INTERPOLATED DRIFT IMPLICIT EULER MLMC METHOD FOR BARRIER OPTION PRICING AND APPLICATION TO CIR AND CEV MODELS MOUNA BEN DEROUICH AND AHMED KEBAIER

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INTERPOLATED DRIFT IMPLICIT EULER MLMC METHOD FOR BARRIER

OPTION PRICING AND APPLICATION TO CIR AND CEV MODELS

MOUNA BEN DEROUICH AND AHMED KEBAIER

Abstract. Recently, Giles et al. [14] proved that the eﬃciency of the Multilevel Monte Carlo

(MLMC) method for evaluating Down-and-Out barrier options for a diﬀusion process (Xt)t∈[0,T ]

with globally Lipschitz coeﬃcients, can be improved by combining a Brownian bridge technique and

a conditional Monte Carlo method provided that the running minimum inft∈[0,T ]Xthas a bounded

density in the vicinity of the barrier. In the present work, thanks to the Lamperti transformation

technique and using a Brownian interpolation of the drift implicit Euler scheme of Alfonsi [2], we

show that the eﬃciency of the MLMC can be also improved for the evaluation of barrier options for

models with non-Lipschitz diﬀusion coeﬃcients under certain moment constraints. We study two

example models: the Cox-Ingersoll-Ross (CIR) and the Constant of Elasticity of Variance (CEV)

processes for which we show that the conditions of our theoretical framework are satisﬁed under

certain restrictions on the models parameters. In particular, we develop semi-explicit formulas for

the densities of the running minimum and running maximum of both CIR and CEV processes which

are of independent interest. Finally, numerical tests are processed to illustrate our results.

1. Introduction

Barrier options are one of the most widely traded exotic options in the ﬁnancial markets. Pricing

and hedging such path-dependent option can quickly become very challenging especially when we

need to achieve a good precision for the approximation. Evaluating barrier options by a classic

Monte Carlo method introduces a systematic bias when approximating the continuous running

maximum (resp. minimum) in the crossing-barrier indicator by a discrete running maximum (resp.

minimum). To overcome this diﬃculty, several numerical strategies exist in the literature among

them the popular Brownian bridge technique introduced in [3] well known for its eﬃciency and ease of

use (see also [16] for related reﬁnements). The Brownian bridge technic uses an analytic expression

for the probability of hitting the barrier between two known values in a simulated path of the

underlying asset. More recently, a combination of the Multilevel Monte Carlo (MLMC) method with

the Brownian bridge technique has been developed in [14] for pricing barrier options. The Multilevel

Monte Carlo method introduced in Giles [12] as an extension of the two-level Monte Carlo method of

[20], signiﬁcantly reduces the time complexity of the classical Monte Carlo method. More precisely,

for a given precision ε > 0and a Lipschitz payoﬀ function, if the underlying asset (Xt)t∈[0,T ]is

approximated using a discretization scheme (¯

Xt)t∈[0,T ]with time step h > 0satisfying E|Xt−¯

Xt|2=

O(hβ)and |E[Xt−¯

Xt]|=O(hα)with α≥1

2, then the time complexity of the MLMC methods is:

O(ε−2)when β > 1,O(ε−2(log ε)2)when β= 1 and O(ε−2−1−β

α)when β∈(0,1). However, for

the same precision ε > 0the optimal time complexity of a classic Monte Carlo method is O(ε−3).

As the payoﬀ function of a barrier option is not Lipschitz, Giles et al. [14] take advantage of the

Brownian bridge to run the MLMC method for pricing such options, since this technique substitutes

Date: September 17, 2024.

2010 Mathematics Subject Classiﬁcation. 60H10, 60H35, 65C05, 65C30, 33C15, 41A60.

Key words and phrases. Multilevel Monte Carlo, Stochastic diﬀerential equations with singular diﬀusion coeﬃ-

cients, drift implicit Euler scheme, Lamperti transformation, Conﬂuent hypergeometric functions, Asymptotic ap-

proximations, Computational ﬁnance.

arXiv:2210.00779v2 [math.PR] 16 Sep 2024

the barrier-crossing indicators by the probabilities that the approximation scheme (¯

Xt)t∈[0,T ]hits

the barrier between each two consecutive discretization times tiand ti+1 and which are represented

as smooth functions of the realized points ¯

Xtiand ¯

Xti+1 . More precisely, Giles et al. [14] consider an

underlying asset solution to a one-dimensional stochastic diﬀerential equation (SDE) with globally

Lipschitz smooth coeﬃcients that is approximated by a high order strong approximation scheme

namely the Milstein scheme (¯

XMilstein

t)t∈[0,T ]that satisﬁes E|Xt−¯

XMilstein

t|2=O(h2). For this case,

they prove that the MLMC method reaches its optimal time complexity O(ε−2)for pricing a Down-

and-Out barrier option 1provided that inft∈[0,T ]Xthas a bounded density in the neighborhood

of the barrier. This latter condition cannot be easily checked even when the SDE coeﬃcients are

Lipschitz except for very speciﬁc cases.

