A FINITE DIFFERENCE SCHEME FOR OPTION PRICING IN JUMP ...rama/papers/pide.pdf · A FINITE DIFFERENCE SCHEME FOR OPTION PRICING IN JUMP DIFFUSION AND EXPONENTIAL LEVY MODELS´ ∗

SIAM J. NUMER. ANAL. c© 2005 Society for Industrial and Applied MathematicsVol. 43, No. 4, pp. 1596–1626

A FINITE DIFFERENCE SCHEME FOR OPTION PRICING INJUMP DIFFUSION AND EXPONENTIAL LEVY MODELS∗

RAMA CONT† AND EKATERINA VOLTCHKOVA†

Abstract. We present a finite difference method for solving parabolic partial integro-differentialequations with possibly singular kernels which arise in option pricing theory when the random evo-lution of the underlying asset is driven by a Levy process or, more generally, a time-inhomogeneousjump-diffusion process. We discuss localization to a finite domain and provide an estimate for thelocalization error under an integrability condition on the Levy measure. We propose an explicit-implicit finite difference scheme which can be used to price European and barrier options in suchmodels. We study stability and convergence of the scheme proposed and, under additional condi-tions, provide estimates on the rate of convergence. Numerical tests are performed with smooth andnonsmooth initial conditions.

Key words. parabolic integro-differential equations, finite difference methods, Levy process,jump-diffusion models, option pricing, viscosity solutions

AMS subject classifications. 47G20, 65M06, 65M12, 49L25, 60H30, 60G51

DOI. 10.1137/S0036142903436186

1. Introduction. The shortcomings of diffusion models in representing the riskrelated to large market movements have led to the development of various option pric-ing models with jumps, where large returns are represented as discontinuities in pricesas a function of time. Models with jumps allow for a more realistic representation ofprice dynamics and greater flexibility in modelling and have been the focus of muchrecent work [11].

Exponential Levy models, where the market price of an asset is represented asthe exponential St = exp(rt + Xt) of a Levy process Xt, offer analytically tractableexamples of positive jump processes which are simple enough to allow a detailed studyboth in terms of statistical properties and as models for risk-neutral dynamics, i.e.,option pricing models. Option pricing with exponential Levy models is discussed in[11, 17, 24]. The flexibility of choice of the Levy process X allows us to calibrate themodel to market prices of options and reproduce a wide variety of implied volatilityskews/smiles [12]. The Markov property of the price allows us to express pricesof European and barrier options in terms of solutions of partial integro-differentialequations (PIDEs) that involve, in addition to a (possibly degenerate) second-orderdifferential operator, a nonlocal integral term that requires specific treatment at boththe theoretical and numerical levels [13].

In this paper, we propose a finite difference scheme for solving such PIDEs. Ournumerical solution is based on splitting the operator into a local and a nonlocalpart: we treat the local term using an implicit step and the nonlocal term using anexplicit step. This idea, previously used for nonlinear PDEs [3], allows for an efficientnumerical implementation. Some difficulties arise due to the nonlocal character of the

∗Received by the editors October 13, 2003; accepted for publication (in revised form) October 1,2004; published electronically October 31, 2005. This work was presented at the Bachelier Seminar(Paris, 2003), Workshop on Levy Processes and Partial Integro-Differential Equations (Palaiseau,France, 2003), IFIP 2003 (Nice, France), Workshop on Computational Finance (Zurich, 2003), andECCOMAS 2004 (Jyvaskyla, Finland).

http://www.siam.org/journals/sinum/43-4/43618.html†CMAP-Ecole Polytechnique, F 91128 Palaiseau, France ([email protected],

[email protected]).

1596

FINITE DIFFERENCE SCHEMES FOR PIDEs 1597

integral operator, nonsmoothness of initial conditions, the singularity at zero of theintegral kernel, and the possible degeneracy of the diffusion coefficient. We resolvethese difficulties in the framework of viscosity solutions and provide error estimates ineach case, under assumptions which are easily verified on the Levy density. We studythe consistency and stability of this scheme, show its convergence to the solution ofthe PIDE, and study its numerical performance in two examples, the Merton modelwith Gaussian jumps and the infinite activity variance Gamma model. Our schemecan be used for European and barrier options and can also be extended to the case ofnonconstant coefficients.

1.1. Relation to previous literature. Various numerical methods for solvingsuch parabolic integro-differential equations have been proposed in the recent litera-ture [2, 25, 16, 31]. In the case where the characteristic function of the log-price isknown analytically, the fast Fourier transform of Carr and Madan [9] can be used forpricing European options. Though our finite difference method requires more opera-tions than the fast Fourier transform [9], our method does not require a closed formexpression for the characteristic function of the log-price and can also handle barrieroptions (i.e., boundary value problems).

Finite difference schemes for PIDEs have been proposed in [2, 16, 30], but arigorous analysis of consistency, stability, and convergence is absent from these studies.By appealing to the formalism of viscosity solutions, our analysis allows fairly generalhypotheses on the model and applies to models based on pure-jump Levy processessuch as the variance Gamma model [23], hyperbolic models [17], and the normalinverse Gaussian (NIG) model [7].

For jump-diffusion models with finite jump intensity, Andersen and Andreasen [2]proposed an operator splitting method where the differential part is treated using aCrank–Nicholson step and the jump integral is computed using an explicit time step.Our method applies more generally to models with infinite activity, i.e., singular inte-gral kernels; in addition, we propose an analysis of the convergence of our algorithm,which is absent in [2].

Using a variational formulation of the integro-differential equation, Zhang [31]studied a finite difference scheme in the case of jump-diffusion models having finiteintensity and possessing all exponential moments (see also [16]). These conditions ruleout all models in the literature except the Merton model: our analysis does not requiresuch restrictive conditions. The variational formulation has been recently extendedby Matache, von Petersdorff, and Schwab [25] to the infinite activity case using awavelet Galerkin method. While the approach of [25] is more general than the aboveapproaches, it does not allow us to treat singular cases such as the variance Gammamodel [23].

1.2. Outline. Section 2 starts by recalling facts about Levy processes and ex-ponential Levy models. In section 3 we briefly discuss, following [13], the characteri-zation of prices of European and barrier options in exponential Levy models in termsof viscosity solutions of PIDEs.

Solving such PIDEs by finite difference methods involves several approximations:localization of the equation to a bounded domain, treatment of the singularity due tosmall jumps, discretization of the equation in space, and iteration in time. We discusslocalization errors in section 4 and provide an estimate for the localization errors underan integrability condition on the Levy measure. In section 5 we propose an explicit-implicit finite difference scheme and study consistency, stability, and convergence ofthe scheme proposed. Convergence properties of the scheme are studied in section

1598 RAMA CONT AND EKATERINA VOLTCHKOVA

6: we show that the scheme is monotone, unconditionally stable, and consistent andexhibit conditions for its convergence to the solution of the PIDE. Under furtherconditions on the scheme, we are able to give an estimate of the rate of convergencein section 6.4. Finally, in section 7, numerical tests are performed for smooth andnonsmooth initial conditions to assess the effect of various numerical parameters onthe accuracy of the scheme.

2. Exponential Levy models. We consider here the class of exponential Levymodels: the risk-neutral dynamics of the underlying asset is given by St = exp(rt +Xt), where Xt is a time-homogeneous jump-diffusion (Levy) process.

2.1. Levy processes: Definitions. A Levy process is a stochastic process Xt

with stationary independent increments which is continuous in probability. Withoutloss of generality we assume that X0 = 0. The characteristic function of Xt has thefollowing form, called the Levy–Khinchin representation [27]:

(2.1)

E[eizXt ] = e−tψ(z) = exp

{t

(−σ2z2

2+ iγz +

∫ ∞

−∞(eizx − 1 − izx1|x|≤1)ν(dx)

)},

where σ > 0 and γ are real constants and ν is a positive measure verifying∫ +1

−1

x2ν(dx) < ∞,

∫|x|>1

ν(dx) < ∞.(2.2)

The random process X can be interpreted as the superposition of a Brownian mo-tion with drift and an infinite superposition of independent (compensated) Poissonprocesses with various jump sizes x, ν(dx) being the intensity of jumps of size x.In general ν is not a finite measure:

∫ν(dx) need not be finite. In the case where

λ =∫ν(dx) < +∞, the measure ν can be normalized to define a probability measure

μ, which can now be interpreted as the distribution of jump sizes:

μ(dx) =ν(dx)

λ.

The jumps of X are then described by a compound Poisson process with λ as jumpintensity (average number of jumps per unit time) and μ(.) as jump size distribution.In this case the truncation of small jumps is not needed, and the Levy–Khinchinrepresentation reduces to

E[eizXt ] = exp

{t

(−σ2z2

2+ iγ0(ν)z +

∫ ∞

−∞(eizx − 1)ν(dx)

)}.

A Levy process is a Markov process; its infinitesimal generator LX : f → LXf isan integro-differential operator defined by the expression

(2.3) LXf(x) = limt→0

E[f(x + Xt)] − f(x)

t

=σ2

2

∂2f

∂x2+ γ

∂f

∂x+

∫ν(dy)

[f(x + y) − f(x) − y1{|y|≤1}

∂f

∂x(x)

],

which is well defined for f ∈ C2(R) with compact support.


2.2. Exponential Levy models. Let (St)t∈[0,T ] be the price of a financial assetmodelled as a stochastic process on a filtered probability space (Ω,F ,Ft,Q). Underthe hypothesis of absence of arbitrage there exists a measure equivalent to Q underwhich (St) is a martingale. We will assume in what follows that Q is a martingalemeasure.

Exponential Levy models assume that the (risk-neutral) dynamics of St under Q

is represented as the exponential of a Levy process:

St = S0ert+Xt .(2.4)

Here Xt is a Levy process with characteristic triplet (σ,γ,ν), and the interest rater is included for ease of notation. Different exponential Levy models proposed inthe financial modelling literature simply correspond to different choices for the Levymeasure ν; see [11, Chap. 3] for a review. The absence of arbitrage then imposes thatSt = Ste

−rt = expXt is a martingale, which is equivalent to the following conditionson the triplet (σ,γ,ν):∫

|y|>1

ν(dy)ey < ∞,(2.5)

γ = γ(σ, ν) = −σ2

2−∫

(ey − 1 − y1|y|≤1)ν(dy).(2.6)

We will assume this relation holds in what follows. The infinitesimal generator LX

then becomes

(2.7) LXf(x) =σ2

2

[∂2f

∂x2− ∂f

∂x

]+

∫ ∞

−∞ν(dy)

[f(x + y) − f(x) − (ey − 1)

∂f

∂x(x)

].

