on the probability of hitting a constant or a time ...- to provide exact analytical formulae for the...

Applied Mathematical Sciences, Vol. 8, 2014, no. 20, 989 - 1009HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2014.311631

On the Probability of Hitting a Constant or a Time-

Dependent Boundary for a Geometric Brownian

Motion with Time-Dependent Coefficients

Tristan Guillaume

Laboratoire Thema, Université de Cergy-Pontoise33 boulevard du Port 95011 Cergy-Pontoise Cedex, France

Copyright © 2014 Tristan Guillaume. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

Abstract

This paper presents exact analytical formulae for the crossing of a constant one-sided or two-sided boundary by a geometric Brownian motion with time-dependent, non-random, drift and diffusion coefficients, under the assumption thatthe drift coefficient is a constant multiple of the diffusion coefficient, as well asapproximate analytical formulae for general time-dependent, non-random, driftand diffusion coefficients and general time-dependent, non-random, boundaries.The numerical implementation of these formulae is very simple.

Mathematics Subject Classification: 60J65, 60J60; 58J35

Keywords: geometric Brownian motion; first passage time; moving boundary;Fokker Planck equation; change of probability measure.

1 Introduction

The question of the hitting time of a diffusion process to an absorbing boundary isof central importance in many applied mathematical sciences. It appears as aclassical modelling framework in various branches of physics but it is also at the

990 Tristan Guillaume

core of more recent subjects such as quantitative finance. Typically, one needs tocompute the probability that some random dynamics that can be modelled as adiffusion process will remain under or above some critical threshold over a giventime interval. Almost all known exact results rely on the assumption that both thecoefficients of the diffusion process and the boundary are constant, whereas inpractice time-dependent diffusion process coefficients and boundaries are requiredfor modeling purposes. As a result, slow and tedious numerical schemes need tobe implemented, the accuracy of which may be dubious.Only two cases of time-dependent boundaries are easily handled from amathematical point of view : the crossing of a linear barrier by a Brownian motionwith constant drift and diffusion coefficients (see Doob [7] and Sheike [24] for aslight generalization) and the crossing of the exponential of a linear barrier by ageometric Brownian motion with constant drift and diffusion coefficients (seeKunitomo & Ikeda [13]) .Some papers provide semi-analytical results for boundaries that are fairly generalfunctions of time. Durbin [8], Ferebee [9] and Park & Schuurmann [19] thusderive integral equations, which must be numerically solved. Salminen [23]obtains a formula involving the expected value of a Brownian functional requiringto solve a non-time homogeneous Schrödinger equation with boundary and initialconditions. Nonlinear boundaries are studied in a general framework in Jennen &Lerche [11], Daniels [6], as well as Alili & Patie [1].Other contributions deal with specific forms of the boundary. Breiman [3] thusstudies the case of a square root boundary and relates it to the question of the firsthitting time of a constant boundary by an OU (Ornstein-Uhlenbeck) process,while Groeneboom [10] examines the case of a quadratic function of time andshows that the first passage density can be written as a functional of a Besselprocess of dimension 3.Alternatively, some authors focus on diffusion processes other than standardBrownian motion. Changsun & Dougu [5] thus evaluate the first passage timedensities of OU process to exponential boundaries and of Brownian bridge to twolinearly shrinking boundaries. Alili & Patie [2] study the problem of the firsthitting time of an OU process in general. Lo & Hui [15] derive a formula for thefirst passage density of a time-dependent OU process to a parametric class ofmoving boundaries.Finally, a collection of papers focuses on applications to financial mathematics,more specifically to option pricing, which shifts the focus to geometric Brownianmotion, as the latter serves as a the building block for modeling asset prices.Kunitomo & Ikeda [13] show that exponential boundaries are a simple extensionto the standard two-sided barrier option pricing problem with constant boundaries.Roberts & Shortland [20] obtain tight bounds on the prices of barrier options withgeneral moving boundaries and non constant coefficients by means of a hazardrate tangent approximation involving some numerical integration. Rogers & Zane[22] use a trinomial tree approach combined with a transformation of thestatespace and a time change that turn smoothly-moving barrier option pricingproblems into fixed barrier problems. Their result builds on a binomial lattice

