðx, 0Þ ¼ u0ðxÞ, in U� f0g,
unþ1ðx, tÞ ¼ ’Uðx, tÞ, in @U� ð0, ~TÞ:
8><>: ð2:4Þ
Note that the boundary conditions are identical to theoriginal problem (2.3), but by using the known solutionfrom the previous step in the G term the main equationloses its integral term and becomes a regular PDE.
. Under the hypotheses given, each of systems(2.4) has a unique solution unþ1 2W2,1
p ðVÞ,where
W2,1p ðVÞ ¼ fv 2 Lp : vxi , vxixj 2 Lp, vt 2 Lpg
(see, e.g., Krylov (1996) and Lieberman (1996)).. We prove that � un�, 8n using the special
properties of L and G.. We show that {un} is a Cauchy sequence in
W2,1p ðVÞ using the fact that G is a completely
continuous operator, the previous step and theLebesgue’s dominated convergence theorem.
. We conclude that un! u is in the W2,1p -norm,
and then u is a strong solution of the problem.
3. Algorithms for numerical solutions
We propose two algorithms to find the solution ofproblem (2.3).
The first algorithm is new and was inspired by theconstructive proof of the theorem. It can be consideredthat this algorithm handles the integral part of the PIDEusing an implicit method because, after convergence, the uappearing in the integral term is evaluated at the currentvalue of t.
The second algorithm is a more standard explicitapproximation, where the u appearing in the integral is
yIn this general model, the case in which g is increasing with respect to u and all jumps are positive corresponds to the evolution of acall option near a crash.
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evaluated at the previous time step. We mention the studyof Cont and Voltchkova (2005), who provide a similarapproach for a more specific PIDE in only one spatialdimension and using a finite-difference scheme (we usefinite elements). We have no proof of convergence for thisalgorithm and indeed we observed that, for stability, thetime step had to be less than a threshhold value, whichdecreases as � increases. This is not surprising, since whileboth algorithms use implicit methods for the partialdifferential equations, the second algorithm handles theintegral part of the PIDE using an explicit scheme.
The purpose of presenting this algorithm is twofold.First, we want to compare the solution provided by thisalgorithm with the solution provided by the otheralgorithm in the hope of showing that, under reasonableparameter values and discretizations, the two solutionsare similar. Secondly, this algorithm is much faster thanthe other algorithm (in our tests, 15 times faster) and thusthere is interest in applying it in financial applicationswhere computer time is very important.
3.1. Algorithm 1
The new algorithm we propose comes from the proof oftheorem 2.1. We first find the ( ) lower and ( ) uppersolutions of the problem and we ensure that the operatorsL and G verify the hypotheses of the theorem.
. We start with u0¼ the lower solution.
. For every n5 0 we solve system (2.4). The PDEin this system cannot be solved analytically andwe implement a finite-element scheme for it.
. We take a three-dimensional grid int2 [0,T ], S2 [0,Smax], � 2 [0, �max]. Theupper bounds are suitably chosen largenumbers. This grid is kept the same for alliterations.
. We approximate the integrals in G(t, un)using a midpoint rule, with the requiredvalues of un(S, �, t) interpolated from thevalues saved at all the points on the grid,using quadratic interpolation in S and � andlinear interpolation in t.
. Since the boundary condition is usually at Twe solve the resulting partial differentialequation backwards in time using a PDEsolver (we use PDE2D in our experiments).
. The result is unþ1 calculated at all grid pointsand to be used in the next iteration.
. The un sequence converges to the solution of themain system (2.3). Thus the algorithm stopswhen the maximum difference between twoconsecutive iterations at all points on the grid issmall.
3.2. Algorithm 2
Algorithm 2 may be viewed as an explicit numericalscheme. It does not use the discretization (2.4), but
instead works directly with the given system (2.3). Itworks backward in time from t¼T.
. It starts with the final condition at T.
. When a PDE solver (we use PDE2D) is used tosolve (2.3) backwards from tiþ1 to ti, G(t, u(t)) isreplaced by G(tiþ1, u(tiþ1)), the integral isapproximated using a midpoint rule, and therequired values of u at tiþ1 are approximatedusing quadratic interpolation to the valuessaved at the S, � grid points.
. When t¼ 0, we stop. The solution approxima-tion is obtained for all grid points.
4. Numerical solutions using PDE2D
In order to test the two algorithms we consider the PIDEresulting from the model combining stochastic volatilityand jumps (system (2.2)). In this system we use thefollowing parameter values.
Parameters characterizing the stock: initial valueS0¼ 100, risk-free rate r¼ 0.05, volatility of volatility¼ 0.4, and correlation �¼� 0.3. The drift parameters �and do not enter into the PIDE.