We will also use the notation Yt = rt + Xt. The infinitesimal generator of Yt is

Lf = LXf + r∂f

∂x.(2.8)

3. Partial integro-differential equation for option prices. The value of anoption is defined as a discounted conditional expectation of its terminal payoff HT un-der the risk-adjusted martingale measure (sometimes called risk-neutral probability)Q:

Ct = E[e−r(T−t)HT |Ft].

For a European call or put, HT = H(ST ). From the Markov property, Ct =C(t, S), where

C(t, S) = E[e−r(T−t)H(ST )|St = S].(3.1)

Introducing the change of variable τ = T − t, x = ln(S/S0), and defining h(x) =H(S0e

x) and u(τ, x) = erτC(T − τ, S0ex), then

u(τ, x) = E[h(x + Yτ )].(3.2)

If u is sufficiently smooth—for example, u ∈ C1,2 with bounded derivatives—thenby applying Ito’s formula to u(t,Xt) between 0 and T one can show [8] that it is aclassical solution of the Cauchy problem:

∂u

∂τ= Lu on (0, T ] × R, u(0, x) = h(x), x ∈ R.(3.3)


Barrier options lead to initial-boundary value problems. Consider, for instance, anup-and-out call option with maturity T , strike K, and (upper) barrier U > S0. Theterminal payoff is given by

HT = (ST −K)+1T<θ,

where θ = inf{t ≥ 0 | St ≥ U}, the first moment when the barrier is crossed. Due tothe strong Markov property of Levy processes, it is possible to express the value ofthe option Ct = e−r(T−t)E[HT |Ft] as a deterministic function of time t and currentstock value St before the barrier is crossed. Namely, for any (t, S) ∈ [0, T ] × (0,∞)we can define

Cb(t, S) = e−r(T−t) E[H(SeYT−t)1T<θt ],(3.4)

where H(S) = (S−K)+, {Ys−t, s ≥ t} is a Levy process, and θt = inf{s ≥ t | SeYs−t ≥U}, the first exit time after t. Then,

Ct = Cb(t, St)1t≤θ(3.5)

for all t ≤ T . Note that outside of the set {t ≤ θ} the objects Ct and Cb(t, St) aredifferent: if the barrier has already been crossed, Ct will always be zero, but Cb(t, St)may become positive if the stock returns to the region below the barrier. By going tothe log variables we define

ub(τ, x) = erτCb(T − τ, S0ex).(3.6)

Again, if ub is smooth the Ito formula can be used to show [8] that ub is a solution ofthe following initial-boundary value problem:

∂u

∂τ= Lu on (0, T ] × (−∞, log(U/S0)),

u(0, x) = h(x), x < log(U/S0); u(τ, x) = 0, x ≥ log(U/S0).

Prices of down-and-out or double barrier options are defined similarly. In the caseof pure jump models where σ = 0, these smoothness conditions can fail to hold;counterexamples are given in [13]. In this case the option price should be seen as aviscosity solution of the PIDE, as discussed below.

3.1. Viscosity solutions for integro-differential equations. Existence anduniqueness of (classical) solutions for the PIDEs considered above in Sobolev–Holderspaces have been studied in [8, 18] in the case where the diffusion component isnondegenerate: for a Levy process this simply means σ > 0, but more generally theseresults apply to jump diffusion where the diffusion coefficient is bounded away fromzero. However, many of the models in the financial modelling literature are purejump models with σ = 0, for which such results are not available. In fact, in purejump models with finite variation (3.3) is formally a first order in the price variableso the effect of the jump term is more like a convection term rather than a diffusionterm. A notion of solution that yields existence and uniqueness for such equationswithout requiring nondegeneracy of coefficients or a priori knowledge of smoothnessof solutions is the notion of viscosity solution, introduced by Crandall and Lions forPDEs (see [14]) and extended to integro-differential equations of the type consideredhere in [1, 4, 26, 28, 29].


Denote by USC (respectively, LSC) the class of upper semicontinuous (respec-tively, lower semicontinuous) functions u : (0, T ] × R → R and by C+

p ([0, T ] × R) theset of measurable functions on [0, T ] × R with polynomial growth of degree p at +∞and bounded on [0, T ] × R−:

ϕ ∈ C+p ([0, T ] × R) ⇐⇒ ∃C > 0, |ϕ(t, x)| ≤ C(1 + |x|p 1x>0).(3.7)

Let O = (l, u) ⊆ R be an open interval, ∂O = {l, u} its boundary, and g ∈ C+p ([0, T ]×

R \O) a continuous function. Consider the following initial-boundary value problemon [0, T ] × R:

∂u

∂τ= Lu on (0, T ] ×O,(3.8)

u(0, x) = h(x), x ∈ O; u(τ, x) = g(τ, x), x /∈ O.(3.9)

Definition 3.1 (viscosity solution). A function u ∈ USC is a viscosity subso-lution of (3.8)–(3.9) if for any test function ϕ ∈ C2([0, T ] × R) ∩ C+

p ([0, T ] × R) andany global maximum point (τ, x) ∈ [0, T ] × R of u − ϕ, the following properties areverified:

if (τ, x) ∈ (0, T ] ×O,

(∂ϕ

∂τ− Lϕ

)(τ, x) ≤ 0,(3.10)

if τ = 0, x ∈ O, min

{(∂ϕ

∂τ− Lϕ

)(τ, x), u(τ, x) − h(x)

}≤ 0,

if τ ∈ (0, T ], x ∈ ∂O, min

{(∂ϕ

∂τ− Lϕ

)(τ, x), u(τ, x) − g(τ, x)

}≤ 0,

if x /∈ O, u(τ, x) ≤ g(τ, x).(3.11)

A function u ∈ LSC is a viscosity supersolution of (3.8)–(3.9) if, for any test functionϕ ∈ C2([0, T ]× R) ∩C+

p ([0, T ]× R) and any global minimum point (τ, x) ∈ [0, T ]× R

of u− ϕ, we have

if (τ, x) ∈ (0, T ] ×O,

(∂ϕ

∂τ− Lϕ

)(τ, x) ≥ 0,

if τ = 0, x ∈ O, max

{(∂ϕ

∂τ− Lϕ

)(τ, x), u(τ, x) − h(x)

}≥ 0,

if τ ∈ (0, T ], x ∈ ∂O, max

{(∂ϕ

∂τ− Lϕ

)(τ, x), u(τ, x) − g(τ, x)

}≥ 0,

if x /∈ O, u(τ, x) ≥ g(τ, x).

A function u ∈ C+p ([0, T ] × R) is called a viscosity solution of (3.8)–(3.9) if it is

both a subsolution and a supersolution. This function is then continuous on (0, T ]×R.Note that the initial and boundary conditions are verified in a viscosity sense.

The definition also includes the case of initial value problems: O = R. Existence anduniqueness of viscosity solutions for such parabolic integro-differential equations inthe case O = R are discussed in [1] in the case where ν is a finite measure and in [4]and [26] for general Levy measures. Growth conditions other than u ∈ C+

p can beconsidered (see, e.g., [1, 4]) with additional conditions on the Levy measure ν. Themain tool for showing uniqueness is the comparison principle: if u, v are viscosity


solutions and u(0, x) ≥ v(0, x), then for all τ ∈ [0, T ], u(τ, x) ≥ v(τ, x). This propertycan be extended to subsolutions and supersolutions in the following sense [1, 20].

Proposition 3.2 (comparison principle for semicontinuous solutions [1, 20]). Ifu ∈ USC is a subsolution and v ∈ LSC is a supersolution of (3.8)–(3.9) with O = R

and h is a continuous function, then u ≤ v on [0, T ] × R.

Proofs and extensions can be found in [1] for the case where ν is a boundedmeasure; the case of a general Levy measure has recently been treated in [20].

3.2. Option prices as viscosity solutions of PIDEs. The following result,whose proof in a more general setting is given in [13], shows that values of Euro-pean and barrier options with Lipschitz payoff function can be expressed in terms of(viscosity) solutions of (3.8)–(3.9).

Proposition 3.3 (option prices as viscosity solutions). Let the payoff functionH verify the Lipschitz condition on its domain of definition,

|H(S1) −H(S2)| ≤ C|S1 − S2| ∀S1, S2 ∈ (S0el, S0e

u),(3.12)

and let h(x) = H(S0ex) have polynomial growth at infinity. Then the following hold:

1. The forward value of a European option u(τ, x) defined by (3.2) is the uniqueviscosity solution of the Cauchy problem (3.3).

2. If the forward value ub(τ, x) of a knockout (single or double) barrier optiondefined by (3.6) is continuous, then it is a viscosity solution of (3.8)–(3.9)(with g ≡ 0).

The assumptions on the payoff function apply to put options, single-barrier knock-out puts, double barrier knockout options and also to the log-contract. One can thenretrieve call options by put-call parity. For barrier options, continuity holds in par-ticular if ν(R) < ∞ and σ > 0 but also in pure jump models with infinite activity[13]. For barrier options with rebate, the zero boundary condition has to be replacedby the value of the rebate, as in the case of diffusion models.

4. Localization estimates. In order to solve numerically the PIDE, we firstlocalize the variables and the integral term to bounded domains. This section discussesestimates for the localization error using a probabilistic approach.

4.1. Localization to a bounded domain. To solve numerically the initial-boundary value problem (3.8)–(3.9) in the case of an unbounded domain O, we firsttruncate the domain to an interval x ∈ (−A,A). Usually, this leads us to definesome boundary conditions at x = −A and x = A. As noted above, the operator L isnonlocal: computing the integral term at a point x ∈ (−A,A) requires knowledge ofu(τ, ·) on {x+ y | y ∈ supp ν}, which in most examples is equal to the whole real lineR. In the case of knock-out barrier options, a natural boundary condition is given bythe zero extension (or the rebate). In other cases, this extension is done by imposing anumerical boundary condition. Choosing u(τ, x) = g(τ, x) for some given continuousfunction g with polynomial growth will lead to a probabilistic interpretation of thesolution of the localized problem.