Geometric Brownian motion with time-dependent coefficients 991

approach by Rogers & Stapleton [21]. Novikov et al. ([16], [17]) computepiecewise linear approximations for one-sided and two-sided boundary crossingprobabilities using repeated numerical integration and apply this method to thepricing of time-dependent barrier options. Thompson [25] derives upper andlower bounds in the form of double integrals that prove to be slightly tighter butharder to compute than those of Roberts & Shortland [20]. Lo et al. [14] alsoprovide tight bounds for barrier option prices by means of a multistageapproximation scheme in terms of multivariate normal distribution functionswhich involve non-trivial numerical integration issues.For all this rather abundant published research, there is no known formula for thecrossing of a time-dependent absorbing boundary by a diffusion process withtime-dependent drift and diffusion coefficients in general. The purpose of thisarticle is two-fold :- to provide exact analytical formulae for the crossing of a constant one-sided ortwo-sided boundary by a geometric Brownian motion with time-dependent, non-random, drift and diffusion coefficients, under the assumption that the driftcoefficient is a constant multiple of the diffusion coefficient- to provide approximate analytical formulae for general time-dependent, non-random, drift and diffusion coefficients and general time-dependent, non-random,boundariesThe method used is a combination of partial differential equations and changes ofprobability measure. All formulae are very easily implemented and yieldinstantaneous numerical values.Section 2 provides exact analytical results under restrictive assumptions on thetime-dependent diffusion process coefficients. Section 3 provides approximateanalytical results for general time-dependent diffusion process coefficients andboundaries. Section 4 aims at testing the accuracy of the approximation formulaegiven in Section 3.

2 Exact solutions in particular cases

This section presents two exact closed form results that hold when the time-dependent drift coefficient is a constant multiple of the time-dependent diffusioncoefficient and the boundary is constant. The first result is called Proposition 1. Itprovides the joint cumulative distribution function of not hitting a one-sidedconstant boundary during a finite time interval 0,T and of being under a given

point at time T . The second result is called Proposition 2. It provides the jointcumulative distribution function of not hitting a two-sided constant boundaryduring a finite time interval 0,T and of being above or under a given point at

time T .

PROPOSITION 1Let B t be a standard Brownian motion with natural filtration t .


Let ts be a piecewise continuous, non-random, positive real function such that :

2

0

0

T

t dts , : 0T T (1)

Let m be any real constant. Let X t be a geometric Brownian motion such that

00 0X x . Under a given measure , X t is driven by the following

stochastic differential equation : 2dX t m t X t dt t X t dB ts s (2)

Let 0h be a positive real constant such that 0 0h x and k be a positive real

constant such that 0k h . Let .N denote the cumulative distribution function of

a standard normal random variable.Then, the probability that X t will not hit 0h during the finite time interval

0,T and that it will be below k at time T is given by :

0

0sup ,

t TX t h X T k

2 2021

0 00 0 0 0

0 02 2

0 0

1 1ln ln

2 2

T T

m m

T T

kxkt dt t dt

x hx hN N

h xt dt t dt

s s

s s

(3)

Proof of Proposition 1As a consequence of the theory of absorbed diffusions in general and of theFokker-Planck equation in particular (see, e.g., Oksendal [18]), the soughtprobability, denoted by ,p x t , is the solution of the following IBVP (initial

boundary value problem) (4)-(5) :

2 2

2 222

p p t pm t x x

t x x

ss

, 0 0 00 , 0 ,t T x h h x (4)

0

lim , 1x

p t x

, 0, 0p t h , 0,p x k x (5)

where . stands for the Heaviside functionThe following change of the space coordinate :

0

lnx

yx

(6)

yields a new IBVP (7)-(8) :

2 22 0

20

1, 0 , ln

2 2

hp p t pt m t T y

t y xy

ss

(7)

lim , 1y

p t y

, 0

0

, ln 0h

p tx

, 00, expp y k x y (8)

The following change of function:


2

0

1 1, exp ,

2 8

t

p y t m y s ds w y ts

(9)

along with the following change of variable :

