CUBO A Mathematical Journal

Vol.10, N
o
¯ 03, (57–64). October 2008

The Extension of the Formula by Dupire

Shunsuke Kaji

Department of Mathematics, Graduate School of Science,

Osaka University, Machikaneyamachou 1-1,

Toyonaka, Osaka Japan 560-0043,

Osaka Japan 560-0043

email: kaji@math.sci.osaka-u.ac.jp

ABSTRACT

We provide the extension of Dupire’s PDE, as the partial integro-differential equations

of market prices of call options with many maturities and strike prices for jump diffusion

model.

RESUMEN

Nosotros damos la extensión de Dupire PDE, como las ecuaciones parciales integro-

diferenciales de precios de mercado de opciones de llamada con muchos vencimientos y

golpe de precios para modelos de difusión con saltos.

Key words and phrases: Dupire, PDE, Jump-diffusion model.

Math. Subj. Class.: 35K15, 60H30.



58 Shunsuke Kaji CUBO
10, 3 (2008)

1 Introduction

Let (Ω,F,P) be a probability space. On the space (Ω,F,P) we set a standard Brownian motion

W = {Wt}t∈[0,T ] from W0 = 0 and a Poission random measure N(dtdx) on (0,T ]×R with intensity

measure dtν(dx), where T ∈ (0,∞) and the measure ν on R satisfies

∫

R

(1 + e2z) ∧ z2ν(dz) < ∞. (1)

We consider a risk-neutral price process {Sxt }t∈[0,T ] of a risk asset satisfing

dSxt = σ(t,S
x
t )S

x
t dWt + (r − δ)S

x
t dt;

Sx
0

= x ∈ (0,∞),

where r ≥ 0 denotes the interest rate and δ ≥ 0 the dividend rate. The function σ : [0,T ]×(0,∞) →

[0,∞) has the Lipschiz condition and is often called the volatility of the asset’s price. According

to the well-known discussion of option pricing model, if for each T,K ∈ (0,∞) we have a unique

solusion u(t,x,T,K) to the parabolic equation and boundary condition

∂u

∂t
+

1

2
σ(t,x)

2
x2
∂2u

∂x2
+ (r − δ)x

∂u

∂x
− ru = 0, (t,x) ∈ [0,T ) × (0,∞);

u(t,x,T,K)|t=T = (x − K)
+, x ∈ (0,∞),

then a price of a call option with maturity T and strike price K is given by

u(t,x,T,K)|t=0 = e
−rT E[(SxT − K)

+].

Dupire[1] found that u(t,x,T,K) as a function of (T,K) satisfies the following dual equation to

the last parabolic equation:

∂u

∂T
=

1

2
σ(T,K)

2
K2

∂2u

∂K2
− (r − δ)K

∂u

∂K
− δu, (T,K) ∈ (t,∞) × (0,∞).

But his approach is not enough mathematically. There are some works justifing rigorously his idea,

for example, Klebaner[4] etc. Klebaner[4] gives the last equation by the Meyer-Tanaka formula.

On the other hand, there are also works on option pricing model for jump-diffusion processes, for

example, geometric Lévy processes by Fujiwara and Miyahara[2]. Recently, Jourdain[3] provides

the extension of Dupire’s work for jump-diffusion processes by stochastic flow approch.

Now, we consider the following risk-neutral evolusion {Xxt }t∈[0,T ] for the underlying risk asset’s

prices:

Xxt = x +

∫ t

0

a(u,Xxu )X
x
udWu + (r − δ)

∫ t

0

Xxudu

+

∫

(0,t]×R

Xxu−(e
z − 1){N(dudz) − duν(dz)}, t ∈ [0,T ]



CUBO
10, 3 (2008)

The Extension of the Formula by Dupire 59

where a(t,x) : [0,T ]×(0,∞) → [0,∞) satisfies the Lipschiz condition and has the second derivative

with respect to x. Then {Xxt }t∈[0,T ] has the extended diffusion operator(see Yoshida[5] p.408)

(Atf)(x) =
1

2
a(t,x)

2
x2f′′(x) + (r − δ)xf′(x)

+

∫

R

f(xez) − f(x) − (ez − 1)xf′(x)ν(dz).

For each maturity T and strike price K we denote

C(x,T,K) = e−rT E[(XxT − K)
+] (2)

by a call option price with an asset price x. In particular, in the case a(·, ·) ≡ a the last definition

(2) is justified by Fujiwara and Miyahara[2]. If we moreover assume that a(·, ·) belongs to the class

V =

{

f : [0,T ] × (0,∞) → R| sup
(t,x)∈[0,T ]×(0,∞)

3
∑

k=0

|xk
∂kf

∂xk
(t,x)| < ∞

}

,

then Jourdain[3] provides the following equation of (T,K):

−
∂C

∂T
+ AT C = 0, (T,K) ∈ (0,∞) × (0,∞),

where

(AT f)(K) =
1

2
a(T,K)

2
K2f′′(K) − (r − δ)Kf′(K) − δf(K)

+

∫

R

{f(Ke−z) − f(K) − (e−z − 1)Kf′(K)}ezν(dz).