Although many choices are possible for the boundary condition g, we will considerhere two cases. The simplest choice is g = 0, i.e., extend the solution by zero outsidethe domain. Another extension is given by the payoff function (the initial condition)itself, g(τ, x) = h(x), which is asymptotically close to the solution at infinity. We willsee that both choices lead to a localization error that exponentially decreases with


the domain size. Let us define uA(τ, x) as the solution of the localized problem

∂uA

∂τ= LuA, (0, T ] × (−A,A),(4.1)

uA(0, x) = h(x), x ∈ (−A,A); uA(τ, x) = g(τ, x), x /∈ (−A,A),

where g = 0 or g(τ, x) = h(x).Proposition 4.1. Assume that h is bounded (||h||∞ < ∞) and

∃α > 0,

∫|x|>1

eα|x|ν(dx) < ∞.(4.2)

Let u(τ, x) be the solution of (3.3) and let uA(τ, x) be the (viscosity) solution of (4.1)with boundary condition g = 0 or g(τ, x) = h(x). Then

|u(τ, x) − uA(τ, x)| ≤ 2Cτ,α||h||∞e−α(A−|x|) ∀x ∈ (−A,A),(4.3)

where the constant Cτ,α does not depend on A.Proof. The proof is based on the probabilistic representation [8, 13] of the solutions

of (5.1) and (4.1). Let us define Mxτ = supt∈[0,τ ] |Yt + x|. Then

u(τ, x) = E[h(Yτ + x)],

uA(τ, x) = E[h(Yτ + x)1{Mxτ <A}] if g = 0,

or uA(τ, x) = E[h(Yτ + x)1{Mxτ <A} + h(Yθ(x) + x)1{Mx

τ ≥A}],

where θ(x) = inf{t ≥ 0, |Yt + x| ≥ A} is the first exit time of Yt + x from [−A,A].Subtracting uA from u gives

|u(τ, x) − uA(τ, x)| = |Eh(Yτ + x)1{Mxτ ≥A}|

≤ ||h||∞Q(Mxτ ≥ A) for g = 0,

and in the case g(τ, x) = h(x) we obtain

|u(τ, x) − uA(τ, x)| ≤ E|h(Yτ + x)1{Mxτ ≥A}| + E|h(Yθ(x) + x)1{Mx

τ ≥A}|≤ 2||h||∞Q(Mx

τ ≥ A).

So, in both cases

|u(τ, x) − uA(τ, x)| ≤ 2||h||∞Q(Mxτ ≥ A).(4.4)

Theorem 25.18 of [27] together with (4.2) implies

Cτ,α = EeαM0τ < ∞.(4.5)

Therefore, Chebyshev’s inequality applies, and we obtain

Q(M0τ ≥ A) ≤ Cτ,αe

−αA.(4.6)

Now, to pass from M0τ to Mx

τ , we use the following implications:

sup |Yt + x| ≤ sup |Yt| + |x|⇒ (sup |Yt + x| ≥ A ⇒ sup |Yt| + |x| ≥ A)

⇒ Q(Mxτ ≥ A) ≤ Q(M0

τ ≥ A− |x|)≤ Cτ,αe

−α(A−|x|) by (4.6).


Combining the last inequality with (4.4) gives the desired result.Remark 1. In the case of a put option, ‖h‖∞ < ∞ and Proposition 4.1 applies.

In other examples, h may be unbounded; in this case it is still possible to obtain anexponentially decreasing localization error under additional restrictions on ν. Notethat for the call option, although the payoff grows exponentially, one can transformthe problem into pricing a put using put-call parity in order to obtain a smallerlocalization error.

Remark 2. An exponential bound on localization error in the L2-norm is givenin [25] using analytical methods. The advantage of the probabilistic approach is toprovide a local (pointwise) estimate. For instance, our estimate (4.3) reflects theintuitive fact that the localization error is more pronounced near the boundary.

The above result implies that the localization error decreases uniformly on eachclosed subinterval of (−A,A):

|u(τ, x) − uA(τ, x)| ≤ ke−αδA for |x| ≤ (1 − δ)A,

where 0 < δ < 1.Assumption (4.2) means that the tails of ν have to decrease exponentially, which

is true in all examples considered in the option pricing literature (except Carr andWu’s log-stable model [10]). Note that in an exponential Levy model we already have∫ +∞1

eαxν(dx) < ∞ for all α ≤ 1, because of the martingale condition, so (4.2) is acondition on the negative jumps.

4.2. Truncation of the integral. To compute numerically the integral term,we need to reduce the region of integration to a bounded interval. In terms of thejump process, this amounts to the truncation of large jumps. We will now give anestimate for the error resulting from this approximation. Recall that the solution ofthe Cauchy problem (3.3) (in the European case where O = R) for a Lipschitz payofffunction H is

u(τ, x) = E[H(S0ex+rτ+Xτ )],(4.7)

where Xτ is a Levy process with the triplet (γ, σ, ν). Let us define a new process Xτ

characterized by the Levy triplet (γ, σ, ν1x∈[Bl,Br]), where γ is such that exp(rt+ Xt)remains a martingale:

γ = −σ2

2−∫ Br

Bl

(ey − 1 − y1|y|≤1)ν(dy).

We now define

u(τ, x) = E[H(S0ex+rτ+Xτ )](4.8)

and we estimate the difference between u and the true solution u.Proposition 4.2. Let H be Lipschitz: |H(S1)−H(S2)| ≤ c|S1−S2|. Assume that

there exists αr, αl > 0, such that∫∞1

e(1+αr)yν(dy) < ∞ and∫ −1

−∞ |y|eαl|y|ν(dy) < ∞.If u and u are defined by (4.7) and (4.8), respectively, then

|u(τ, x) − u(τ, x)| ≤ 2c S0ex+rττ(C1e

−αl|Bl| + C2e−αr|Br|).(4.9)

Proof. Let us denote Rτ=Xτ − Xτ , Rτ ⊥⊥ Xτ . We have

|u(τ, x) − u(τ, x)| = |E[H(S0ex+rτ+Xτ+Rτ )] − E[H(S0e

x+rτ+Xτ )]|≤ c S0e

x+rτE[eXτ |eRτ − 1|] = c S0ex+rτE|eRτ − 1|.


By construction, E[eRτ − 1] = 0. Since |eRτ − 1| = (eRτ − 1) + 2(1 − eRτ )+ and(1 − eRτ )+ ≤ |Rτ |, we obtain

|u(τ, x) − u(τ, x)| ≤ 2c S0ex+rτE|eRτ |.(4.10)

The Levy triplet of Rτ is (γ − γ, 0, ν1x/∈[Bl,Br]) with

γ − γ = −∫y/∈[Bl,Br]

(ey − 1)ν(dy).

One can write Rτ = Pτ + Nτ , where Pτ and Nτ are characterized by (∫ Bl

−∞(1 −ey)ν(dy), 0, ν1x>Br ) and (−

∫∞Br

(ey − 1)ν(dy), 0, ν1x<Bl), respectively. We assume

without loss of generality that Bl < −1, Br > 1.1 Since Pτ has nonnegative drift, noBrownian component, and only positive jumps bounded from below by Br > 0, wehave Pτ ≥ 0 (recall that P0 = 0). Conversely, Nτ has only negative jumps (boundedfrom above by Bl < 0) and nonpositive drift. In consequence, Nτ ≤ 0. Therefore,

E|Rτ | ≤ E|Pτ | + E|Nτ | = EPτ − ENτ

= τ

[∫ Bl

−∞(1 − ey − y)ν(dy) +

∫ ∞

Br

(ey − 1 + y)ν(dy)

]

≤ τ

[2

∫ Bl

−∞|y|ν(dy) + 2

∫ ∞

Br

eyν(dy)

].(4.11)

Using the hypotheses on ν, we obtain

E|Rτ | ≤ τ

(2e−αl|Bl|

∫ Bl

−∞|y|eαl|y|ν(dy) + 2e−αr|Br|

∫ ∞

Br

e(1+αr)yν(dy)

)

≤ τ(C1e−αl|Bl| + C2e

−αr|Br|),

which we substitute into (4.10).

Remark 3. The hypotheses on ν in Proposition 4.2 are a little stronger than (4.2).We require them to obtain an exponential decay of the truncation error. However, wecan use estimate (4.11) directly. In other words, existence of the integrals in (4.11)suffices to obtain a convergence of u to u as |Bl| and |Br| grow to infinity, but thisconvergence does not necessarily occur at an exponential rate.

Remark 4. The requirements are different for the left and right tails of ν. Forexample, in the variance Gamma model with ν(x) = a exp(−η±|x|)/|x| one needs η+

to be greater than 1, and η− only positive. Proposition 4.9 then applies with αl < η−and αr < η+ − 1.

Using Propositions 4.1 and 4.2 we can fix in advance [−A,A] and [Bl, Br] to havea given bound on the respective errors. In what follows we will assume this has beendone and concentrate on the numerical solution of the localized problem.

1Clearly, if (4.9) is true for such values, it is true for all Bl, Br, up to change of the constants.On the other hand, this estimate is not useful if ν has a bounded support. For example, if there areonly negative jumps, we will take Br = 0, but in this case ν1x≤Br = ν, and there is no truncationerror due to Br.


5. An explicit-implicit finite difference scheme. We now present a numer-ical procedure for solving the PIDE:

∂u

∂τ= Lu, (0, T ] ×O,(5.1)

u(τ, x) = g(τ, x), x ∈ Oc,(5.2)

u(0, x) = h(x), x ∈ O,(5.3)

where L is defined by (2.8), Oc = R \ O, and g ∈ C+p ([0, T ] × R \ O) is a continuous

function.Our method is based on splitting the operator L into two parts:

∂u

∂τ= Du + Ju,

where D and J stand for the differential and integral parts of L, respectively. Wereplace Du with a finite difference approximation DΔu and Ju with the trapezoidalquadrature approximation JΔu and use the following explicit-implicit time-steppingscheme:

un+1 − un

Δt= DΔun+1 + JΔun.

We treat the integral part in an explicit time stepping in order to avoid the inversionof the nonsparse matrix JΔ. We show that this does not affect the stability of thescheme: it is unconditionally stable like the fully implicit scheme but does not requireus to invert the dense matrix JΔ. We first describe the space discretization and thetime-stepping scheme in the case of a jump-diffusion model where the jump intensityis finite. Next, we deal with the singular case ν(R) = +∞ using an approach similarto the “vanishing viscosity” method [14].

5.1. Explicit-implicit scheme: Finite intensity case. We suppose here thatν(R) = λ < +∞. Then the integro-differential operator can be written as Lu ≡Du + Ju, where

Du =σ2

2

∂2u

∂x2−(σ2

2− r + α

)∂u

∂x− λu, Ju =

∫ Br

Bl

ν(dy)u(τ, x + y),(5.4)

and α =∫ Br

Bl(ey − 1)ν(dy).

We introduce a uniform grid on [0, T ] × [−A,A]: τn = nΔt, n = 0, . . . ,M ,xi = −A + iΔx, i ∈ {0, . . . , N}, with Δt = T/M , Δx = 2A/N . Let {un

i } be thesolution of the numerical scheme, to be defined below.