In the current paper, we are interested in studying the MLMC method for pricing barrier options

when the underlying asset is solution to a SDE with a non-Lipschitz diﬀusion coeﬃcient such as

the popular Cox-Ingersoll-Ross (CIR) and the Constant of Elasticity of Variance (CEV) processes.

Only few works exist in the literature that studied the problem of pricing path-dependent options

under such singular models (see e.g. [8]). To analyze the performance of the MLMC method, we

consider in Section 2a general framework of models with non-Lipschitz diﬀusion coeﬃcients and

use a Lamperti transformation to focus our study on a new process (Yt)t∈[0,T ]with an additive

noise diﬀusion but in counterpart with a possibly singular drift coeﬃcient L. Then, we introduce

a Brownian interpolation scheme (¯

Yt)t∈[0,T ]associated to the drift implicit Euler scheme of Alfonsi

[2] for which we prove a strong convergence result with order one (see Theorem 2.1). In Section 3,

we use the Brownian bridge technic that substitutes the crossing-indicators with smooth functions

of realized points in the path of the scheme (¯

Yt)t∈[0,T ]to build the corresponding MLMC estimator.

Next, under suitable assumptions on the drift L, we prove that the obtained MLMC method for

pricing Down-and-Out (resp. Up-and-Out) reaches its optimal time complexity O(ε−2)provided

that inft∈[0,T ]Yt(resp. supt∈[0,T ]Yt) has a bounded density in the neighborhood of the barrier (see

Theorem 3.3 and Remark 3.4). In Sections 4and 5, we provide two examples of processes satisfying

Theorem 3.3 conditions, namely the CIR and the CEV models. It turns out that under additional

constraints on the parameters of these two models ensuring the existence of ﬁnite negative moments

up to a certain order, the MLMC method behaves exactly like a classical unbiased Monte Carlo

estimator despite the use of approximation schemes. To show that the conditions of our theoretical

framework are satisﬁed for these two models, we develop using ﬁne asymptotic properties of conﬂuent

hypergeometric type functions, semi-explicit formulas for the densities of the running minimum and

running maximum of both CIR and CEV processes which are of independent interest (see Theorems

4.1,4.2,5.2 and 5.3). Finally, we proceed to several numerical tests illustrating our results.

2. General framework

Let us consider a process (Xt)t∈[0,T ]solution to

dXt=b(Xt)dt +σ(Xt)dWt, X0=x, (1)

where (Wt)t≥0is a standard Brownian motion, b:R→Rand σ:R→R∗

+are locally Lipschitz-

functions such that 1

σis locally integrable. For ϕ(y) = Ry

σ(x)dx, if σ∈ C1then by the Lamperti

transform Yt=ϕ(Xt)solves the stochastic diﬀerential equation

dYt=L(Xt)dt +dWt, Y0=ϕ(x),

1A Down-and-Out barrier Call (resp. Put) is the option to buy (resp. sell), at maturity T, the underlying with a

ﬁxed strike if the underlying value never falls below the barrier before time T.

with L(x) = b

σ−σ′

2(ϕ−1(x)). In this work, we are interested in approximating barrier option

prices such as the Down-and-Out (D-O) and the Up-and-Out (U-O) barrier options

πBD=Ehf(XT){inft∈[0,T ]Xt>BD}iand πBU=Ehf(XT){supt∈[0,T ]Xt<BU}i.

The other types of barrier options such as the Down-and-In and the Up-and-In can be easily deduced

from the price of the vanilla option E[f(XT)]. As the function ϕis monotonic, by the Lamperti

transformation we reduce ourselves to a pricing problem with the process (Yt)t∈[0,T ]. More precisely,

we get πBD=πDand πBU=πUwhere

πD=Ehg(YT){inft∈[0,T ]Yt>D}i, πU=Ehg(YT){supt∈[0,T ]Yt<U}i,

g(x) = f◦ϕ−1(x),D=ϕ(BD)and U=ϕ(BU). In the sequel, we consider the general setting given

in [2] and let (Yt)t≥0denote the SDE deﬁned on I= (0,+∞)solution to

dYt=L(Yt)dt +γdWt, t ≥0, Y0=y∈I, with γ∈R∗,(2)

where the drift coeﬃcient Lis supposed to satisfy the following monotonicity assumption:

L:I−→ Ris C2,such that ∃κ > 0,∀y, y′∈I, y ≤y′, L(y′)−L(y)≤κ(y′−y).(3)

In addition, for an arbitrary point d∈I, we assume that

v(x) = Zx

dZy

exp −2

γ2Zy

L(ξ)dξdzdy satisﬁes lim

x→0+v(x) = +∞.(H1)

On the one hand, by the Feller’s test (see e.g. [19]), (3) and (H1) ensure that the SDE (2) admits

a unique strong solution (Yt)t≥0on Ithat never reaches the boundaries 0and +∞. On the other

hand, under these two conditions the below drift implicit continuous scheme introduced in [2],

t=b

ti+L(b

t)(t−ti) + γ(Wt−Wti),with t∈(ti, ti+1], ti=iT

n,0≤i≤n−1,(4)

0=y.

is well deﬁned and for all t∈[0, T ],Yn

t∈I. Besides, if in addition we assume that for p≥1, we

have

EhZT

0|L′(Yu)L(Yu) + γ2

2L′′(Yu)|dupi<∞and EhZT

(L′(Yu))2dup

2i<∞,(H2)

then by [2], there exists a positive constant Kpsuch that

p"sup

t∈[0,T ]|b

t−Yt|p#≤Kp

For our purpose, we rather focus on a slightly diﬀerent interpolated version of the drift implicit

scheme. More precisely, we ﬁrst introduce the discrete version of the drift implicit scheme given by







ti+1 =Yn

ti+L(Yn

ti+1 )T

n+γ(Wti+1 −Wti),with ti=iT

n,0≤i≤n−1,

0=y.

(5)

and introduce the following interpolated drift implicit scheme

t=Yn

ti+L(Yn

ti+1 )(t−ti) + γ(Wt−Wti),for t∈[ti, ti+1[,0≤i≤n−1.(6)

The main advantages of this Brownian interpolation is that it preserves the rate of strong conver-

gence of the original drift implicit scheme (4) and allows at the same time the use of the Brownian

bridge technique for pricing Barrier options (see Section 3.1 below). In what follows, we strengthen

our assumption on the drift coeﬃcient Las follows:

L:I→Ris C2such that: Lis decreasing on (0, A)for A > 0,

and L′the ﬁrst derivative of Lsatisﬁes ∃L′

A>0s.t. ∀y∈(A, ∞),|L′(y)| ≤ L′

A.(H3)

Theorem 2.1. Assume that conditions (H2)and (H3)hold true for a given p > 1and with

L′

A<n

2T. Then, there exists a constant Kp>0such that

phsup

t∈[0,T ]|Yn

t−Yt|pi≤Kp

Proof. At ﬁrst, for p≥1and t∈[0, T ], we denote et=Yn

t−Yt. By (2), we have for 0≤i≤n−1,

eti+1 =eti+L(Yn

ti+1 )(ti+1 −ti)−Zti+1

L(Ys)ds

since for all 0≤i≤n−1,Yn

ti∈I. As Lis of class C2there exists a point ξti+1 lying between

Yti+1 and Yn

ti+1 such that L(Yn

ti+1 )−L(Yti+1 ) = βti+1 (Yn

ti+1 −Yti+1 )with βti+1 =L′ξti+1 . Besides,

according to the proof [2, Proposition 3], we know that

Ehsup

1≤i≤n|eti|pi≤KT

npEhZT

0|L′(Yu)L(Yu) + γ2

2L′′(Yu)|dupi

+|γ|pEhZT

(L′(Yu))2dup

2i,(7)

where Kis a positive constant that depends on Tand p. On the one hand, we ﬁrst use (6) to write

eti+1 =eti+L(Yn

ti+1 )(ti+1 −ti)−Zti+1

L(Ys)ds

=eti+hL(Yn

ti+1 )−L(Yti+1 )i(ti+1 −ti) + Zti+1

tiL(Yti+1 )−L(Ys)ds

=eti+βti+1 eti+1 (ti+1 −ti) + Zti+1

tiL(Yti+1 )−L(Ys)ds.

It follows that

1−βti+1 (ti+1 −ti)eti+1 =eti+Zti+1

tiL(Yti+1 )−L(Ys)ds. (8)

On the other hand, we have for all t∈[ti, ti+1)

t=Yn

ti+L(Yn

ti+1 )(t−ti) + γ(Wt−Wti)

=Yn

ti+L(Yn

ti+1 )(ti+1 −ti) + γ(Wti+1 −Wti)−L(Yn

ti+1 )(ti+1 −t)−γ(Wti+1 −Wt)

=Yn

ti+1 −L(Yn

ti+1 )(ti+1 −t)−γ(Wti+1 −Wt).

