International Journal of Analysis and Applications

Volume 19, Number 4 (2021), 494-502

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-19-2021-494

PRICING OPTIONS IN A DELAYED MARKET DRIVEN BY LE’VY NOISE

ISMAIL HAMED ELSANOUSI∗

Department of Mathematics, Faculty of Sciences Al-Baha University,P.O.Box-1988, Alaqiq,

Al-Baha-65431, Saudi Arabia

∗Corresponding author: i elsanousi@hotmail.com

Abstract. In this paper we studied stochastic delayed differential equations driven by Le’vy noise. The

analogue of Itô formula is considered. The Black-Scholes formula analogue for Vanilla call option price

formula is derived.

1. Introduction

In this paper we studied the Stochastic Delay Differential Equations driven by Le’vy noise which arise

in many applications of stochastic analysis in finance specifically in pricing of options security markets. As

known such systems are quite hard to study due to their lack of Markovianity which is a key property for

the study of option prices. Basically, the difficulties arises from the fact that delay systems have, in general,

an infinite dimensional nature.

The model for the stock price ζ(t) that we consider satisfies a stochastic delay differential equation driven

by Le’vy noise with volatility σ depending on time t and the path ζt = {ζ(t + θ),θ ∈ [−τ, 0]} called a level

and past-dependent volatility. An analogue of Itô’s formula for such a stochastic systems is obtained.

Received March 29th, 2021; accepted April 23rd, 2021; published May 11th, 2021.

2010 Mathematics Subject Classification. 60G40, 34K50.

Key words and phrases. stochastic delay equations; Black-Scholes formula; (B,S)-securities market.

©2021 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

494

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-19-2021-494


Int. J. Anal. Appl. 19 (4) (2021) 495

An option price value of the form

G(t,ζt) =

∫ 0
−τ
erθF(ζ(t + θ),ζ(t), t)dθ

is studied when F is in C0,1,2(R×R×R3). A special case of G(t,ζt) of the form

G(t,ζt) = g1(ζ(t), t) +

∫ 0
−τ
erθg2(ζ(t + θ), t)dθ

when g1(ζ(t), t) is a classical Black-Scholes call option is studied. The partial differential equation of Black-

Scholes type is derived for such option.

2. Notations and Preliminaries

Let us consider a probability space (Ω,z,ρ) on which is defined ((B(t))t≥0, (η(t))t≥0) where ((B(t))t≥0

and (η(t))t≥0) are independent stochastic processes

-((B(t))t≥0 is a standard Brownian motion with respect to its natural filtration,

-(η(t))t≥0) is a pure jump Le’vy process.

The Poisson random measure N(t) of the process η is defined by

N(t,A) =
∑
ζ∈[0,t]

IA(η(s) −η(s−)), A ⊂ R.

The Le’vy measure ν of the process η is supposed to be
∫
R z

2ν(dz) < +∞, since (η(t))t≥0) is a pure jump

ν({0}) = 0.

Define the measure-valued process (Ñ(t))t≥0 by

Ñ(t,dz) := N(t,dz) −ν(dz)

so that the compensated poisson random measure is

Ñ(dt,dz) := N(dt,dz) −ν(dz)dt

Let (zt)t≥0 be the filtration generated by the process B(t) and η(t) as defined above. Since B and η

are assumed independent, then B(.) and Ñ(.,A) are still Brownian motion and square integrable martingale

with respect to the filtration (zt)t≥0.



Int. J. Anal. Appl. 19 (4) (2021) 496

2.1. Stochastic Delay Differential Equations Driven By Le’vy Noise

Consider the following Sdde:

dx(t) =µ(t,xt)dt + σ(t,xt)dw(t) +

∫
R
γ(x−t ,z)Ñ(dt,dz), t ∈ [0,T]

x0 = x(θ), θ ∈ [−r, 0]
(2.1)

where µ : [0,T] ×D → Rn,σ : [0,T] ×D → Rnd,γ : D ×D → R are predictable processes.

For the unknown process (x(t))t∈[−r,T] in R,T < ∞ is a fixed finite time, x(t) is the value of x at t ∈ [0,T]

and xt its segment, i.e. its value in the past time interval [t − r,t] i.e. xt(.) : [−r, 0] → R defined by

xt(θ) : x(t + θ) for all θ ∈ [−r, 0].