To approximate the integral terms we use the trapezoidal quadrature rule withthe same step Δx. Let Kl, Kr be such that [Bl, Br] ⊂ [(Kl − 1/2)Δx, (Kr +1/2)Δx].Then

∫ Br

Bl

ν(dy)u(τ, xi + y) ≈Kr∑

j=Kl

νjui+j ,λ ≈ λ =

Kr∑j=Kl

νj ,

α ≈ α =

Kr∑j=Kl

(eyj − 1)νj , where νj =

∫ (j+1/2)Δx

(j−1/2)Δx

ν(dy).(5.5)


The space derivatives are discretized using finite differences:(∂2u

∂x2

)i

≈ ui+1 − 2ui + ui−1

(Δx)2,(5.6)

(∂u

∂x

)i

≈{ ui+1−ui

Δx if σ2/2 − r + α < 0ui−ui−1

Δx if σ2/2 − r + α ≥ 0.(5.7)

The choice of approximation for the first-order derivative is determined by stabilityrequirement and will be discussed later (section 6). Since the two cases are treatedsimilarly, let us suppose without loss of generality that σ2/2 − r + α < 0.

Using (5.5)–(5.7) we obtain Du ≈ DΔu, Ju ≈ JΔu, where

(DΔu)i =σ2

2

ui+1 − 2ui + ui−1

(Δx)2−(σ2

2− r + α

)ui+1 − ui

Δx− λui,(5.8)

(JΔu)i =

Kr∑j=Kl

νjui+j .(5.9)

Finally, we replace problem (4.1) with the following time-stepping scheme:

Initialization:

u0i = h(xi), i ∈ {0, . . . , N},(5.10)

u0i = g(0, xi) otherwise.(5.11)

(S) For n = 0, . . . ,M − 1,

un+1i − un

i

Δt= (DΔun+1)i + (JΔun)i if i ∈ {0, . . . , N},(5.12)

un+1i = g( (n + 1)Δt, xi) if i /∈ {0, . . . , N}.(5.13)

5.2. Explicit-implicit scheme: Infinite intensity case. If ν(R) = +∞, theabove method cannot be applied directly. The idea is to come down to a nonsingularcase by approximating the process Xτ by an appropriate finite activity process witha modified diffusion coefficient.

The procedure is similar to the one described in section 4.2, but this time we dealwith small jumps. Given ε > 0 let us define a process Xε

τ characterized by the Levytriplet (γ(ε),

√σ2 + σ2(ε), ν1|x|≥ε), where

σ2(ε) =

∫ ε

−ε

y2ν(dy),

and γ(ε) is determined by the martingale condition

γ(ε) = −σ2 + σ2(ε)

2−∫|y|≥ε

(ey − 1 − y1|y|≤1)ν(dy).

This means that we replace the jumps of size smaller than ε by a Brownian motionσ(ε)Wτ . Therefore, Xε

τ has jumps of finite intensity. The function uε defined as

uε(τ, x) = E[h(x + rτ + Xετ )] ≡ E[h(x + Y ε

τ )](5.14)


satisfies the following Cauchy problem:

∂uε

∂τ= Lεuε, (0, T ] × R,(5.15)

uε(0, x) = h(x), x ∈ R,

where

(5.16) Lεf =σ2 + σ2(ε)

2

∂2f

∂x2−

(σ2 + σ2(ε)

2− r + α(ε)

)∂f

∂x− λ(ε)f(x)

+

∫|y|≥ε

ν(dy)f(x + y),

and α(ε) =∫|y|≥ε

(ey − 1)ν(dy), λ(ε) =∫|y|≥ε

ν(dy).

Note that even if σ = 0, Lε contains a nonzero diffusion term, as in the vanish-ing viscosity method, with the difference that we also have additional terms in thefirst- and zeroth-order terms in order to conserve the martingale property. The nexttheorem gives an estimate of the rate of convergence for this “compensated vanishingviscosity” approximation.

Theorem 5.1. Let h be Lipschitz: |h(x) − h(y)| ≤ c |x − y|. Let u and uε bedefined by (4.7) and (5.14), respectively. Then

|u(τ, x) − uε(τ, x)| ≤ C

∫ ε

−ε|y|3ν(dy)

σ2(ε).(5.17)

Proof. We essentially use [11, Proposition 6.2] with the only difference being thatwe also adjust the drift parameter γ to preserve the martingale property. Let us defineZτ = Yτ − (γ − γ(ε))τ . Then,

(5.18) |u(τ, x) − uε(τ, x)| = |E[h(x + Yτ )] − E[h(x + Y ετ )]|

≤ |E[h(x + Zτ )] − E[h(x + Y ετ )]| +

+ |E[h(x + Zτ + (γ − γ(ε))τ)] − E[h(x + Zτ )]|.

Since h is Lipschitz, it is almost everywhere differentiable with |h′| ≤ c. By [11,Proposition 6.2] we have

|E[h(x + Zτ )] − E[h(x + Y ετ )]| ≤ K c

∫ ε

−ε|y|3ν(dy)

σ2(ε)(5.19)

with K < 16.5. The second term may be estimated as follows:

|E[h(x + Zτ + (γ − γ(ε))τ)] − E[h(x + Zτ )]| ≤ c |γ − γ(ε)|τ,

where

(5.20) |γ − γ(ε)| =

∣∣∣∣∣σ2(ε)

2−∫|y|<ε

(ey − 1 − y)ν(dy)

∣∣∣∣∣=

∣∣∣∣∣12∫|y|<ε

ν(dy)

∫ y

0

es(y − s)2ds

∣∣∣∣∣ ≤ eε

6

∫ ε

−ε

|y|3ν(dy).


Since σ2(ε) → 0 as ε → 0, (5.20) converges faster than (5.19) and therefore may beneglected.

Remark 5. If limx→0 ν(x)|x|1+β = a > 0 , with 0 ≤ β < 2, then (5.17) gives

|u(τ, x) − uε(τ, x)| ≤ C(β)ε,

so the approximation error is proportional to ε. This case includes all practical ex-amples used in option pricing such as variance Gamma, NIG, and tempered stableprocesses.

6. Convergence. We study in this section the convergence of the finite differ-ence scheme presented above. In the usual approach to the convergence of finitedifference schemes for PDEs, consistency and stability ensure convergence under reg-ularity assumptions on the solution such as uniform boundedness of the derivatives.This approach is not feasible here because, even for a European put option, the secondderivative (Gamma of the option) is never uniformly bounded in t.

6.1. Monotonicity. Monotonicity is an important property for pricing appli-cations: it guarantees that a (discrete) comparison principle holds for the numericalsolution so arbitrage inequalities will be verified exactly and not only as Δt,Δx → 0,leading to arbitrage-free approximations. The following result shows that the schemeabove is monotone.

Proposition 6.1 (monotonicity). Scheme (S) is monotone: if u0, v0 are twobounded initial conditions, then

u0 ≥ v0 ⇒ ∀n ≥ 1, un ≥ vn.

Proof. We start by rewriting (5.12) in the following form:

−cΔtun+1i−1 + (1 + aΔt)un+1

i − bΔtun+1i+1 = un

i + Δt∑j

νjuni+j ,(6.1)

where2

a =σ2

(Δx)2−(σ2

2− r + α

)1

Δx+ λ ≥ 0,

b =σ2

2(Δx)2−(σ2

2− r + α

)1

Δx≥ 0,

c =σ2

2(Δx)2≥ 0.(6.2)

Notice that a = b + c + λ.

Let un and vn be two solutions of (S) corresponding to the initial conditions h(x)and f(x), respectively, and let h(x) ≥ f(x) for all x ∈ R. Let us show by inductionthat wn = un − vn ≥ 0 for all n ≥ 0. We have w0

i = h(xi) − f(xi) ≥ 0 for alli ∈ Z. Let wn ≥ 0 and suppose that infi∈Z wn+1

i < 0. Since for all i ∈ Z\{0, . . . , N},

2We recall that the case σ2/2 − r + α < 0 is considered. If σ2/2 − r + α ≥ 0, we change theapproximation of the first-order derivative (see (5.7)) to have a, b, c ≥ 0, which is needed for stabilityand monotonicity.


wn+1i = h(xi) − f(xi) ≥ 0, this implies that there exists i0 ∈ {0, . . . , N}, such that

wn+1i0

= infi∈Z wn+1i . Using (6.1) we obtain that

infi∈Z

wn+1i = wn+1

i0= −cΔtwn+1

i0+ (1 + aΔt)wn+1

i0− bΔtwn+1

i0− λΔtwn+1

i0

≥ −cΔtwn+1i0−1 + (1 + aΔt)wn+1

i0− bΔtwn+1

i0+1

= wni0 + Δt

∑j

νjwni0+j ≥ 0,(6.3)

which contradicts the assumption. Therefore, infi∈Z wn+1i ≥ 0, and, consequently

wn+1 ≥ 0.

It is convenient to rewrite the scheme as follows:

u(τn, xi) = FΔ[u(τn − Δt, ·)](xi), n = 1, . . . ,M, i ∈ {0, . . . , N},(6.4)

u(τn, xi) = g(τn, xi), n = 0, . . . ,M, i /∈ {0, . . . , N},u(0, xi) = h(xi), i ∈ {0, . . . , N}.

Let uΔ be the solution of the scheme (6.4) defined on the grid QΔ = {(τn, xi) | n =0, . . . ,M, i ∈ Z}. One can define super- and subsolutions for the scheme by analogywith the definitions in section 3.

Definition 6.2. A function wΔ defined on QΔ is a supersolution of the scheme(6.4) if

wΔ(τn, xi) ≥ FΔ[wΔ(τn − Δt, ·)](xi), n = 1, . . . ,M, i ∈ {0, . . . , N},wΔ(τn, xi) ≥ g(τn, xi), n = 0, . . . ,M, i /∈ {0, . . . , N},

wΔ(0, xi) ≥ h(xi), i ∈ {0, . . . , N}.

A function zΔ on QΔ is a subsolution of (6.4) if

zΔ(τn, xi) ≤ FΔ[zΔ(τn − Δt, ·)](xi), n = 1, . . . ,M, i ∈ {0, . . . , N},zΔ(τn, xi) ≤ g(τn, xi), n = 0, . . . ,M, i /∈ {0, . . . , N},

zΔ(0, xi) ≤ h(xi), i ∈ {0, . . . , N}.

In particular, the scheme is unconditionally stable in the sup norm. The followingresult extends the discrete comparison principle to sub- and supersolutions.

Lemma 1. For any supersolution w and any subsolution z of (6.4) we have

zΔ ≤ uΔ ≤ wΔ on QΔ.

Proof. For n = 0 or i /∈ {0, . . . , N} the above inequalities are satisfied by def-inition. For n > 0, i ∈ {0, . . . , N}, they follow directly from the monotonicity ofthe scheme. Indeed, if zΔ(τn − Δt, ·) ≤ uΔ(τn − Δt, ·) ≤ wΔ(τn − Δt, ·), then, fori ∈ {0, . . . , N}, we obtain

zΔ(τn, xi) ≤ FΔ[zΔ(τn − Δt, ·)](xi) ≤ FΔ[uΔ(τn − Δt, ·)](xi) = uΔ(τn, xi)

= FΔ[uΔ(τn − Δt, ·)](xi) ≤ FΔ[wΔ(τn − Δt, ·)](xi) ≤ wΔ(τn, xi).