Here notice that the assumption a(·, ·) ∈ V satisfies the Lipschiz condition. In this note we provide

the same result of the above without a(·, ·) ∈ V by using not only stochastic flow approch but also

another one.

2 Main result

We fix x ∈ (0,∞) as follows. We have the following main theorem.

Theorem 2.1. C(x,T,K) as a function of (T,K) satisfies

−
∂C

∂T
+ AT C = 0, (T,K) ∈ (0,∞) × (0,∞)

in weak sense; that is,
∫

∞

0

∫

∞

0

C(x,T,K)

{

∂ϕ

∂T
(T,K) + A∗T ϕ(T,K)

}

dTdK = 0, ∀ϕ ∈ C∞
0

((0,∞)
2
),

where
∫

∞

0

∫

∞

0

ψ(T,K)A∗T ϕ(T,K)dTdK =

∫

∞

0

∫

∞

0

AT ψ(T,K)ϕ(T,K)dTdK,

∀ϕ,∀ψ ∈ C∞
0

((0,∞)
2
).



60 Shunsuke Kaji CUBO
10, 3 (2008)

2.1 Lemmas

Lemma 2.1. It follows that

0 ≤ C(x,T,K) ≤ e−δT x, (T,K) ∈ (0,∞) × (0,∞). (3)

For every ϕ(·) ∈ C2
0
((0,∞))

e−rT E[ϕ(XxT )] =

∫

∞

0

C(x,T,K)ϕ′′(K)dK, T ∈ (0,∞) (4)

holds.

Remark 2.1. It follows from (3) that C(x,T,K) as a function of (T,K) is locally integrable on

(0,∞) × (0,∞). Thus the right-hand side of (4) is well-defined.

proof: By (2) we have

0 ≤ erT C(x,T,K) ≤ E[XxT ].

Moreover, since {e−(r−δ)tXxt }t∈[0,T ] is a nonnegative local martingale with initial value x, the

right-hand side of the last inequality is

≤ e(r−δ)T x.

Hence we get (3). Finally, we compute from (2) that the right-hand side of (4) is

=

∫

∞

0

e
−rT

E[(XxT − K)
+]ϕ′′(K)dK

= e−rT E

[
∫

∞

0

(XxT − K)
+
ϕ
′′(K)dK

]

= e−rT E[ϕ(XxT )].

Hence we get (4).

Before we moreover introduce lemmas, for every ϕ(·, ·) ∈ C∞
0

((0,∞)
2
) we set a family {Φh}h>0

of all functions

Φh(T,x) =
1

h
{E[ϕ(T,XxT +h)] − E[ϕ(T,X

x
T )]}, (T,K) ∈ (0,∞) × (0,∞).

Lemma 2.2.

limh↓0

∫

∞

0

Φh(T,x)dT = −

∫

∞

0

∫

∞

0

e
rT
C(x,T,K)

∂3ϕ

∂T∂K2
(T,K)dTdK.

proof: First, we set C̃(x,T,K) = erT C(x,T,K). By using (4), we have

∫

∞

0

Φh(T,x)dT =

∫

∞

0

{

∫

∞

0

C̃(x,T + h,K) − C̃(x,T,K)

h

∂2ϕ

∂K2
(T,K)dK

}

dT,



CUBO
10, 3 (2008)

The Extension of the Formula by Dupire 61

where h > 0. Moreover we compute that the right-hand side of the last equality is

=

∫

∞

0

{

∫

∞

0

C̃(x,T + h,K) − C̃(x,T,K)

h

∂2ϕ

∂K2
(T,K)dT

}

dK

=

∫

∞

0

{
∫

∞

0

C̃(x,T,K)
1

h

(

∂2ϕ

∂K2
(T − h,K) −

∂2ϕ

∂K2
(T,K)

)

dT

}

dK,

and so we have
∫

∞

0

Φh(T,x)dT =

∫

∞

0

∫

∞

0

C̃(x,T,K)
1

h

(

∂2ϕ

∂K2
(T − h,K) −

∂2ϕ

∂K2
(T,K)

)

dTdK.

Then, by using the dominated convergence theorem, ϕ(·, ·) ∈ C∞
0

((0,∞)
2
) and (3) imply that the

right-hand side of the last equality converges to

−

∫

∞

0

∫

∞

0

C̃(x,T,K)
∂3ϕ

∂T∂K2
(T,K)dTdK

as h ↓ 0. Hence we get the desired result.