Parameters characterizing the jump component: jumpfrequency �¼ 5 (expected five jumps per year) and theexpected jump percentage k¼� 0.15, which can becalculated from the jump distribution. We use aGaussian mixture model for g(Y)¼ 0.75g1(Y)þ0.25g2(Y), where g1(Y) and g2(Y) are Normal densitieswith means 0.7 and 1.3, respectively, and variances equalto 0.01. Thus, if a jump occurs the stock either goes downwith probability 0.75 or goes up with probability 0.25.The parameters of the distributions are chosen to ensurethat a jump is noticeable and separable from thestochastic variability of the model. Fan and Wang(2007) show that, in the presence of both jumps andstochastic volatility, it is hard to separate small-sizedjumps from the large stock variability.
Both algorithms are solved for the value of a Europeancall in this model. Specifically, the option maturity wasT¼ 1 and the strike price K¼ 100. The boundary condi-tions characterizing this option type wereF(S, �,T )¼max(S�K, 0), F(0, �, t)¼ 0, and (@F/@S)(Smax, �, t)¼ 1. We had no boundary conditions at �¼ 0or �¼ �max.
The algorithms were implemented using PDE2D, ageneral-purpose partial differential equation solver(Sewell 2005, 2010) available from Rogue Wave, Inc.(www.roguewave.com/pde2d), and has been used to solvemany mathematical finance applications (Topper 2005).PDE2D solves linear or nonlinear, steady-state, time-dependent and eigenvalue problems, in 1D intervals,general 2D regions (with curved boundaries), and a widerange of simple 3D regions. It has a sophisticated GUIinterface that makes it extremely easy to use.
It can solve 1D, 2D or 3D problems similar to theproblems presented (equations (2.1) and (2.2)) using a
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collocation finite-element method with cubic Hermite
basis functions to discretize the spatial derivatives and
adaptive finite-difference methods to discretize the time
derivatives. Without the integral term, a problem such as
(2.2) is a very straightforward PDE2D application, and
can be solved with little user effort. The integral term in
(2.2), however, requires additional effort.The two algorithms were implemented using PDE2D,
and we present below the specific details of each
implementation.For both algorithms we use Smax¼ 400 and �max¼ 1.
A 101� 101� 101 equally spaced grid was constructed on
[0,T ]� [0,Smax]� [0, �max]. A simple modification of
example 2.3 of Florescu and Mariani (2010) shows that
the operators L and G would satisfy the hypotheses of the
theorem if �min40, and that upper and lower solutions
exist with the lower solution � 0.
Algorithm 1 implementation: The iteration (2.4) is
repeatedly solved backward in time from a final condition
using PDE2D. The first iteration uses u0¼ ¼ 0, so that
no integral term appears, and the solution is saved for
each point on the grid. Each iteration thereafter, (2.4) is
solved with the F in the integral term approximated by
interpolating the solution saved in the previous iteration,
using quadratic interpolation in S and �, and linear
interpolation in time. For our examples, g(Y) was
negligible outside 05Y52, so the integral limits in (2.4)
were taken to be [0, 2], and a midpoint rule was used to
numerically approximate the integral. Since Y can be as
large as 2, however, values of F are needed at points SY
beyond Smax, where they have not been calculated. For
these values, a value of F is extrapolated using the
boundary condition @F/@S¼ 1, that is
FðSY, �, tÞ FðSmax, �, tÞ þ SY� Smax:
Algorithm 2 implementation: This time, (2.2) is onlysolved once using PDE2D, and at each time pointtiþ1¼ (iþ 1)D t (Dt¼T/100), PDE2D saves the solution
on the S� � grid. Then when it integrates backward fromtiþ1 to ti, the terms F(SY, �, t) and F(S, �, t) appearingin the integral are approximated by F(SY, �, tiþ1) andF(S, �, tiþ1), and these values are obtained using quadratic
interpolation to the solution saved at the grid points att¼ tiþ1. The numerical approximation to the integral isthe same as in algorithm 1.