6.2. Consistency. We now show the consistency of the scheme in the uniformnorm for the following class of test functions v : [0, T ] × R �→ R:

H =

{v ∈ C1,2((0, T ] ×O), v uniformly continuous on[0, T ] × R,

∂v

∂τ,∂v

∂x,∂2v

∂x2uniformly continuous on(0, T ] ×O

}.(6.5)

Proposition 6.3 (consistency in the uniform norm). For any v ∈ H and anyε > 0,

∃Δ > 0,

∣∣∣∣v(τn, xi) − v(τn−1, xi)

Δt− LΔv(τn, xi) −

(∂v

∂τ− Lv

)(τ, x)

∣∣∣∣ < ε(6.6)

for all Δt, Δx > 0, (τ, x) ∈ (0, T ]×O, n ≥ 1, i ∈ {0, . . . , N}, such that sup{Δt,Δx, |τn−τ |, |xi − x|} < Δ.

Proof. Let sup{Δt,Δx, |τn − τ |, |xi − x|} < Δ. We have to prove that theexpression in (6.6) is bounded by α(Δ) independently of (τ, x), (τn, xi) ∈ (0, T ] × O,such that α(Δ) → 0 as Δ → 0. We have

(6.7)

∣∣∣∣v(τn, xi) − v(τn−1, xi)

Δt− ∂v

∂τ(τ, x)

∣∣∣∣ =

∣∣∣∣∣ 1

Δt

∫ τn

τn−1

(∂v

∂τ(t, xi) −

∂v

∂τ(τ, x)

)dt

∣∣∣∣∣≤ sup

t∈(τn−1,τn)

∣∣∣∣∂v∂τ (t, xi) −∂v

∂τ(τ, x)

∣∣∣∣ ≤ supt, τ ∈ (0, T ], y, x ∈ O|t − τ| ≤ 2Δ|y − x| ≤ Δ

∣∣∣∣∂v∂τ (t, y) − ∂v

∂τ(τ, x)

∣∣∣∣ Δ↓0−→ 0,

since ∂v∂τ is uniformly continuous by assumption.

Consider now the terms in Dv (the differential part of Lv). Using Taylor expan-sion up to the second order we obtain∣∣∣∣v(τn, xi−1) − 2v(τn, xi) + v(τn, xi+1)

Δx2− ∂2v

∂x2(τ, x)

∣∣∣∣=

∣∣∣∣∣ 1

Δx2

∫ xi+1

xi−1

(∂2v

∂x2(τn, y) −

∂2v

∂x2(τ, x)

)(Δx− |xi − y|)dy

∣∣∣∣∣≤ 2 sup

y∈(xi−1,xi+1)

∣∣∣∣∂2v

∂x2(τn, y) −

∂2v

∂x2(τ, x)

∣∣∣∣≤ 2 sup

t, τ ∈ (0, T ], y, x ∈ O|t − τ| ≤ Δ|y − x| ≤ 2Δ

∣∣∣∣∂2v

∂x2(t, y) − ∂2v

∂x2(τ, x)

∣∣∣∣ Δ↓0−→ 0,

as ∂2v∂x2 is uniformly continuous on (0, T ] ×O. In the same way, we show that

∣∣∣∣v(τn, xi+1) − v(τn, xi)

Δx− ∂v

∂x(τ, x)

∣∣∣∣ =

∣∣∣∣ 1

Δx

∫ xi+1

xi

(∂v

∂x(τn, y) −

∂v

∂x(τ, x)

)dy

∣∣∣∣≤ sup

t, τ ∈ (0, T ], y, x ∈ O|t − τ| ≤ Δ|y − x| ≤ 2Δ

∣∣∣∣∂v∂x (t, y) − ∂v

∂x(τ, x)

∣∣∣∣ Δ↓0−→ 0.


Since |1 − eyj−y| ≤ Δx if Δx ≤ 1 and |yj − y| ≤ Δx/2, we also have

(6.8) |α− α| =

∣∣∣∣∣∣∫ Br

Bl

(ey − 1)ν(dy) −Kr∑

j=Kl

(eyj − 1)νj

∣∣∣∣∣∣=

∣∣∣∣∣∣Kr∑

j=Kl

∫ (j+1/2)Δx

(j−1/2)Δx

(1 − eyj−y)eyν(dy)

∣∣∣∣∣∣ ≤ Δx

∣∣∣∣∣∣Kr∑

j=Kl

∫ (j+1/2)Δx

(j−1/2)Δx

eyν(dy)

∣∣∣∣∣∣= Δx

∫ Br

Bl

eyν(dy) = (α + λ)Δx.

Therefore, using previous estimates and uniform boundedness of ∂v∂x on (0, T ]×O, we

obtain

|(DΔv)(τn, xi) − (Dv)(τ, x)| =

∣∣∣∣σ2

2

[v(τn, xi−1) − 2v(τn, xi) + v(τn, xi+1)

Δx2− ∂2v

∂x2(τ, x)

]

+

(σ2

2− r + α

)[v(τn, xi+1) − v(τn, xi)

Δx− ∂v

∂x(τ, x)

]+ (α− α)

∂v

∂x(τ, x)

∣∣∣∣ Δ↓0−→ 0.

The integral part can be estimated as follows:

(6.9)

|(JΔv)(τn−1, xi) − (Jv)(τ, x)| =

∣∣∣∣∣∣Kr∑

j=Kl

v(τn−1, xi + yj)νj −∫ Br

Bl

v(τ, x + y)ν(dy)

∣∣∣∣∣∣=

∣∣∣∣∣∣Kr∑

j=Kl

∫ (j+1/2)Δx

(j−1/2)Δx

(v(τn−1, xi + yj) − v(τ, x + y))ν(dy)

∣∣∣∣∣∣≤ λΔ sup

θ1, θ2 ∈ [0, T ], ξ1, ξ2 ∈ R

|θ1 − θ2| ≤ 2Δ|ξ1 − ξ2| ≤ 3Δ/2

|v(θ1, ξ1) − v(θ2, ξ2)|Δ↓0−→ 0,

since v is uniformly continuous on [0, T ] × R. Combining all these estimates gives(6.6).

6.3. Convergence. A general approach for obtaining convergence of finite dif-ference schemes in the case of nonsmooth solutions was developed by Barles andSouganidis [6] who showed that, when subsolutions/supersolutions verify a strongcomparison principle, the solution of any monotone, locally consistent scheme con-verges uniformly on compact sets to the solution of the limit equation. The idea of[6] is to show that the pointwise limits

u(τ, x) = lim inf(Δt,Δx)→0 (τn,xi)→(τ,x)

u(Δt,Δx)(τn, xi),(6.10)

u(τ, x) = lim sup(Δt,Δx)→0 (τn,xi)→(τ,x)

u(Δt,Δx)(τn, xi)(6.11)

define sub-/supersolutions and then apply the comparison principle to conclude u =u = u. As noted above, our scheme (5.12) is monotone and locally consistent, but thecomparison principles [1, 20] available for the PIDEs considered here require uniform


continuity properties which may not hold for u and u defined as in (6.10)–(6.11).Therefore we cannot directly apply the Barles–Souganidis result to obtain conver-gence.3

We avoid this problem by considering smooth sub-/supersolutions and derivinginequalities linking them with u, u.

Definition 6.4. A function w ∈ H is a smooth supersolution of the problem(5.1)–(5.3) if it verifies the following inequalities:

∂w

∂τ(τ, x) − Lu(τ, x) ≥ 0, (τ, x) ∈ (0, T ] ×O,(6.12)

w(τ, x) ≥ g(τ, x), x ∈ Oc,(6.13)

w(0, x) ≥ h(x), x ∈ O.(6.14)

A function z ∈ H is a smooth subsolution of the problem (5.1)–(5.3) if it satisfies(6.12)–(6.14) with the reverse inequalities.

Lemma 2. Let w(τ, x) be a smooth supersolution and let z(τ, x) be a smoothsubsolution of the problem (5.1)–(5.3). Then for all ε > 0, there exists Δ > 0 suchthat

∀Δt,Δx ≤ Δ,∀n ≥ 0,∀i ∈ Z, z(τn, xi) − ε < uΔ(τn, xi) < w(τn, xi) + ε.(6.15)

Proof. Choose b such that 0 < b(T + 1) < ε and let w(τ, x) = w(τ, x) + b(τ + 1).If i /∈ {0, . . . , N}, we have

w(τn, xi) = w(τn, xi) + b(τn + 1) ≥ g(τn, xi) + b(τn, xi) ≥ g(τn, xi).(6.16)

If i ∈ {0, . . . , N},

w(0, xi) = w(0, xi) + b ≥ h(xi) + b ≥ h(xi).(6.17)

If n ≥ 1, i ∈ {0, . . . , N}, we obtain by Proposition 6.3

(6.18)w(τn, xi) − w(τn − Δt, xi)

Δt− LΔw(τn, xi)

=w(τn, xi) − w(τn − Δt, xi)

Δt− LΔw(τn, xi) + b(1 + λΔt)

→(∂w

∂τ(τ, x) − Lu(τ, x)

)+ b > 0,

as Δt,Δx → 0, (τn, xi) → (τ, x), uniformly on (0, T ] × O. Therefore, for any suffi-ciently small Δ > 0, for all Δt,Δx ≤ Δ, we have

w(τn, xi) − w(τn − Δt, xi)

Δt− LΔw(τn, xi) ≥ 0

or, equivalently,

w(τn, xi) ≥ FΔ[w(τn − Δt, ·)](xi) ∀n ≥ 1, ∀i ∈ {0, . . . , N}.(6.19)

3Note also that the Barles–Souganidis method does not yield a rate of convergence either: thisissue is treated in section 6.4 below.


By (6.16), (6.17), and (6.19), the function w is a supersolution of (6.4). So, Lemma 1implies that

uΔ(τn, xi) ≤ w(τn, xi) ≤ w(τn, xi) + b(T + 1) < w(τn, xi) + ε ∀n ≥ 0, ∀i ∈ Z,

which is the desired property. The lower bound z(τn, xi) − ε can be proved in thesame manner.

Corollary 6.5. Let u and u be the functions defined by (6.10)–(6.11). For anysmooth subsolution z(τ, x) and any supersolution w(τ, x) of the problem (5.1)–(5.3),we have, for all (τ, x) ∈ [0, T ] × R,

z(τ, x) ≤ u(τ, x) ≤ u(τ, x) ≤ w(τ, x).(6.20)

Proof. By the definition of upper and lower limits, (6.15) implies (6.20).