0

0

lnh

z yx

(10)

then yields the new IBVP : 2 2

2, 0 , 0

2

w t wt T z

t z

s

(11)

, 0 0w t , 00

0

0, exp exp2

h zw z k h z

x (12)

By separation of variables, the following function verifies equation (11) and theboundary condition in (12) :

2

2

0 0

, exp sin2

t

w z t A s ds z dl

l s l l

(13)

As a consequence of the initial condition in (12), the function

00

0

exp exp2

h zf z k h z

x (14)

can be identified as the sine Fourier transform of A l . Then, applying Fubini’s

theorem, classical trigonometric identities and the following Fourier cosinetransform :

2

2 2

0

1exp cos exp , ,

2 4

abz az dz a b

b b

p

(15)

one can obtain :

0ln /2

0

0 22

00

1, exp

222

k h

tt

h v v zw z t dv

xs dss ds sp s

(16)

0ln /2

0

0 22

00

1exp

222

k h

tt

h v v zdv

xs dss ds sp s

Straightforward calculations yield :

2

020 0

0 0 2

0

1ln

21, exp

2 8

t

t

t

kz s ds

hh zw z t s ds N

xs ds

s

s

s

(17)


2

020 0

0 0 2

0

1ln

21exp

2 8

t

t

t

kz s ds

hh zs ds N

xs ds

s

s

s

It can be checked that the function ,w z t given in (17) verifies (11) and (12).

Reverting to the initial function and variables, Proposition 1 ensues.

Hitting times of two-sided boundaries are considered next.

PROPOSITION 2Let B t , ts and X t be defined as in Proposition 1 and, without loss of

generality, let us set 1m in (3). Let 0d and 0u be two positive real constants

such that 0 0 0d x u and let k be a positive real constant such that 0k d .

Then, the probability that X t will hit neither 0d nor 0u during the finite time

interval 0,T and that it will be above k at time T is given by :

0 00 0

inf , sup ,t T t T

X t d X t u X T k

0 0

0

0ln /

20

10 0 0 00 ln /

21

exp exp sin8 2ln / ln /

u dt

n k d

d

x w ns ds w dw

u d u d

ps

22 0

00 0 0 00

1exp sin ln

2 ln / ln /

txn n

s dsdu d u d

p ps

(18)

Corollary :

0 00 0

inf , sup ,t T t T

X t d X t u X T k

0

0 0

0ln /

20

10 0 0 00 ln /

21

exp exp sin8 2ln / ln /

k ut

n d u

u

x w ns ds w dw

u d u d

ps

22 0

00 0 0 00

1 1exp sin ln

2 8ln / ln /

txn n

s dsuu d u d

p ps

(19)

Proof of Proposition 2Following the same steps as in the proof of Proposition 1, one can show that theprobability under consideration is the function ,w z t solving the following

IBVP (20)-(21) : 2 2

02

0

, 0 , 0 ln2

uw t wt T z

t dz

s (20)


, 0 0w t , 0

0

, ln 0u

w td

, 0

00

0, exp exp2

d zw z d z k

x (21)

where :

0

0

lnd

z yx

,

0

lnx

yx

, 2

0

1, exp ,

2 8

ty

p y t s ds w y ts

(22)

Using separation of variables, the boundary and initial conditions in (21) and thesuperposition principle, one easily obtains :

22

1 0 0 0 00

1, exp sin

2 ln / ln /

t

nn

n nw z t A s ds z

u d u d

p ps

(23)

where :

0 0

0

0ln /

0

0 0 0 0ln /

2

exp sin2ln / ln /

u d

n

k d

d

x w nA w dw

u d u d

p (24)

Then, reverting to the initial variables and function, and substituting 0x x ,

Proposition 2 ensues.