The initial data x(θ) is assumed to be in the space D := D([−r, 0],R) of Càdlàg random variables from

[−r, 0] to R.

Hypotheses 2.1 (i). There exists constant L > 0 such that for all t ∈ [0,T] and for all x1,x2 ∈ D,

|µ(t,x1) −µ(t,x2)| + |σ(t,x1) −σ(t,x2) +
∫
R
|γ(x1,z) −γ(x2,z)|ν(dz) ≤ L|x1 −x2|.

(ii). The functions µ,σ,γ satisfy the linear growth condition, i.e. there exists a constant K > 0 such that

for all x ∈ D,

|µ(t,x)| + |σ(t,x)| +
∫
R
|γ(x,z)|ν(dz) ≤ K(1 + |x|).

Theorem 2.1. Suppose that hypotheses (i) and (ii) hold. Then there exists a unique Ca′dla′g adapted

solution to equation (2.1). For proof we refer to [7].

The infinitesimal operator of the solution of equation (2.1)

In reformulation of equation (2.1) in infinite dimension, the following linear stochastic evolution equation

in the space H

dx(t) = Ax(t)dt+ < σ,x(t) > −n̂dw(t) +
∫
R
< r(z),x(t−) > −n̂Ñ(dt,dz) (2.1)′

will represents equation (2.1) in the sense that

x(t) = (x0(t),x1(t)) = (x(t),x(t + θ) : θ ∈ [−T, 0],∀t ≥ 0



Int. J. Anal. Appl. 19 (4) (2021) 497

where A is defined on

D(A = {y = (y0,y1(.)) ∈ H; y1(.) ∈ W 1,2([−1, 0],R),y0 = y1(0)}

by Ay = (µ0y0 +
∫ 0
−T µ1(θ)y1(θ)dθ + µ2y1(−T),y

′
1(.)) is the generator of a strongly continuous semigroup

(s(t))t≥0 on H. Here n̂ = (1, 0) ∈ H and x(t−) := lims↑t x(s) = lims↑t(x(s),s(s+.)) with the limit taken in H.

Theorem 2.1. Let y ∈ C and let x(.; y),x(.; y) representing the solutions of (2.1) and (2.1)′ respectively.

Then x(.; y) represents s(.; y) in the sense that

x(t; y) = (x(t; y),x(t + θ; y) : θ ∈ [−T, 0]),∀t ≥ 0.

For proof see [11].

The infinitesimal operator of the process x(., 0) is formally defined as

[Lφ](y) :=< Ay,ϕy(y) > +
1

2
< σ,y > ϕy0y0 (y) +∫

R
[ϕ(y+ < r(z),y > n̂) −ϕ(y) −ϕy0 (y) < r(z),y >]ν(dz), y ∈ D(A),ϕ ∈ C

2(H; R).

3. Option Price Formula

Assume that the stock price satisfy the following stochastic delay differential equation of the form

ds(t) =rs(t)dt + σ(t,st)dw(t) +

∫
R
γ(s−t ,z)Ñ(dt,dz), t ∈ [0,T]

s(0) = y0,s(θ) = y1(θ), θ ∈ [−r, 0]
(3.1)

where y := (y0,y1(.)) ∈ C is positive. Here C is the subspace of the Hilbert space H := R × L2−r :=

R×L2([−r, 0],R) whose inner product < .,. > is

< .,. >=< .,. >R + < .,.,>L2r

C := y ∈ H : y1(.) admits a Càdla′g representative.

In (3.1) r ∈ R,σ := (σ0,σ1(.) ∈ H,γ(.) := (γ0,γ1(.)(.)) ∈ L2(R,ν; H) are functional parameters.

Analogue of Black-Scholes formula for Vanilla call option price:

Suppose that the financial market under consideration as follows: (i). A risk free asset given by

ds0(t) = r(s0)(t)dt; t ∈ [0.T].

(ii). A risky asset given by equation (3.1)-(3.2).



