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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

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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.

References

Andersen, L. and Andreasen, J., Jump-diffusion processes:Volatility smile fitting and numerical methods for optionpricing. Rev. Deriv. Res., 2000, 4, 231–262.

Barndorff-Nielsen, O.E., Jensen, J.L. and Sorensen, M., Somestationary processes in discrete and continuous time.Adv. Appl. Probab., 1998, 30(4), 989–1007.

Black, F. and Scholes, M., The valuation of options andcorporate liability. J. Polit. Econ., 1973, 81, 637–654.

Chesney, M. and Scott, L., Pricing European currency options:A comparison of the modified Black–Scholes model and arandom variance model. J. Financ. Quant. Anal., 1989, 24,267–284.

Cont, R. and Tankov, P., Financial Modeling with JumpsProcesses, 2003 (Chapman & Hall: London).

Cont, R. and Voltchkova, E., A finite difference scheme foroption pricing in jump diffusion. SIAM J. Numer. Anal.,2005, 43(4), 1596–1626.

Delbaen, F. and Schachermayer, W., A general version of thefundamental theorem of asset pricing. Math. Ann., 1994, 300,463–520.

D’Halluin, Y., Numerical methods for real options in tele-communications. PhD thesis, University of Waterloo, 2004.

Duffie, D., Dynamic Asset Pricing Theory, 3rd ed., 2001(Princeton University Press: Princeton).

Fan, J. and Wang, Y., Multi-scale jump and volatility analysisfor high-frequency financial data. J. Am. Statist. Assoc., 2007,102, 1349–1362.

Florescu, I. and Mariani, M.C., Solutions to integro-differentialparabolic problems arising in the pricing of financial optionsin a Levy market. Electron. J. Differ. Eqns, 2010, 2010(62),1–10.

Florescu, I. and Viens, F., Stochastic volatility: Option pricingusing a multinomial recombining tree. Appl. Math. Finance,2008, 15(2), 151–181.

Fouque, J.-P., Papanicolaou, G. and Sircar, K.R., Derivatives inFinancial Markets with Stochastic Volatility, 2000 (CambridgeUniversity Press: Cambridge).

Geman, H., Pure jump Levy processes for asset price modeling.J. Bank. Finance, 2002, 26, 1297–1316.

Hagan, P., Kumar, D., Lesniewski, A. and Woodward, D.,Managing smile risk.Wilmott Mag., 2002, September, 84–108.

Harrison, M. and Pliska, S., Martingales and stochasticintegrals in the theory of continuous trading. Stochast.Process. Applic., 1981, 11, 215–260.

Heston, S.L., A closed-form solution for options with stochasticvolatility with applications to bond and currency options.Rev. Financ. Stud., 1993, 6(2), 327–343.

Hull, J.C., Options, Futures, and Other Derivatives, 7th ed., 2008(Prentice Hall: Engelwood Cliffs, NJ).

Hull, J.C. and White, A.D., The pricing of options on assetswith stochastic volatilities. J. Finance, 1987, 42(2), 281–300.

Ikeda, N., Stochastic Differential Equations and DiffusionProcesses, 2nd ed., 1989 (North-Holland: Amsterdam).

Jarrow, R.A., Modelling Fixed Income Securities and InterestRate Options, 2nd ed., 2002 (Stanford Economics andFinance).

Krylov, N.V., Lectures on Elliptic and Parabolic Equations inHolder Spaces, 1996 (American Mathematical Society).

Lieberman, G.M., Second Order Parabolic DifferentialEquations, 1996 (World Scientific: Singapore).

Madan, D.B., Carr, P.P. and Chang, E.C., The variance gammaprocess and option pricing. Eur. Finance Rev., 1998, 2(1),79–105.

Mariani, M.C., Florescu, I., Beccar Varela, M.P. andNcheuguim, E., Long correlations and Levy Models appliedto the study of memory effects in high frequency (tick) data.Physica A, 2009, 388(8), 1659–1664.

Merton, R.C., Continuous-time Finance, 1992 (Wiley–Blackwell:New York).

Scott, L.O., Option pricing when the variance changesrandomly: Theory, estimation, and an application.J. Financ. Quant. Anal., 1987, 22(4), 419–438.

Sewell, G., The Numerical Solution of Ordinary and PartialDifferential Equations, 2nd ed., 2005 (Wiley: New York).

Sewell, G., Solving PDEs in non-rectangular 3D regions using acollocation finite element method. Adv. Engng Softw., 2010,41, 748–753.

Stein, E.M. and Stein, J.C., Stock price distributions withstochastic volatility: An analytic approach. Rev. Financ. Stud.,1991, 4(4), 727–752.

Topper, J., Financial Engineering with Finite Elements, 2005(Wiley: New York).

Wiggins, J.B., Option values under stochastic volatility: Theoryand empirical estimates. J. Financ. Econ., 1987, 19(2),351–372.

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