We denote by the following operator depended on time t ∈ [0,∞) :

(Ãtf)(x) =
1

2
a(t,x)

2
x

2
f
′′(x) + {

∂

∂x
(a(t,x)

2
x

2) + (r − δ)x}f′(x)

+

{

1

2

∂2

∂x2
(a(t,x)

2
x

2) + (r − 2δ)

}

f(x)

+

∫

R

e
2z
f(xez) − (2ez − 1)f(x) − (ez − 1)xf′(x)ν(dz).

Lemma 2.3.

limh↓0

∫

∞

0

Φh(T,x)dT =

∫

∞

0

∫

∞

0

e
rT
C(x,T,K)

(

ÃT
∂2ϕ

∂K2
(T,K)

)

dTdK.

proof: First, we divide A· into two parts as follows:

(A·f)(x) =

{

1

2
a(·,x)

2
x2f′′(x) + (r − δ)xf′(x)

+

∫

|z|<1

f(xez) − f(x) − zxf′(x)ν(dz)

−

∫

|z|<1

(ez − 1 − z)xf′(x)ν(dz) −

∫

|z|≥1

f(x) + (ez − 1)xf′(x)ν(dz)

}

+

∫

|z|≥1

f(xez)ν(dz)

= (A0
·
f)(x) +

∫

|z|≥1

f(xez )ν(dz).



62 Shunsuke Kaji CUBO
10, 3 (2008)

Since ϕ(·, ·) ∈ C∞
0

((0,∞)
2
) we can choose subintervals I1 = [α1,β1] and I2 = [α2,β2] of (0,∞)

such that supp ϕ ⊂ I1 ×I2. We pick δ > 0 and set Ĩ1 = {x|α1 ≤ x ≤ β1 + δ} and Ĩ2 = {x|α2e
−1 ≤

x ≤ β2e}. We denote by ‖ f ‖C(Γ) = supx∈Γ|f(x)|, where Γ is a compact subset of (0,∞)
2

and

f ∈ C(Γ) = {f is a real-valued continuous function on Γ}. Then observe that A0
T
ϕ(T, ·) belongs

to C2((0,∞)), since ϕ(·, ·) ∈ C∞
0

((0,∞)
2
) and a(T, ·) has the second derivative, and

|A0uϕ(T,K)| ≤
1

2
‖ a2 ‖

C(Ĩ1×I2)
‖ K2

∂2ϕ

∂K2
‖

C(I1×I2)
1I1×I2 (T,K)

+ |r − δ| ‖ K
∂ϕ

∂K
‖

C(I1×I2)
1I1×I2 (T,K)

+

∫

|z|<1

z2 ν(dz) ‖ K2
∂2ϕ

∂K2
+ K

∂ϕ

∂K
‖

C(I1×Ĩ2)
1

I1×Ĩ2
(T,K)

+

∫

|z|<1

ez − 1 − z ν(dz) ‖ K
∂ϕ

∂K
‖

C(I1×I2)
1I1×I2 (T,K)

+ ν(|z| ≥ 1) ‖ ϕ ‖
C(I1×I2)

1I1×I2 (T,K)

+

∫

|z|≥1

|ez − 1| ν(dz) ‖ K
∂ϕ

∂K
‖

C(I1×I2)
1I1×I2 (T,K)

≤

{

1

2
‖ a2 ‖

C(Ĩ1×I2)
‖ K2

∂2ϕ

∂K2
‖

C(I1×I2)

+ (|r − δ| +

∫

|z|<1

ez − 1 − z ν(dz) +

∫

|z|≥1

|ez − 1| ν(dz))‖ K
∂ϕ

∂K
‖

C(I1×I2)

+

∫

|z|<1

z2 ν(dz)‖ K2
∂2ϕ

∂K2
+ K

∂ϕ

∂K
‖

C(I1×Ĩ2)
+ ν(|z| ≥ 1)‖ ϕ ‖

C(I1×I2)

}

× 1
I1×Ĩ2

(T,K)

= C1 × 1I1×Ĩ2 (T,K), ∀u ∈ Ĩ1,

where C1 < ∞ holds since ϕ(·, ·) ∈ C
∞

0
((0,∞)

2
), (1), and a(·, ·) is continuous. Moreover, it is easy

that we have
∣

∣

∣

∣

∣

∫

|z|≥1

ϕ(T,Kez)ν(dz)

∣

∣

∣

∣

∣

≤ C21I1 (T ),

where C2 is a positive constant not depending on T and K. Therefore the inequality of the

observation and the last inequality imply

|Auϕ(T,K)| ≤ (C1 + C2)1I1 (T ), ∀u ∈ Ĩ1.