We are now ready to give our main result on the convergence of the scheme. Weassume σ > 0 since, as explained in section 5.2, the scheme introduces a viscosityterm σ(ε) > 0 even in the pure jump case where σ = 0.

Theorem 6.6. In the European case (O = R) with σ > 0, if h is a boundedpiecewise continuous function on R, then for all τ > 0 and all x ∈ R, the discretesolution converges to the solution of the continuous problem:

lim(Δt,Δx)→0 (τn,xi)→(τ,x)

u(Δt,Δx)(τn, xi) = u(τ, x).

Proof. If h, h ∈ C∞(R) are such that h ≤ h ≤ h, then z(τ, x) = E[h(x + Yτ )]and w(τ, x) = E[h(x + Yτ )] are solutions in H of (5.1) and therefore, respectively, asubsolution and a supersolution of the Cauchy problem (5.1), (5.3). By Corollary 6.5,we obtain (6.20). If w(τ, x) − u(τ, x) and u(τ, x) − z(τ, x) could be made arbitrarilysmall, this would imply that there exists a limit of u(Δt,Δx)(τn, xi) equal to u(τ, x).So, it remains to construct appropriate smooth approximations h and h of h.

Let ξ1, . . . , ξI be the discontinuity points of h. We will suppose that the jumps ofh are bounded by K. Given ε > 0, we choose h, h ∈ C∞(R) such that

h(x) ≤ h(x) ≤ h(x) ∀x ∈ R,

|h(x) − h(x)| ≤ ε ∀x /∈I⋃

i=1

(ξi − ε, ξi + ε),

|h(x) − h(x)| ≤ K ∀x ∈I⋃

i=1

(ξi − ε, ξi + ε).

Then we have

(6.21) w(τ, x) − z(τ, x) = E[h(x + Yτ ) − h(x + Yτ )]

≤ εQ

(x + Yτ /∈

I⋃i=1

(ξi − ε, ξi + ε)

)+ KQ

(x + Yτ ∈

I⋃i=1

(ξi − ε, ξi + ε)

)

≤ ε + KQ

(x + Yτ ∈

I⋃i=1

(ξi − ε, ξi + ε)

).


Denoting Ωε = {x + Yτ ∈⋃I

i=1(ξi − ε, ξi + ε)} we obtain⋂

ε>0 Ωε = {x + Yτ ∈{ξ1, . . . , ξI}}. Since σ is strictly positive, Yτ has an absolutely continuous distributionfor τ > 0, so we have Q(x + Yτ ∈ {ξ1, . . . , ξI}) = 0. Consequently,

Q

(x + Yτ ∈

I⋃i=1

(ξi − ε, ξi + ε)

)ε↓0−→ 0.

We have shown that the expression in (6.21) goes to zero as ε → 0. The inequalities(6.20) together with z(τ, x) ≤ u(τ, x) ≤ w(τ, x) then imply that u(τ, x) = u(τ, x) =u(τ, x). The proof is complete.

Remark 6. If τ = 0, we obtain

Q

(x + Yτ ∈

I⋃i=1

(ξi − ε, ξi + ε)

)ε↓0−→ Q(x ∈ {ξ1, . . . , ξI}) = 1{x∈{ξ1,...,ξI}}.

So, the scheme does not converge to the initial condition at the discontinuity pointsof h. However, this has no practical importance since we do not need to calculate thesolution numerically at τ = 0.

Together with Proposition 5.1, Theorem 6.6 allows us to compute option prices inall the examples of exponential Levy models given in section 2. Theorem 6.6 does notyield the rate of convergence; we now use a different approach to obtain an estimateon the rate of convergence under a further condition on the scheme.

6.4. Rate of convergence. In the viscosity solution framework, results on con-vergence rates for numerical schemes have been obtained by Crandall and Lions [15]for first-order equations and by Krylov [21, 22] for second-order parabolic PDEs; seealso [5]. Our approach is inspired by [22] but is somewhat simpler and yields betterbounds, given that we are dealing with a linear equation.

First, let us write the scheme in the following form: u(τn, xi) = u(τn−1, xi) +LΔu(τn, xi)Δt, where (in the case σ2/2 − r + α < 0)

LΔu(τ, x) =σ2

2

u(τ, x + Δx) − 2u(τ, x) + u(τ, x− Δx)

Δx2

−(σ2

2− r + α

)u(τ, x + Δx) − u(τ, x)

Δx− λu(τ, x) +

Kr∑j=Kl

νju(τ − Δt, x + jΔx).

As previously, we assume σ > 0.4

Consider the grid {(τn, xi), n = 0, . . . ,M, i ∈ N}, where τn = nΔt, xi = x0 +iΔx. Given J ⊂ {0, . . . ,M}, we denote by BJ×N the space of bounded functions on{(τn, xi), n ∈ J, i ∈ N} with the norm

‖v‖J = supn∈J, i∈N

|v(τn, xi)|.(6.22)

For (v(xi), i ∈ N) we write

‖v‖ = supi∈N

|v(xi)|.(6.23)

The following (see appendix) is a slightly improved version of a result in [22].

4The infinite activity pure-jump case being reduced to this one; see section 5.2.


Lemma 3. Consider, for ξ ∈ B{1,...,M}×N, the problem

v(0, xi) = h(xi), i ∈ N,(6.24)

v(τn, xi) = v(τn−1, xi) + LΔv(τn, xi)Δt + ξ(τn, xi), n = 1, . . . ,M, i ∈ N.(6.25)

(i) For any ξ ∈ B{1,...,M}×N the problem (6.24)–(6.25) has a unique solutionv(ξ) ∈ B{0,...,M}×N.

(ii) If vi = v(ξi), where ξi ∈ B{1,...,M}×N, i = 1, 2, then

‖v1(τn, ·) − v2(τn, ·)‖ ≤n∑

k=1

‖ξ1(τk, ·) − ξ2(τk, ·)‖, n = 1, . . . ,M.(6.26)

We will consider payoff functions h which verify the following conditions.Assumption 6.1. h : R �→ R is continuous on R and there exists ξ1, . . . , ξI ∈ R

such that h is C∞ on ]ξi, ξi+1[ for i = 0, . . . , I − 1 and for all n ≥ 0, for all x /∈{ξ1, . . . , ξI}, |h(n)(x)| ≤ K.

For example, the payoff of a put option h(x) = (1−ex)+ verifies these conditions.Lemma 4. Let h verify Assumption 6.1 and let u(τ, x) = E[h(x + Xτ )] be the

viscosity solution of the problem

∂u

∂τ(τ, x) = Lu(τ, x), (τ, x) ∈ (0, T ] × R,(6.27)

u(0, x) = h(x), x ∈ R.(6.28)

Then, for all τ > 0 and m,n ∈ N, m + n > 0, we have∥∥∥∥ ∂m+nu

∂τm∂xn(τ, ·)

∥∥∥∥ ≤ C

(√τ)2m+n−1

.(6.29)

The constant C depends only on m, n, K, T , and coefficients of the operator L (σ,r, α, and λ).

Using this lemma we prove the following consistency result.Lemma 5. If u solves (6.27)–(6.28), then, for all k ≥ 1,

(6.30) ‖Lu(τk, ·) − LΔu(τk, ·)‖ ≤ C

[Δx2

τ3/2k

+Δx

τ1/2k

+ Δx + 2(√τk −√

τk−1)

],

with C = C(K,T, r, σ, λ, α).Under a CFL-type condition, we obtain the following rate of convergence.Theorem 6.7. Assume that the initial condition h verifies Assumption 6.1. Let

u be the unique solution of (6.27)–(6.28) and let uΔ be the solution of (6.24)–(6.25)with ξ = 0. If c1 ≤ Δt/Δx2 ≤ c2, then

‖u− uΔ‖{0,...,M} ≤ CΔx,(6.31)

where C depends only on T , K, and coefficients of the operator L (σ, r, α, and λ).Proof. The function u(τn, xi) solves (6.24)–(6.25) with ξ(τk, xi) =

∫ τkτk−1

Lu(s, xi)ds−LΔu(τk, xi)Δt. From Lemma 3,

(6.32) ‖u(τn, ·) − uΔ(τn, ·)‖ ≤n∑

k=1

∥∥∥∥∥∫ τk

τk−1

Lu(s, ·)ds− LΔu(τk, ·)Δt

∥∥∥∥∥≤

n∑k=1

∥∥∥∥∥∫ τk

τk−1

(Lu(s, ·) − Lu(τk, ·))ds∥∥∥∥∥ +

n∑k=1

‖Lu(τk, ·) − LΔu(τk, ·)‖Δt.


Let us look at the first term:∣∣∣∣∣∫ τk

τk−1

(Lu(s, x) − Lu(τk, x))ds

∣∣∣∣∣ =

∣∣∣∣∣∫ τk

τk−1

ds

∫ τk

s

∂2u

∂τ2(θ, x)dθ

∣∣∣∣∣≤ C

∫ τk

τk−1

ds

∫ τk

s

dθ

θ3/2≤ CΔt

∫ τk

τk−1

ds

s3/2

by Lemma 4. Therefore,

n∑k=1

∥∥∥∥∥∫ τk

τk−1

(Lu(s, ·) − Lu(τk, ·))ds∥∥∥∥∥ ≤ C

[∫ Δt

0

ds

∫ Δt

s

dθ

θ3/2+ Δt

∫ T

Δt

ds

s3/2

]

= C

[2Δt + Δt

2(√T −

√Δt)√

T√

Δt

]≤ 4C

√Δt.(6.33)

To estimate the second term in (6.32), note that

n∑k=1

Δt√τk

=

n∑k=1

∫ τk

τk−1

ds√τk

≤n∑

k=1

∫ τk

τk−1

ds√s

= 2√τn

andn∑

k=1

Δt

τ3/2k

=1√Δt

n∑k=1

1

k3/2≤ C√

Δt,

since the series∑∞

k=1 1/k3/2 converges. Using Lemma 5, we obtain

(6.34)n∑

k=1

‖Lu(τk, ·) − LΔu(τk, ·)‖Δt ≤ C[Δx2/

√Δt + 2

√τnΔx + τnΔx + 2

√τnΔt

]

≤ C[Δx2/

√Δt + Δx + Δt

].

If c1 ≤ Δt/Δx2 ≤ c2, the estimates (6.32), (6.33), and (6.34) imply (6.31).