Int. J. Anal. Appl. 19 (4) (2021) 498

A portfolio in such market is an Ft predictable process Π(t) representing the number of units held at

time t of the assets number 0, 1, . . . ,n respectively, then the wealth process x(t) = xΠ(t) associated to the

portfolio Π is defined to be:

xΠ(t) = Π(t)s(t) =

n∑
i=1

Πi(t)si(t).

Absence of arbitrage

In order to have no arbitrage opportunities in the considered market, the return from the portfolio must

be risk-free with interest rate r.

In what follows we adopt that Π(t) is riskless during [t,t + dt] and instantaneously earn the same rate of

return as other short-term risk-free assets.

These assumptions on Π(t) gives dΠ(t) = rΠ(t)dt.

Let the option price value has the form

(3.2) G(t,st) =

∫ 0
−T

e−rθF(s(t + θ),s(t), t)dθ

where F ∈ C0,2,1(R×R2 ×R+).

Lemma 3.1. (Itô Formula)

Suppose s(t) is given by (3.1) and a functional G : R+ ×C → R has the form

(3.3) G(t,st) =

∫ 0
−τ
g(θ)F(st(θ),st(0), t)dθ,

where F ∈ C0,2,1(R×R×R+) and g ∈ C1([−τ, 0],R).

Hence in view of the classical Itô formula, we have

G(t,st) =G(0,ζ) +

∫ t
0

AG(s,Ss)ds +

∫ t
0

σ(s,Ss)S(s)BG(s,Ss)dw(s)

+

∫ t
0

∫
R
{F(t,s(t−) + γ(s,z)) −F(t,s(t−)) −F(t,S(t)−)}N(dt,dz)

(3.4)

where for (t,x) ∈ R+ ×C.

AG(t,y) =g(0)F(x0,x0, t) −g(−τ)F(x(−τ),x0, t)−∫ 0
−τ
g′(θ)F(x(θ),x0, t)dθ +

∫ 0
−τ
g(θ)LF(x(θ),x0, t)dθ

+

∫
R
{F(t,x(t−) + γ(x,z)) −F(t,x(t−))}N(dt,dz),



Int. J. Anal. Appl. 19 (4) (2021) 499

with

LF(x(θ,x(0), t) =rx(0)F ′2(x(θ,x(0), t)+

σ2(t,x)x2(0)

2
F ′′22(x(θ,x(0), t) + F

′
3(x(θ,x(0), t)

+

∫
R
{F(x + γ(x,z)) −F(x) −F ′2.γ(x,z)}ν(dz),

where F ′i, i = 1, 2, 3 represents the derivative of F with respect to the i
th argument.

Portfolio concepts for financial markets driven by Le’vy process:

Theorem 3.1. The option price value given by (3.2) satisfies the equation

0 = F |θ=0 −erθF|θ=−τ +
∫ 0
−τ
e−rθ{(F ′3 + rs(t)F

′
2 +

1

2
σ2(t,s(t))s2(t)F ′′22)dθ +∫

R
{F(s + γ(s,z)) −F(s)) −F ′2γ(s,z)}ν(dz)}dθ.

Proof. We sketch the proof as in [12] as follows:

By considering a portfolio consists of -1 derivative and BG(t,st) shares. Then if Π(t) represents the

portfolio value we have

Π(t) = −G(t,st) + BG(t,st)s(t) = −dG + d(BGS) = d(BG)S + BGdS −dF.

Hence

dΠ(t) = −dG + BGdS.

Since we assume BG is held constant during the time-step dt yields d(BG) equal zero.

Substituting for dG and dS from equation (3.4) and (3.1) we get

(3.5) dΠ = −AGdt−σsBGdw + BG(rsdt + σsdw)

where

A = A−
∫ t
R
{F(t,s(t−) + γ(s,z)) −F(t,s(t−)) −F(t,S(t)−)}N(dt,dz).

By considering risk-free during the time dt gives

(3.6) dΠ = rΠdt.



Int. J. Anal. Appl. 19 (4) (2021) 500

By equating the last equations we get

AG(t,st) = rG(t,st).

which gives an equation for F(S(t + θ),S(t), t) in the form

0 = F|θ=0 −erθF|θ=−τ +
∫ 0
−τ
e−rθ{(F ′3 + rs(t)F

′
2 +

1

2
σ2(t,s(t))s2(t)F ′′22)dθ +∫

R
{F(s + γ(s,z)) −F(s)) −F ′2γ(s,z)}ν(dz)}dθ.