3 Approximate analytical solutions for general time-dependent, non-random boundary, drift and diffusion coefficients

This section provides approximate closed form analytical results for general non-random, time-dependent boundaries and process coefficients.The first result is called Proposition 3. It provides an approximate formula for thecumulative distribution function of not hitting a one-sided time-dependent, non-random boundary during a finite time interval 0,T . The second result is called

Proposition 4. It provides an approximate formula for the joint cumulativedistribution function of not hitting a two-sided constant boundary during a finitetime interval 0,T and of being below a given point at time T . The final result is

called Proposition 5 and provides the joint cumulative distribution function of nothitting a one-sided constant boundary during a finite time interval 0,T and of

being under or above a given point at time T .

PROPOSITION 3Let B t and ts be defined as in Proposition 1. Let tm be a piecewise

continuous non-random real function such that :

2

20

Tt

dtt

m

s , : 0T T (25)

Under a given measure , the process X t is driven by the following stochastic

differential equation :


dX t t X t dt t X t dB tm s (26)

where the usual growth conditions apply to tm and ts for equation (26) to be

properly defined (Oksendal [18])Let h t be a continuous, once differentiable real function, such that

00h x

and such that if there is a coefficient multiplying t , then this coefficient isnegative.Under these assumptions, the probability that X t will not hit h t during the

finite time interval 0,T can be approximated by the following formula :

,X t h t t T

22

02

02

0

0ln

2T

T

h t h tt

x h tN dt

tt dt

sm

ss

(27)

22

20

02

0

22

0exp ln

T

T

t h tt

h tdt

t h

xt dt

sm

s

s

220

202

0

ln20

T

T

t h txt

h thN dt

tt dt

sm

ss

Remark : if we denote by h tt the first passage time of the process X t to the

time-dependent boundary h t in 0,T , that is :

inf 0, :h t t T X t h tt (28)

then the density of h tt can be approximated by

,h t dt X s h s s tt

t

(29)

where an approximation of ,X s h s s t is given by (27)

Proof of Proposition 3 :

As a consequence of the theory of absorbed diffusions in general and of theFokker-Planck equation in particular (see, e.g., Oksendal [18]), the probability


under consideration, denoted by ,p x t , is the solution of the following IBVP

(30)-(31) :

2 2

222

p p t pt x x

t x x

sm

,

00 , 0 , 0t T x h t h x (30)

0

lim , 1x

p t x

,

, 0x h tp t x , 0, 1p x (31)

The next transformations of the space coordinate :

0

lnx

yx

,

0

lnh t

z yx

(32)

yield a new IBVP (33)-(34) :

2 2 2

2, 0 , 0

2 2

p t h t p t pt t T z

t h t z z

s sm

(33)

lim , 1z

p t z

, , 0 0p t , 0, 1p z (34)

If the function ,p z t solving (33)-(34) can be obtained, then the sought

probability will be equal to 0ln 0 / ,p z h x t T . Unfortunately, the

IBVP (33)-(34) cannot be solved exactly by known methods. A probabilisticapproach can nevertheless be applied to get an analytical approximation of thefunction ,p z t . Set :

2

2

t h tt t

h t

sl m

(35)

Let a new process Y t be started from the origin, whose motion is driven by : dY t t dt t dB tl s (36)

Then, solving IBVP (33)-(34) is the same as calculating the probability, under themeasure , that Y t will remain below the real number 0 0ln 0 /y h x

during the finite time interval 0,T .

Let be the measure under which the stochastic differential of Y t is given

by : dY t t dW ts (37)

where W t is a standard Brownian motion.

Then, using classical results on the hitting times of a standard Brownian motion(see, e.g., Karatzas & Shreve [12]), a simple time change of W t yields :

0 00

02 2

0 0

supT Tt T

y yY t y N N

t dt t dts s

(38)

The - measure is equivalent to and its Radon-Nikodym density is given by :

2

0 0

1exp

2t

t t

t

d s sdW s ds L

d s s

l l

s s

(39)

Existence of the integrals in (39) is guaranteed by the assumptions made on thefunctions


tl et ts .