7. Numerical results. We now illustrate the performance of the scheme pro-posed above with two examples. The computations were done in variance Gammamodels with Levy density

ν(x) = aexp(−η±|x|)

|x|

and two sets of parameters: a = 6.25, η− = 14.4, η+ = 60.2 (VG1) and a = 0.5, η− =2.7, η+ = 5.9 (VG2), and a Merton model with Gaussian jumps in log-price withvolatility σ = 15% and Levy density

ν(x) = 0.1e−x2/2

√2π

for a put of maturity of T = 1 year. Both of these models satisfy the hypothesesof sections 4.1 and 4.2: the Levy densities have exponentially decreasing tails. TheBlack–Scholes implied volatilities computed from prices of put options are shown inFigure 7.1: as expected, they display a “smile” (convex) feature as a function of the


0.5

1

1.50.2

0.4

0.6

0.8

1

10

20

30

40

50

60

Ke-rT/S0

T

0.5

1

1.5

0.2

0.4

0.6

0.8

1

20

40

60

80

Ke-rT/S0

T

Fig. 7.1. Black–Scholes implied volatilities for put options, as a function of strike and maturity.Left: variance Gamma model. Right: Merton jump-diffusion model.

2.5 3 3.5 4 4.5 5 5.5 6 6.5 70.008

0.009

0.01

0.011

0.012

0.013

0.014

0.015

0.016

A/[T*(σ2 + ∫ y2 ν(dy))]1/2

rela

tive

erro

r ε(

T,x

) in

%

|ε(T,0)|sup

2/3≤ ex ≤ 2

|ε(T,x)|

Initial data: sin(x)

2.5 3 3.5 4 4.5 5 5.5 6 6.5 70.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

A / [T*(σ2+ ∫ y2ν(dy))]1/2

impl

ied

vola

tility

err

or ε

(τ,x

) in

%

|ε(T,0)|sup

{2/3 ≤ ex ≤ 2}

|ε(T,x)|

Initial data: (1 − ex)+

Fig. 7.2. Influence of domain size on localization error for the explicit-implicit finite differencescheme. Left: smooth initial condition h(x) = sin(x) in Merton jump-diffusion model. Right: putoption in Merton jump-diffusion model.

strike of the option, which flattens out with increasing maturity. These two modelshave been chosen since in each case there is an alternative method for computingthe solution of the PIDE: in the Merton model the solution can be expressed as aseries expansion, and in the variance Gamma model a closed form expression for thecharacteristic function is available, so one can use Carr and Madan’s FFT method[9]. Comparing our finite difference solution to these alternative solutions (computedwith high precision) allows us to study the behavior of various error terms in relationwith the parameters of the scheme.

The error metric which is relevant from the point of view of financial applicationsis the error in terms of Black–Scholes implied volatility:

ε(τ, x) = |ΣPIDE(τ, x) − ΣFFT(τ, x)| in %,

where Σ denotes the Black–Scholes implied volatility computed by inverting theBlack–Scholes formula with respect to the volatility parameter and applying it tothe computed option price. We have computed both pointwise errors at x = 0(i.e., forward at-the-money options) and uniform errors on the computational rangex ∈ [log(2/3), log(2)]. This range contains all options prices quoted on the market.

The localization error is shown in Figure 7.2 for the Merton model: domainsize A is represented in terms of its ratio to the standard deviation of XT . An


1 1.5 2 2.5 3 3.5 4 4.5 5

4

5

x 10−4

Domain size as multiple of total variance

TSMertonVG

Fig. 7.3. Influence of domain size on localization error for the explicit-implicit finite differencescheme: put option in tempered stable, Merton, and variance Gamma models.

5 10 15 20 25 30 35

0.3

0.4

0.5

0.6

0.7

0.8

M

impl

ied

vola

tility

err

or in

%

|ε(T,0)|sup

{2/3 ≤ ex ≤ 2}

|ε(T,x)|

Initial data: (1 − ex)+

50 100 150 200 250 300 350 400 4500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

N

impl

ied

vola

tility

err

or in

%

|ε(T,0)|sup

{2/3 ≤ ex ≤ 2}

|ε(T,x)|

Initial data: (1 − ex)+Δ x = 0.1

Δ x = 0.06

Δ x = 0.03 Δ x = 0.016 Δ x = 0.008

Fig. 7.4. Numerical accuracy for a put option in the Merton model. Left: influence of numberof time steps M , Δx = 0.05, Δt = T/M . Right: influence of number of space steps N , Δx = 2A/N ,Δt = 0.02.

acceptable level is obtained for values of order � 5. Notice that as soon as this ratiois greater than or equal to 3, the uniform and pointwise errors are quite close to eachother, indicating that we are out of the zone of influence of the numerical boundaryconditions. Figure 7.3 shows the same analysis for the variance Gamma model.

Figure 7.4 illustrates the decay of numerical error when Δt,Δx → 0, i.e., whenthe number of time/space steps is increased. The behavior is quite similar to the caseof the Black–Scholes model.

Figure 7.5 illustrates the behavior of the error (for a fixed grid size) as a func-tion of maturity for two initial conditions: a smooth one (forward contract) and a


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

T

impl

ied

vola

tility

err

or in

%

|ε(T,0)|sup

{2/3 ≤ ex ≤ 2}

|ε(T,x)|

Initial condition: (1 − ex)+

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8x 10

−5

T

|ε(T,0)|sup

{0.5 ≤ ex ≤ 1.5}

|ε(T,x)|

Initial condition: ex

ε(t,x) = unum(t,x) − uexact(t,x)

Fig. 7.5. Decrease of error with maturity in the Merton model. Left: nonsmooth initial condi-tion (put option) h(x) = (1 − ex)+. Right: smooth initial condition h(x) = ex.

1 2 3 4 5 6 7 8 9 10

0.5

1

1.5

2

2.5

3

x 10−3

ε/Δ x

sup

of a

bsol

ute

erro

r on

{2/

3 ≤

ex ≤ 2

}

T = 1, VG1T = .25, VG1 T = .25, VG2 T = 1, VG2

Δ x = 0.01

Fig. 7.6. Influence of truncation of small jumps on numerical error in various variance Gammamodels. Put option.

nonsmooth one (put option). We observe that a nonsmooth initial condition leadsto a lack of accuracy for small T . This phenomenon, which is not specific to modelswith jumps, can be overcome using an irregular time-stepping scheme which exploitsthe smoothness in time of the solution. Matache, von Petersdorff, and Schwab [25]have suggested using irregularly (logarithmically) spaced time stepping, more refinednear maturity, in order to improve this convergence. Note that scheme introduces a“numerical viscosity” σ(ε), so even when the underlying model is a pure-jump Levyprocess this numerical diffusion term has a regularizing effect.

In the case of infinite activity models, an additional parameter which influencesthe solution is the truncation parameter ε for the small jumps. Whereas the error inProposition 5.1 vanishes when ε → 0, for a fixed Δx the discretization error increasesas ε → 0: the constant in (6.31) is of order λ(ε), which increases with ε. This suggeststhat there is an optimal choice ε(Δx) > 0 for a given Δx. The optimal choice of ε isnot universal and depends on the growth of the Levy density near zero. Figure 7.6


0 200 400 600 800 1000 1200 14000.032

0.034

0.036

0.038

0.04

0.042

0.044

T = 1, S0 = K = 1; L = 0.8, H = 1.2

N

P(t

=0,

S=

1)

At−the−money double−barrier Put price. Merton model.

0.6 0.7 0.8 0.9 1 1.1 1.2 1.30

0.005

0.01

0.015

S

C(0

,S)

Up−and−out Call price. Merton model: λ = 0.1, μ=0, γ=1.

T=1, K=1, H=1.2

Fig. 7.7. Left: at-the-money double-barrier put price as a function of the number of spacesteps. Barrier levels: L = 0.8, H = 1.2. Right: up-and-out call price in the Merton model. Barrierlevel: H = 1.2.

Table 7.1

Examples of numerical values for at-the-money option prices. S = K = 100, T = 1, A =

5√

σ2 +∫

y2ν(dy), dt = 0.02, dx = 0.01. The truncation parameter ε is chosen according to

Figure 7.6.

Model Put t Up-and-out call t Double-barrier put tsec. H = 120 sec. L = 80, H = 120 sec.

VG1 6.72 0.5 2.73 0.2 2.42 0.1VG2 8.38 0.9 3.34 0.5 1.68 0.1

Merton 11.04 1.2 1.17 0.5 3.35 4

illustrates this phenomenon for the variance Gamma model.

The last two figures give examples of boundary value problems for barrier optionsin the Merton model. Figure 7.7 (right) shows the price of an up-and-out call. Fig-ure 7.7 (left) illustrates the numerical convergence of a double-barrier put price as thenumber N of space steps increases.

In Table 7.1, we give some examples of option values obtained with our numericalscheme as well as the corresponding computation time in seconds.

Appendix A. Proofs of lemmas.

A.1. Proof of Lemma 3. To prove (i), we rewrite (6.24)–(6.25) as v(τ, x) =ΨΔv(τ, x), where ΨΔ is a contraction, and apply the fixed point theorem. Define

pΔ =1

Δt+

σ2

Δx2+

∣∣∣∣σ2

2− r + α

∣∣∣∣ + λ + 1 ≥ 1

such that ΦΔv(τn, xi) = −v(τn,xi)−v(τn−1,xi)Δt +LΔv(τn, xi) + pΔv(τn, xi) is monotone:

if v ≥ 0, then ΦΔv ≥ 0. ΦΔ verifies

ΦΔe−2τ = e−2τ

[pΔ − (1 − e−2Δt)

(1

Δt+ λ

)]≤ e−2τ (pΔ − 1)(A.1)


for small Δt (for example, this property holds as soon as Δt ≤ 0.7). We now definev(τ, x) = e2τv(τ, x), ξ(τ, x) = (Δt)−1p−1

Δ e2τξ(τ, x) and rewrite (6.24)–(6.25) as

v(0, xi) = h(xi), i ∈ N,

v(τn, xi) = p−1Δ e2τnΦΔ[e−2τn v(τn, xi)] + ξ(τn, xi), n = 1, . . . ,M, i ∈ N,(A.2)

or, equivalently, v(τn, xi) = ΨΔv(τn, xi), n = 0, . . . ,M, i ∈ N, with

ΨΔv(τn, xi) = 1n∈{1,...,M}(p−1Δ e2τnΦΔ[e−2τn v(τn, xi)] + ξ(τn, xi)) + 1n=0h(xi).