4. Price Formula for European Call Option:

Theorem 4.1.(Black-Scholes PDE Type)

In view of (3.3) the option price value G(T,ST ) will has the form

(4.1) G(T,ST ) = max(S(T) −K, 0).

For simplification we assume that G(T,St) takes the form

(4.2) G(t,st) = g1(s(t), t) +

∫ 0
−τ
e−rθg2(s(t + θ), t)dθ,

where g1(s(t), t) is a classical Black-Scholes call option price with variance assumed equal to a long-run

variance rate V, then

(4.3) g1(s(t), t) = s(t)N(d1) −ke−r(T−t)N(d2),

where

N(x) =
1
√

2π

∫ x
−∞

e−
x2

2 dx

d1 = ln(
s(t)

k
) + (r +

V

2
)(T − t)

d2 = d1 −
√
V (T − t).

Then G(t,St) satisfies the following equation:

∂G

∂t
+ rS(t)

∂g1
∂S

+
1

2
[σ2(t,st)]s

2(t)
∂2g1
∂s2

+∫
R
{F(s + γ(s,z)) −F(s)) −F ′2γ(s,z)}ν(dz) = rG.



Int. J. Anal. Appl. 19 (4) (2021) 501

Proof. By substituting (4.1) into (3.5) we obtain the equation for g2 as follows

g2(S(t), t) −erτg2(S(t− τ), t)+
∫ 0
tau

e−rθ
∂g2
∂t

dθ

=
1

2
(V −σ2(t,S(t)))S2(t)

∂2g1
∂S2

+∫
R
{F(s + γ(s,z)) −F(s)) −F ′2γ(s,z)}ν(dz).

Since g1 satisfies the classical Black-Scholes PDE we have:

∂g1
∂t

+ rS(t)
∂g1
∂S

+
1

2
[V ]S2(t)

∂2g1
∂s2

= rg1.

By combining the last equation we get the following equation G(t,St) of the form :

∂G

∂t
+ rS(t)

∂g1
∂S

+
1

2
[σ2(t,st)]s

2(t)
∂2g1
∂s2

+∫
R
{F(s + γ(s,z)) −F(s)) −F ′2γ(s,z)}ν(dz) = rG,

which is an integro-differential equation.

Remark. For pricing American Put Option we give the following discussion :

Suppose the functional (3.3) has the form S(t)−
∫ 0
−τ S(t+ξ)dξ then the pricing formula for the European

Put Option is to minimize the following functional overall Ft stoping time, τ < ∞ a.s. for all t ≥ 0:

F(τ,Sτ ) = E[(K − (S(τ) −
∫ 0
−T

S(t + ξ)dξ))+]

where K > 0.

The dynamics of the stock prices are driven by the following stochastic differential equation:

dS(t) =µ(S(t) −S(t−T))dt + α(S(t) −µ
∫ 0
−T

S(t + ξ))dw(t)+

β

∫
R

(S(t−) −µ
∫ 0
−T

S(t− + ξ)dξ)zÑdtdz

with the initial conditions S(0) = y0; S(θ) = y1(θ),θ ∈ [−T, 0), where µ,α,β are constants and y :=

(y0,y1(.) ∈ C.

The assumptions y0 −µ
∫ 0
−T y1(ξ)dξ > 0 and ν ≡ 0 on (−∞, 0] are imposed to ensure that positivity of

the solution and which it is exist and the allowance only for positive jumps.



Int. J. Anal. Appl. 19 (4) (2021) 502

The problem of finding the optimal exercise time of

F(τ,Sτ ) = E[(K − (S(τ) −
∫ 0
−T

S(t + ξ)dξ))+]

is strictly connected to the corresponding optimal stoping problem of such stochastic system. This prob-

lem was studied by [13] where the solution is obtained by rewriting the problem in infinite dimension from

which they found the condition under which the problem is reduced to one dimensional case which yields

explicit solution.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

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	1. Introduction
	2. Notations and Preliminaries
	3. Option Price Formula
	4. Price Formula for European Call Option:
	References