Girsanov’s theorem (see, e.g., Karatzas & Shreve [12]) allows us to state that theprocess B t defined by :

0

ts

B t W t dss

l

s (40)

is a - Brownian motion

The function

t

t

l

sis non-random. Hence, if t is fixed :

2

0 0

0,

t ts s

dW s dss s

l l

s s

(41)

where ,a b refers to the normal distribution with expectation a and variance b

Thus, if t is fixed, the following equalities hold in law :

2 2law law

0 0 0

t t ts W t s s

dW s ds dss t s s

l l lf

s s s (42)

where f is a standard normal random variable

Using (38), (39) and (42), the sought probability 0

0sup

t TY t y

can now be

approximated as follows :

0

0sup

t TY t y

00sup

t T

tY t y

L

(43)

0

2

0 2 2

0 0

2

20

2

0

1exp

2

21 1exp exp

2 22

T

y

t dtT T

T

t tx dt dt

t t

yxx

t dt

s

l l

s s

ps

dx

(44)

Straightforward computations yield :

0

0sup

t TY t y

0 0

0exp 2y yJ t

N J t y N J tI t I t I t

(45)

with :

2

0

t

I t s dss ,

2

20

ts

J t dss

l

s (46)


The result stated in (45) is just an approximation since (42) only holds for fixed t

and not as a stochastic differential, as shown by the fact that :

2

0

tW t s t

d ds dW tt s t

l l

s s

(47)

It can be checked that the function ,p z t defined by :

, exp 2

z J t zp z t N J t z N J t

I t I t I t (48)

satisfies the required boundary and initial conditions in (34) as well as thefollowing PDE (partial differential equation) :

2 2 2

2,

2 2

p t h t p t pt z t

t h t z z

s sm

0 , 0t T z (49)

The function ,z t is a non-zero term that can be explicitly computed by

substituting the function ,p z t given by (48) into the PDE given by (33). If the

solution (48) were exact, then ,z t would be null. The residue ,z t becomes

more and more negligible as the parameter T in Proposition 3 decreases, so thatthe proposed analytical approximation should be very accurate on relatively“short” time intervals, as will be checked in Section 3.

Next, an approximation formula is provided when the boundary is constant andtwo-sided.

PROPOSITION 4Let B t , tm , ts and X t be defined as in Proposition 3. Let 0d and 0u be

two positive real constants such that 0 0 0d x u and let k be a positive real

constant such that 0k d . Then, the probability that X t will hit neither 0d nor

0u during the finite time interval 0,T and that it will be under k at time T can

be approximated by :

0 00 0

inf , sup ,t T t T

X t d X t u X T k


0 0 0

0 0 02

10 02

0 0

1 0

0

ln 2 ln ln

2 ln ln

exp

lnn

k u dn

x x xN I t

I tu dI t n

x x

I t d

xN

0 0

0 02

1

2 ln lnu d

nx x

I tI t

0 0 02

0 0 0

1

2 ln ln ln

expn

d u dI t n

x x x

I t

(50)

0 0 0 0

0 0 0 02

1

0 0 0

0 0 02

1

ln 2 ln 2 ln ln

ln 2 ln ln

k d u dn

x x x xN I t

I t

d u dn

x x xN I t

I t

where :

2

0

T

I t t dts (51)

22

2 20

2T

tt

I t dtt

sm

s

(52)

Proof of Proposition 4:

Following the same steps as in the proof of Proposition 3, the probability underconsideration, denoted by ,p y t , is the solution of the following IBVP (53)-(54):

2 2 2

0 02

0 0

, 0 , ln ln2 2

d up t p t pt t T y

t y x xy

s sm

(53)

0

0

, ln 0d

p tx

, 0

0

, ln 0u

p tx

, 00, expp y k x y (54)

where0

lnx

yx

Set :


2

2

tt t

sb m (55)

Let a new process Z t be started from the origin, whose motion is driven by : dZ t t dt t dB tb s (56)

Solving IBVP (53)-(54) is the same as calculating the probability, under themeasure , that Z t will remain below the real number 0 0ln /u x and above

the real number 0 0ln /d x during the finite time interval 0,T and that Z t

will be smaller than the real number0

lnk

x

at time T .

Let be the measure under which the stochastic differential of Z t is given

by : dZ t t dW ts (57)

where W t is a standard Brownian motion.