Here, fix T and by using Appendix 3.2 it follows from ϕ(·, ·) ∈ C∞
0

((0,∞)
2
) that

Φh(T,x) =
1

h

∫ T +h

T

E[(Auϕ(T, ·))(X
x
u )]du

=
1

h

∫ T +h

T

E[Auϕ(T,X
x
u )]du,



CUBO
10, 3 (2008)

The Extension of the Formula by Dupire 63

for all 0 < h < δ. Then the last two inequality and equality imply

limh↓0Φh(T,x) = E[AT ϕ(T,X
x
T )];

|Φh(T,x)| ≤ (C1 + C2)1I1 (T ), 0 < ∀h < δ.

According to the dominated convergence theorem, the last two results imply

limh↓0

∫

∞

0

Φh(T,x)dT =

∫

∞

0

E[AT ϕ(T,X
x
T )]dT. (5)

On the other hand, by using (4) we have from the above observation

e−rT E[A0T ϕ(T,X
x
T )] =

∫

∞

0

C(x,T,K)
∂2

∂K2
(A0T ϕ(T, ·))(K)dK

Moreover we have

e−rT E[

∫

|z|≥1

ϕ(T,XxT e
z)ν(dz)] =

∫

|z|≥1

e−rT E[ϕ(T,XxT e
z)]ν(dz)

=

∫

|z|≥1

∫

∞

0

C(x,T,K)
∂2

∂K2
(ϕ(T,Kez))dKν(dz)

=

∫

∞

0

C(x,T,K)

∫

|z|≥1

e2z
∂2ϕ

∂K2
(T,Kez)ν(dz)dK,

where the second line of the last equality holds by (4). Therefore the last two equalities imply

e−rT E[AT ϕ(T,X
x
T )] =

∫

∞

0

C(x,T,K){
∂2

∂K2
(A0T ϕ(T,K))

+

∫

|z|≥1

e
2z ∂

2ϕ

∂K2
(T,Kez)ν(dz)}dK,

and so by computing the right-hand side of the last equality we have

e−rT E[AT ϕ(T,X
x
T )] =

∫

∞

0

C(x,T,K)(ÃT
∂2ϕ

∂K2
(T,K))dK.

Hence (5) and the last equality imply the desired result.

2.2 Proof of Theorem 2.1

First, pick any ψ(T,K) ∈ C∞
0

((0,∞)
2
). According to Lemma 2.2 and 2.3, for all ϕ(T,K) ∈

C∞
0

((0,∞)
2
) such that erT ∂

2
ϕ

∂K2
= ψ, we have

∫

∞

0

∫

∞

0

e
rT
C(x,T,K)

{

∂3ϕ

∂T∂K2
(T,K) + ÃT

∂2ϕ

∂K2
(T,K)

}

dTdK = 0,

and so
∫

∞

0

∫

∞

0

C(x,T,K)

{

∂ψ

∂T
(T,K) + ÃT ψ(T,K)

}

dTdK = 0



64 Shunsuke Kaji CUBO
10, 3 (2008)

holds. On the other hand, we can compute the integral by parts

∫

∞

0

∫

∞

0

ψ(T,K)ÃT ϕ(T,K)dTdK =

∫

∞

0

∫

∞

0

AT ψ(T,K)ϕ(T,K)dTdK,

∀ϕ,∀ψ ∈ C∞
0

((0,∞)
2
).

Hence the last two equalities imply the desired conclusion.

3 Appendix

Appendix 3.1. Let X ⊂ Rd, where d is a positive integer, be a domain and Ck(X), where

k = 0, 1, 2, · · · ,∞, be a class of all real-valued functions on X which have continuous partial

derivatives of order ≤ k if k < ∞; of order < ∞ if k = ∞. Let Ck
0
(X) be a class of all functions

which belong to Ck(X) and compact supports.

Appendix 3.2. (Dynkin’s formula)

For every f ∈ C2
0
((0,∞)),

E[f(Xxt )] = f(x) + E

[
∫ t

0

(Auf)(X
x
u )du

]

, (t,x) ∈ [0,∞) × (0,∞)

holds.

Acknowledgment: The author is grateful to J.Sekine, Assistant Professor for useful advices.

Received: February 2008. Revised: May 2008.

References

[1] Dupire, B., Pricing with a smile, Risk, 7 (1994), pp.18–20.

[2] Fujiwara, T. and Miyahara, Y., The minimal entropy martingale measures for geometric
Lévy processes, Finance Stochast., 7 (2003), pp.509–531.

[3] Jourdain, B., Stochastic flow approach to Dupire’s formula, Finance Stochast., (2007).

[4] Klebaner, F., Option price when the stock is a semimartingale, Elect. Comm. in Probab.,
7 (2002), pp.79–83.

[5] Yoshida, K., Functional analysis, 4th ed.


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