ΨΔ defines an operator on B{0,...,M}×N. Moreover, ΨΔ is a contraction,

|ΨΔv1(τn, xi) − ΨΔv2(τn, xi)| = 1n∈{1,...,M}p−1Δ e2τn |ΦΔ[e−2τn(v1 − v2)](τn, xi)|

≤ 1n∈{1,...,M}p−1Δ e2τn‖v1 − v2‖{0,...,M}ΦΔ[e−2τn ],

since v1 − v2 ≤ ‖v1 − v2‖{0,...,M} and ΦΔ is monotone. Using (A.1), we obtain

‖ΨΔv1 − ΨΔv2‖{0,...,M} ≤ (1 − p−1Δ )‖v1 − v2‖{0,...,M},

so ΨΔ is a contraction and the fixed point theorem entails (i).To prove (ii), first assume that for all k, i, ξ1(τk, xi) ≥ ξ2(τk, xi) and show that in

this case, v1 ≥ v2. Let v = v1−v2. We have v(0, xi) = 0. Suppose that v(τn−1, xi) ≥ 0.Given ε > 0, let us take i(ε) such that v(τn, xi(ε)) ≤ infi v(τn, xi) + ε ≤ v(τn, xi) + ε,for all i. Then, for all i ∈ N, we have

infiv(τn, xi) ≥ v(τn, xi(ε)) − ε

= −cΔtv(τn, xi(ε)) + (1 + aΔt)v(τn, xi(ε)) − bΔtv(τn, xi(ε)) − λΔtv(τn, xi(ε)) − ε

≥ −cΔt(v(τn, xi(ε)−1) + ε) + (1 + aΔt)v(τn, xi(ε)) − bΔt(v(τn, xi(ε)+1) + ε)

−λΔt(infiv(τn, xi) + ε) − ε

= v(τn−1, xi(ε)) + Δt∑j

νjv(τn−1, xi(ε)+j) + (ξ1 − ξ2)(τn, xi(ε)) − λΔt infiv(τn, xi)

−(1 + aΔt)ε ≥ −λΔt infiv(τn, xi) − (1 + aΔt)ε.

Therefore infi v(τn, xi) ≥ − 1+aΔt1+λΔt

ε.

Taking the limit ε → 0, we obtain infi v(τn, xi) ≥ 0, so v = v1 − v2 ≥ 0.Consider now the general case. Let v(τn, xi) = v1(τn, xi) +

∑nk=1 ‖ξ1(τk, ·) −

ξ2(τk, ·)‖. Then v solves (6.24)–(6.25) with ξ(τn, xi) = v(τn, xi) − v(τn−1, xi) −LΔv(τn, xi)Δt. We have

ξ(τn, xi) = ξ1(τn, xi) + ‖ξ1(τn, ·) − ξ2(τn, ·)‖ − LΔ

[n∑

k=1

‖ξ1(τk, ·) − ξ2(τk, ·)‖]

Δt

= ξ1(τn, xi) + (1 + λΔt)‖ξ1(τn, ·) − ξ2(τn, ·)‖ ≥ ξ2(τn, xi).

But as shown above this implies v(τn, xi) ≥ v2(τn, xi), so

v2(τn, xi) − v1(τn, xi) ≤n∑

k=1

‖ξ1(τk, ·) − ξ2(τk, ·)‖.

Interchanging v1 with v2, we obtain (6.26).


A.2. Proof of Lemma 4. By definition,

u(τ, x) = h(x) ∗ pτ (x) = h(x) ∗ pWτ (x) ∗ p(X−W )τ (x),(A.3)

where pτ (x) = pτ (−x) is the density of Xτ , and pWτ (−x), p(X−W )τ (−x) are densities

of the processes σWτ and (Xτ − σWτ ), respectively. Therefore,∥∥∥∥ ∂m+nu

∂τm∂xn(τ, ·)

∥∥∥∥ ≤∥∥∥∥∂m+n(h ∗ pWτ )

∂τm∂xn

∥∥∥∥ .(A.4)

The derivatives of h may have jumps at ξ1, . . . , ξI ; denote these jumps by a(n)i =

h(n)(ξi+) − h(n)(ξi−). Then, for all n ≥ 1, i = 1, . . . , I, |a(n)i | ≤ 2K. Using standard

properties of convolution products we obtain

(A.5)

∂n(h ∗ pWτ )

∂xn(x) = (h(n) ∗ pWτ )(x) +

I∑i=1

[a(n−1)i pWτ (x− ξi) + a

(n−2)i

∂pWτ∂x

(x− ξi) +

· · · + a(1)i

∂n−2pWτ∂xn−2

(x− ξi)

],

where h(n) is the nth pointwise derivative of h. For all n ≥ 0, we have

∂npWτ∂xn

(x) =1

(√τ)n+1

∂npW1∂xn

(x√τ

).(A.6)

Consequently, for all n ≥ 1,∥∥∥∥∂n(h ∗ pWτ )

∂xn

∥∥∥∥ ≤ ‖h(n)‖ + 2KI

(‖pWτ ‖ +

∥∥∥∥∂pWτ∂x∥∥∥∥ + · · · +

∥∥∥∥∂n−2pWτ∂xn−2

∥∥∥∥)

≤ K + 2KI

(1√τ‖pW1 ‖ +

1

(√τ)2

∥∥∥∥∂pW1∂x∥∥∥∥ + · · · + 1

(√τ)n−1

∥∥∥∥∂n−2pW1∂xn−2

∥∥∥∥)

≤ K

(√τ)n−1

[T

n−12 + 2I

(T

n−22 ‖pW1 ‖ + T

n−12

∥∥∥∥∂pW1∂x∥∥∥∥ + · · · +

∥∥∥∥∂n−2pW1∂xn−2

∥∥∥∥)]

=C

(√τ)n−1

,

so (6.29) is verified for m = 0 and n ≥ 1. Proceed by induction on m: assume (6.29)for m− 1 and n ≥ 1. For any f ∈ C∞(R), we have

|Lf(x)| =

∣∣∣∣∣σ2

2

∂2f

∂x2(x) −

(σ2

2− r + α

)∂f

∂x(x) − λf(x) +

∫ Br

Bl

ν(dy)f(x + y)

∣∣∣∣∣≤ σ2

2

∣∣∣∣∂2f

∂x2(x)

∣∣∣∣ + |σ2/2 − r + α|∣∣∣∣∂f∂x (x)

∣∣∣∣ + 2λ‖f‖.

Applying this result to f = ∂m+n−1u(μ)/∂τm−1∂xn and using ∂τu = Lu, we obtain∣∣∣∣ ∂m+nu

∂τm∂xn(τ, x)

∣∣∣∣ =

∣∣∣∣ ∂∂τ(

∂m−1+nu

∂τm−1∂xn

)(τ, x)

∣∣∣∣ =

∣∣∣∣L(

∂m−1+nu

∂τm−1∂xn

)(τ, x)

∣∣∣∣≤ σ2

2

∥∥∥∥ ∂m−1+n+2u

∂τm−1∂xn+2(τ, ·)

∥∥∥∥+

∣∣∣∣σ2

2− r + α

∣∣∣∣∥∥∥∥ ∂m−1+n+1u

∂τm−1∂xn+1(τ, ·)

∥∥∥∥+2λ

∥∥∥∥ ∂m−1+nu

∂τm−1∂xn(τ, ·)

∥∥∥∥≤ C(K,T,m, n, r, σ, λ, α)

(√τ)2(m−1)+(n+2)−1

=C

(√τ)2m+n−1

.


We have thus shown (6.29) for m ≥ 0, n ≥ 1. For m ≥ 1, n = 0, we proceed similarly,by induction on m starting from

∥∥∥∥∂u∂τ (τ, ·)∥∥∥∥ = ‖Lu(τ, ·)‖ ≤ σ2

2

∥∥∥∥∂2u

∂x2(τ, ·)

∥∥∥∥+

∣∣∣∣σ2

2− r + α

∣∣∣∣∥∥∥∥∂u∂x (τ, ·)

∥∥∥∥+2λ ‖u(τ, ·)‖ ≤ C√τ,

since ‖u(τ, ·)‖ = ‖h ∗ pWτ ‖ ≤ ‖h‖ ≤ K.

A.3. Proof of Lemma 5. We have

|Lu(τk, xi) − LΔu(τk, xi)| ≤ |Du(τk, xi) −DΔu(τk, xi)| + |Ju(τk, xi) − JΔu(τk−1, xi)|,

where D and J are the differential and integral parts of L defined by (5.4), and DΔ,JΔ are their approximations given by (5.8)–(5.9). Recall that |α − α| ≤ (α + λ)Δxby (6.8). From Taylor’s formula, there exist ξ1, η1, ξ2 ∈ [xi−1, xi+1] such that

|Du(τk, xi) −DΔu(τk, xi)| =

∣∣∣∣σ2

2

Δx2

24

[∂4u

∂x4(τk, ξ1) +

∂4u

∂x4(τk, η1)

]

+

(σ2

2− r + α

)Δx

2

∂2u

∂x2(τk, ξ2) + (α− α)

∂u

∂x(τk, xi)

∣∣∣∣≤ Δx2

12

σ2

2

∥∥∥∥∂4u

∂x4(τk, ·)

∥∥∥∥ +Δx

2|σ2/2 − r + α|

∥∥∥∥∂2u

∂x2(τk, ·)

∥∥∥∥ + Δx(α + λ)

∥∥∥∥∂u∂x (τk, ·)∥∥∥∥

≤ C[Δx2/τ3/2k + Δx/τ

1/2k + Δx]

by Lemma 4. The integral part can be estimated as follows:

|Ju(τk, xi) − JΔu(τk−1, xi)| =

∣∣∣∣∣∣Kr∑

j=Kl

u(τk−1, xi+j)νj −∫ Br

Bl

u(τk, xi + y)ν(dy)

∣∣∣∣∣∣=

∣∣∣∣∣∣Kr∑

j=Kl

∫ yj+1/2

yj−1/2

[u(τk−1, xi + yj) − u(τk, xi + y)]ν(dy)

∣∣∣∣∣∣≤

∣∣∣∣∣∣Kr∑

j=Kl

[u(τk−1, xi + yj) − u(τk, xi + yj)]νj

∣∣∣∣∣∣ +

∣∣∣∣∣∣Kr∑

j=Kl

∫ yj+1/2

yj−1/2

[u(τk, xi + yj)

− u(τk, xi + y)]ν(dy)

∣∣∣∣∣≤

∣∣∣∣∣∣Kr∑

j=Kl

νj

∫ τk

τk−1

∂u

∂τ(s, xi + yj)ds

∣∣∣∣∣∣ +

∣∣∣∣∣∣Kr∑

j=Kl

∫ yj+1/2

yj−1/2

ν(dy)

∫ xi+y

xi+yj

∂u

∂x(τk, ξ)dξ

∣∣∣∣∣∣≤ λ

∫ τk

τk−1

∥∥∥∥∂u∂τ (s, ·)∥∥∥∥ ds +

λΔx

2

∥∥∥∥∂u∂x (τk, ·)∥∥∥∥ ≤ C

[∫ τk

τk−1

ds√s

+ Δx

]

= C[2(√τk −√

τk−1) + Δx].

Assembling the various terms we obtain (6.30).


Acknowledgments. We thank Yves Achdou, Peter Forsyth, Daniel Gabay,Huyen Pham, Olivier Pironneau, Christoph Schwab, Peter Tankov, and an anony-mous referee for helpful suggestions.

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