Using the classical method of images (Carslaw & Jaeger [4]), one can obtain :

0 0 0 0 00 0inf ln / , sup ln / , ln /t T t T

Z t d x Z t u x Z T k x

0 0 0 0 0

2

0

0 0 0 0 0 0

2

0

ln / 2 ln / ln /

ln / 2 ln / ln /

T

n

T

k x n u x d xN

t dt

d x n u x d xN

t dt

s

s

(58)

0 0 0 0 0 0 0

2

0

0 0 0 0 0 0

2

0

ln / 2 ln / 2 ln / ln /

ln / 2 ln / ln /

T

n

T

k x d x n u x d xN

t dt

d x n u x d xN

t dt

s

s

The - measure is equivalent to and its Radon-Nikodym density is given by :

2

0 0

1exp

2t

t t

t

d s sdW s ds L

d s s

b b

s s

(59)


Along the lines of the proof of Proposition 3, it can then be shown that the soughtprobability can be approximated by :


Z t d x Z t u x Z T k x


tZ t d x Z t u x Z T k x

L

(60)

where the expectation operator in (60) is expanded into integral form using (58)and the following equality in law, for fixed t :

2law

0 0

, 0,1

t ts s

dW s dss s

b bf f

s s (61)

Performing the necessary calculations, the formula in (50) is then obtained.

Eventually, an approximation formula is stated when the boundary is constant andone-sided.

PROPOSITION 5Let B t , tm , ts and X t be defined as in Proposition 3. Let 0h be a

positive real constant such that 0 0h x and k be a positive real constant such

that 0k h . Then, the probability that X t will not hit 0h during the finite time

interval 0,T and that it will be below k at time T can be approximated by :

0

0sup ,

t TX t h X T k

22

02

02

0

ln2

T

T

k tt

xN dt

tt dt

sm

ss

(62)

22

220

2 20 0 0

20 02 2

0 0

22 ln

2exp ln

T

T

T T

tt

kx tdt tt h h

N dtx t

t dt t dt

sm

sm

s

ss s

Corollary : Let 0h be a positive real constant such that 0 0h x and k be a

positive real constant such that 0k h . Then, the probability that X t will not

hit 0h during the finite time interval 0,T and that it will be above k at time T is

given by :


00inf ,t T

X t h X T k

22

0

202

0

ln 2T

T

tx tkN dt

tt dt

sm

ss

(63)

22

22 20

200 0

20 02 2

0 0

22 ln

2exp ln

T

T

T T

tt

h tdt tt kxh

N dtx t

t dt t dt

sm

sm

s

ss s

The proof is similar to that of Proposition 3, so the details are omitted.

4 Numerical results

In this final section, the accuracy of the analytical approximations provided byProposition 3, Proposition 4 and Proposition 5, is numerically tested, bycomparing obtained values with selected benchmarks.The first stage of the procedure is to randomly draw parameters for the diffusionprocess X t defined in Proposition 3. Three different functional forms of

coefficients tm and ts are selected. The linear form reads :

1 1t a b tm ,

2 2t a b ts (64)

The quadratic form reads : 2

1 1 1t a b t c tm , 22 2 2t a b t c ts (65)

The square root form reads :

1 1t a b tm , 2 2t a b ts (66)

The real constants 1 2 1 2 1 2, , , , ,a a b b c c are randomly drawn within wide predefined

ranges. For instance, if the linear form is prescribed for the volatility function ts , the initial volatility

20 as may vary from 5% to 25%, while the

coefficient 2b , measuring the increase in volatility over time, may vary from 30%

to 80%. Depending on the random value of parameter T , final values of thevolatility function, Ts , of up to 85% were observed during the numerical

experiment.The initial value of the process, 0X , is set at 1. To test Proposition 4 and