4.1. Results
The solution is a function of three variables, so it isimpossible to view on a regular 3D plot. For this reasonwe only plot the solutions when t¼ 0 since this is usually
the most relevant value. When t40, the plots and thediscussion are very similar. Recall that, due to the finalboundary conditions (at t¼T ), the solution is simply
F(S, �,T )¼ (S�K)þ.Figure 1 shows the solution from algorithm 1 after the
first iteration. This is the same as the homogeneousproblem with �¼ 0, that is without the integral term,corresponding to the stochastic volatility model where the
jump-diffusion term does not exist. This figure is providedfor comparison, to view the effect of the integral term inthe PIDE. Figure 2 shows the final solution F(S, �, 0)after convergence, and figure 3 shows a more detailed
picture (around the origin) of the same final solution.Algorithm 1 shows significant oscillations near �¼ 0
for the first few iterations, but the oscillations are dampedafter further iteration, and (2.4) converges after 30iterations, using our stopping criterion of maximum
difference53 10�5. In general, increasing the value of� (increasing the importance of the integral term)
0
50
100
150
0
0.5
10
20
40
60
80
100
Sσ
F
Figure 1. F(S, �, 0) for the integral term equal to 0 (algorithm 1, first iteration).
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slows convergence. The oscillations are believed to be due
to the lack of a boundary condition at �¼ 0.Algorithm 2 produces a solution that closely agrees
with that produced by algorithm 1 (which is why we do
not plot it), the maximum difference between the two
solutions on the three-dimensional grid being about 0.21.
Since algorithm 2 performs only one ‘iteration’
(solves (2.2) once only), it is expected to be the more
efficient algorithm, and indeed it requires about 15 times
less computer time for this example. To give more details
about the order of magnitude of the difference between
the two solutions we calculate and plot in figure 4 the root
mean squared error, defined as RMSEðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½1=ðn� 1Þ�
PðF1ðS, �, tÞ � F2ðS, �, tÞÞ2
q, and the mean absolute
deviation, defined as MAD(t)¼ (1/n)PjF1(S, �, t)�
F2(S, �, t)j. F1 and F2 are the solutions obtained using
algorithm 1 and algorithm 2, respectively, and the sum
runs over all grid points. We do not provide a plot when
t¼ 1 since the two solutions coincide with the boundary
value.Finally, since at t¼ 0 we know the S variable but
� is unobservable, we present in figure 5 the option
price evolution depending on the realized volatility
value �0.
0100
200300
400
0
0.5
10
50
100
150
200
250
300
350
Sσ
F
Figure 2. The final solution F(S, �, 0) after convergence of algorithm 1.
0
50
100
150
0
0.5
10
20
40
60
80
100
S
σ
F
Figure 3. The final solution F(S, �, 0); a closeup at the origin.
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4.2. A discussion of the influence of the integral term
The approach presented in the current article dealsspecifically with partial differential equations containingan integral term. As may be observed from any of the
equations presented, �, the parameter that governs thefrequency of the jumps, ends in the resulting PIDEmultiplying the integral term. Thus we chose to presentthe results when � increases in value and thus the integralterm becomes dominant.
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.02
0.04
0.06
0.08
0.10
Times
Dev
iatio
nRoot mean squared errorMean absolute deviation
Figure 4. Two measures of difference between the twosolutions.
0.0 0.2 0.4 0.6 0.8 1.0
3540
45
vol
Opt
ion
Figure 5. The call option value depending on the volatility valueat time t¼ 0.
050
100150
0
0.5
10
20
40
60
80
100(a)(b)
S
l=0
l=10 l=100
l=2
σ
F
050
100150
0
0.5
10
20
40
60
80
100
Sσ
F
050
100
150
0
0.5
10
20
40
60
80
100
(c)
Sσ
F
050
100150
0
0.5
10
50
100
150
(d)
Sσ
F
Figure 6. Changes in the solution at t¼ 0 when the importance of the integral term increases.
Numerical solutions for PIDEs 71451
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In figure 6 we present graphs of the solution at t¼ 0,obtained when � changes in value. These graphswere obtained by running algorithm 2 (faster). Thesolutions obtained running the two algorithms were verysimilar in all cases attempted. We see from figure 6(a) theresult of the regular PDE with no integral term, and as �increases the integral term becomes dominant. Note thatwhen �¼ 100 (figure 6(d)) the solution is almost F¼S,which is to be expected since F¼S is the steady-statesolution when � is infinite (and satisfies the boundaryconditions).
5. Conclusions
In this article we present a completely novel algorithmproven to converge to solve partial differential equationswith an integral term. We compare the new algorithmwith another, more explicit algorithm that is not provento converge. In our numerical example we found thedifference between the two solutions to be minimal. Thisopens up the way for numerical schemes to approximatemore general PIDE produced by Levy models of the stockprice. The algorithms presented should work for theseproblems, although a more detailed study of convergencefor algorithm 2 should be performed. In our numericalexperiments we discovered that increasing � producedinstability, which was fixed by using a smaller time step.This is very similar to the traditional explicit finite-difference scheme and a relation between parameters thatwould guarantee convergence would be very beneficial.
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