Then, it follows that for all t∈[ti, ti+1)

t−Yt=Yn

ti+1 −Yti+1 +Yti+1 −Yt−L(Yn

ti+1 )(ti+1 −t)−γ(Wti+1 −Wt)

et=eti+1 +Zti+1

L(Ys)ds +γ(Wti+1 −Wt)−L(Yn

ti+1 )(ti+1 −t)−γ(Wti+1 −Wt)

=eti+1 −L(Yn

ti+1 )−L(Yti+1 )(ti+1 −t) + Zti+1

L(Ys)−L(Yti+1 )ds

=eti+1 −βti+1 (Yn

ti+1 −Yti+1 )(ti+1 −t) + Zti+1

L(Ys)−L(Yti+1 )ds.

So, we deduce that for all t∈[ti, ti+1)

et= (1 −βti+1 (ti+1 −t))eti+1 +Zti+1

L(Ys)−L(Yti+1 )ds. (9)

By assumption (H3), on (0, A)Lis decreasing, so it is easy to see that 1<1−βti+1 (ti+1 −t)<

1−βti+1 (ti+1 −ti). On (A, ∞), as L′is bounded and since n > 2L′

AT, we have |1−βti+1 (ti+1 −t)| ≤ 3

and 1−βti+1 (ti+1 −ti)>1

2. Then, it follows that 1−βti+1 (ti+1−t)

1−βti+1 (ti+1−ti)≤3.Now, combining (8) and

(9) we easily get

et=1−βti+1 (ti+1 −t)

1−βti+1 (ti+1 −ti)eti+Zti+1

tiL(Yti+1 )−L(Ys)ds+Zti+1

L(Ys)−L(Yti+1 )ds. (10)

Then, by Itô’s formula and Fubini theorem we get

|et| ≤ 3|eti|+Zti+1

tiL(Yti+1 )−L(Ys)ds+Zti+1

L(Ys)−L(Yti+1 )ds

≤3|eti|+T

nZti+1

tiL′(Yu)L(Yu) + γ2

2L′′(Yu)du +|γ|Zti+1

(u−ti)L′(Yu)dWu

nZti+1

tL′(Yu)L(Yu) + γ2

2L′′(Yu)du +|γ|Zti+1

(u−t)L′(Yu)dWu.

Therefore, there exists a positive constant Cpsuch that

|et|p≤Cp"|eti|p+ 2T

npZti+1

tiL′(Yu)L(Yu) + γ2

2L′′(Yu)dup+|γ|pZti+1

(u−ti)L′(Yu)dWup

+|γ|pZti+1

uL′(Yu)dWup+|γ|pTpZti+1

L′(Yu)dWup#

and thus,

sup

t∈[0,T ]|et|p≤Cp"sup

0≤i≤n|eti|p+ 2T

npZT

0L′(Yu)L(Yu) + γ2

2L′′(Yu)dup

+|γ|psup

0≤s≤t≤TZt

(u−tη(u))L′(Yu)dWup+|γ|psup

0≤s≤t≤TZt

uL′(Yu)dWup

+|γ|pTpsup

0≤s≤t≤TZt

L′(Yu)dWup#

≤Cp"sup

0≤i≤n|eti|p+ 2T

npZT

0L′(Yu)L(Yu) + γ2

2L′′(Yu)dup

+ 2p−1|γ|psup

0≤t≤TZt

(u−tη(u))L′(Yu)dWup+ 2p−1|γ|psup

0≤t≤TZt

uL′(Yu)dWup

+ 2p−1|γ|pTpsup

0≤t≤TZt

L′(Yu)dWup#.

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摘要：

INTERPOLATEDDRIFTIMPLICITEULERMLMCMETHODFORBARRIEROPTIONPRICINGANDAPPLICATIONTOCIRANDCEVMODELSMOUNABENDEROUICHANDAHMEDKEBAIERAbstract.Recently,Gilesetal.[14]provedthattheefficiencyoftheMultilevelMonteCarlo(MLMC)methodforevaluatingDown-and-Outbarrieroptionsforadiffusionprocess(Xt)t∈[0,T]withgloballyL...

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INTERPOLATED DRIFT IMPLICIT EULER MLMC METHOD FOR BARRIER OPTION PRICING AND APPLICATION TO CIR AND CEV MODELS MOUNA BEN DEROUICH AND AHMED KEBAIER

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