Proposition 5, the parameters 0 0 0, ,u d h and k are randomly drawn. The

predefined ranges for 0u , 0h and 0d are 0 10%;40%X . The parameter k


may vary between 0 10%X and 0 10%X . To test Proposition 3, a time-

dependent boundary h t is also randomly picked that can take on the functional

forms (64)-(65). Finally, the width of the time interval may vary from 0 to 1 .The next stage in the procedure is to define a benchmark that is assumed to be the“true” value. Possible benchmarks are the numerical solutions of the initialboundary value problems obtained in (33)-(34) and (53-54). This approach wasattempted by means of classical Crank-Nicolson finite differences and led to quiteunreliable results, even with fine meshes. Alternatively, one can resort to theMonte Carlo simulation of diffusion process X t - and of time dependent

boundary h t as far as Proposition 3 is concerned. This approach is both simple

and robust. It was implemented using a standard Euler scheme for thediscretization of the stochastic differential equation (26) with a timestep equal to1/16000 . A rather fine timestep such as the latter is needed, otherwise too many

boundary crossings would be missed. Still, plain Monte Carlo simulation isnotoriously inaccurate. Besides increasing the number of simulations and using areliable random number generator, it is advisable to have exact analyticalbenchmarks at disposal, with which one can test the accuracy of the Monte Carlosimulation. This can be achieved by means of Proposition 1 and Proposition 2 for

the special case in which 2t tm s . So, our numerical procedure is

performed twice for each set of process coefficients and boundaries : (i) drawing ageneral tm in the first place to test the accuracy of Proposition 3, Proposition 4

and Proposition 5, in comparison with a Monte Carlo approximation used as abenchmark ; (ii) setting 2t tm s and using Proposition 1 and Proposition 2

as exact benchmarks to test the accuracy of the Monte Carlo approximation itself.The second stage of the procedure is carried out using the same random numbersthat are generated to compute a Monte Carlo approximation for a general tm in

the first place.Tables 1 - 4 report the obtained results when conducting the numericalexperiment above described. A sample of 100,000 collections of randomly drawnparameters is used in each Table. Three main indicators are reported :- the observed mean absolute value of the error- the observed highest absolute value of the error- the observed proportions of errors that are : less than 0.5% , between 0.5% and1% , between 1% and 2% , between 2% and 3% , and over 3%

In Table 1, the “error” under consideration is the difference between the analyticalapproximations provided by Proposition 4, Proposition 5 and Proposition 3 on theone hand, and Monte Carlo approximations used as benchmarks on the otherhand. The error is expressed as a percentage of the numerical value of the MonteCarlo approximation.In Table 2, the “error” under consideration is the difference between a MonteCarlo approximation and exact analytical benchmarks provided by Proposition 1and Proposition 2.Table 1 shows that the mean absolute value of the error entailed by the use ofProposition 4, Proposition 5 and Proposition 3, is always less than 1%. But, in


practice, it is obviously important to have an idea of the order of magnitude of thegreatest possible error incurred by a single use of an analytical approximation.Over the fairly large and uniformly distributed sample of random parameters thatwas used, it turns out that the maximum absolute value of the error is always lessthan 3% for Proposition 4 and 5, and always less than 5% for Proposition 5. Theseresults are the all the more reassuring as one must bear in mind that the MonteCarlo approximations used as benchmarks in Table 1 are relatively inaccurate,despite the high number of simulations and the fine timestep. Indeed, Table 2shows that the absolute error entailed by the use of a Monte Carlo approximationas a benchmark may be as high as 1.67%. This suggests that the maximumabsolute value of error entailed by Proposition 4, Proposition 5 and Proposition 3may actually be smaller than the figures reported in Table 1.As pointed out in Section 3, the magnitude of the errors entailed by the analyticalapproximations is a function of the width of the time interval 0,T . So, in Tables

3 and 4, three ranges for T are examined: 0.1T , 0.1, 0.25T , and

0.25,1T . Only Proposition 4 and Proposition 3 are tested, as the results

obtained for Proposition 5 are very similar to those obtained for Proposition 4. Itturns out that, when 0.1T , which includes time intervals that may contain asmany as 1,600 time steps, the maximum absolute value of the error entailed byProposition 4 is only 0.58%, while 93% of observed errors are less than 0.5%. Thefigures are roughly the same for Proposition 3. These numerical results suggestthat our analytical approximations should be quite relevant for practical purposesin a variety of applied sciences. Indeed, not only do these analyticalapproximations provide a framework in which the impact and the interactions ofvariables can be mathematically analyzed, but they are also extremely efficientcompared with the slowness of a Monte Carlo simulation. To get a single MonteCarlo estimate in Table 1, one must draw between 80, 000, 000 and 8,000,000,000uniform random numbers as the parameter T varies between 0.01 and 1, whichmay typically take up to half an hour on a single PC. Furthermore, the issue of thereliability of the random number generator becomes more acute as the number ofdraws increases. In contrast, it always takes less than one second to obtain anestimate using Proposition 3, Proposition 4 or Proposition 5, as the functionsinvolved are standard and the number of function evaluations implied by thenumerical integrations is very modest.Eventually, one can also wonder whether the magnitude of the errors entailed bythe analytical approximations may vary according to the functional forms of thediffusion coefficients tm and ts , as well as to the functional form of the time

dependent boundary h t . Numerical tests were implemented in this regard and

showed no significance of that factor, as the orders of magnitude remained thesame for all different functional forms of the diffusion process coefficients and ofthe time dependent boundary.


Table 1 – Numerical assessment of the error entailed by the use of the analyticalapproximations provided by Proposition 4, Proposition 5 and Proposition 3

Proposition 4 Proposition 5 Proposition 3mean absolute value oferror

0.85% 0.77% 0.81%

maximum absolute value oferror

2.76% 2.62% 4.87%

Proportion of

error 0.5%39% 36% 24%

Proportion of

error 0.5%;1%37% 41% 38%

Proportion of

error 1%;2%15% 13% 21%

Proportion of

error 2%;3%9% 10% 11%

Proportion of

error 3%0% 0% 6%

Notes : The error is measured as the divergence from a Monte Carlo approximation used as a benchmark, andexpressed as a percentage of the value of the Monte Carlo approximation. A sample of 100,000 different sets

of parameters for the process X t is considered. Details about the way the parameters are drawn can be

found at the beginning of Section 4. For each set of parameters, a total of 500,000 Monte Carlo simulationsare carried out to obtain a Monte Carlo benchmark, using a timestep equal to 1 / 16 000 for the discretization

scheme of the stochastic differential of X t .

Table 2 – Numerical assessment of the error entailed by the use of a Monte Carloapproximation as a benchmark

Proposition 1 Proposition 2mean absolute value of error 0.54% 0.58%maximum absolute value oferror

1.59% 1.67%

Proportion of

error 0.5%59% 61%

Proportion of

error 0.5%;1%27% 24%

Proportion of

error 1%;2%14% 15%

Proportion of

error 2%;3%0% 0%

Proportion of

error 3%0% 0%

Notes : The error is measured as the difference between the exact analytical results provided by Proposition 1and Proposition 2 and their Monte Carlo approximations, and expressed as a percentage of the exact values.Details about the selected sample and the way Monte Carlo simulation is carried out are the same as in thenotes of Table 1


Table 3 – Numerical assessment of the error entailed by Proposition 4 as afunction of the width of the time interval 0,T

0.1T 0.1, 0.25T 0.25,1T

mean absolute value oferror

0.14% 0.35% 1.08%

maximum absolute valueof error

0.58% 1.21% 2.97%

Proportion of

error 0.5%93% 78% 38%

Proportion of

error 0.5%;1%7% 18% 39%

Proportion of

error 1%;2%0% 4% 13%

Proportion of

error 2%;3%0% 0% 10%

Proportion of

error 3%0% 0% 0%

Notes : same as in Table 1

Table 4 – Numerical assessment of the error entailed by Proposition 3 as afunction of the width of the time interval 0,T

0.1T 0.1, 0.25T 0.25,1T

mean absolute value oferror

0.18% 0.41% 1.36%

maximum absolute valueof error

0.71% 1.45% 5.04%

Proportion of

error 0.5%86% 66% 16%

Proportion of

error 0.5%;1%14% 22% 29%

Proportion of

error 1%;2%0% 12% 28%

Proportion of

error 2%;3%0% 0% 13%

Proportion of

error 3%0% 0% 14%

Notes : same as in Table 1

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Received: November 1, 2013

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