First hitting time and option pricing problem under Geometric

Brownian motion with singular volatility

HAOYAN ZHANG, YECE ZHOU, XUAN LI, YINYIN WU*

College of Science,

Civil Aviation University of China,

Tianjin 300300,

CHINA

2m,

*Corresponding Author

Abstract: In this paper, we discuss the first hitting time and option pricing problem under Geometric Brownian

motion with singular volatility. By solving the Sturm-Liouville equation and introducing probability scheme, we

derive the closed-form solutions to the target problems. At last, numerical results are provided to analyze our

calculations.

Key-Words: Geometric Brownian motion, First hitting time, Laplace transform, Option pricing, Closed-form

solution, Numerical result

Received: July 15, 2023. Revised: October 14, 2023. Accepted: October 29, 2023. Published: November 22, 2023.

1 Introduction

Geometric Brownian motion (GBM) is a simple

continuous-time stochastic process in which the loga-

rithm of the randomly varying quantity of interest fol-

lows a Brownian motion with drift and a special case

of Markov process. Geometric Brownian motion fre-

quently features in mathematical modeling. The ad-

vantage of considering this process lies in its univer-

sality, as it represents an attractor of more complex

models that exhibit non-ergodic dynamics, we refer

the readers to [1], [2], [3]. Geometric Brownian mo-

tion is defined by dSt

St=µdt +σdWt, where µis a

drift term, σis a noise amplitude, and Wtis a Wiener

process. [4] proposed this model without the noise,

i.e. σ= 0, that means the model is simply exponen-

tial growth at rate µ. If σ= 0, it can be interpreted

as exponential growth with a fluctuating growth rate.

[5] put forward the logarithm of random variables sat-

isfying Geometric Brownian motion. The main fac-

tors affecting the change of stock price are the rising

trend of stock price over time and the average volatili-

ty of stock price. The contribution of the former to the

growth of stock prices relies on the length of time; the

latter only depends on the random fluctuations caused

by Brownian motion. In the classical Wiener process,

we assume that the expected drift rate is constant, that

is, µ(·) = µ. Therefore, Geometric Brownian motion,

a widely used model to describe the behavior of stock

prices, is obtained. If the price of the stock at time tis

S, then the drift rate of the underlying model should

be µS, where µis a constant. But in practice, it is

more reasonable to assume that the expected rate of

return is not constant. Compared with Brownian mo-

tion, the expectation of Geometric Brownian motion

is independent of the stock price, which conforms to

the expectation of real market.

In recent years, Geometric Brownian motion has

been widely used in the field of transportation plan-

ning in [6] and management in [7]. See, for instance

that, [8] predicted traffic flow. [9] predicted a mini-

mum future closing price in a company by using GB-

M. Among others, Geometric Brownian motion has

applications in ecology in [10], economics in [11],

[12], etc. More recent work on option pricing prob-

lems, readers may consult the papers in [13], [14],

[15], [16], [17].

In the financial field, Geometric Brownian mo-

tion is the most widely used dynamic. In 1965, based

on Einstein's Brownian motion research method and

Wiener's definition of Brownian motion, [18] estab-

lished a Geometric Brownian motion model to de-

scribe the random fluctuation of stock prices. In 1973,

[19] first assumed that the stock price was subject to

the Geometric Brownian motion model of Samuel-

son, to deduce the famous B-S option pricing formula,

and used the stock price model to solve the reasonable

pricing problem of financial derivatives such as stock-

s, bonds, currencies and commodities. In the same

year, [20] also derived the B-S option pricing formu-

la using the stochastic calculus method, further weak-

ened the assumptions on which the B-S option pricing

formula depends, and added the Poisson jump process

to the Geometric Brownian motion model, expanding

the application scope of the B-S option pricing for-

mula. Later in [21], the paper put forward a simple

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and efficient numerical procedure for valuing option-

s for which premature exercise may be optimal. Al-

so in [22], the paper empirically provided an alterna-

tive pption pricing models by first deriving an option

model that allows volatility, interest rates and jumps

to be stochastic. Further, [23] provided an analytical

treatment of a class of transforms, including various

Laplace and Fourier transforms as special cases, that

allow an analytical treatment of a range of valuation

and econometric problems to solve example applica-

tions including fixed-income pricing models, with a

role for intensity-based models of default, as well as

a wide range of option-pricing applications. It is well

known that the volatility and yield are set as constants

in the Geometric Brownian motion. Economists often

used this Geometric Brownian motion to describe the

stock trend, so it is very meaningful to study it. In

addition, this has attracted the attention and discus-

sion of a large number of scholars. Among them, it is

an important aspect to analyze asset pricing with the

change of financial time series. [24] considered the

research of multivariate GARCH model in explaining

the difference in expected returns between Shanghai

and Shenzhen of China's stock market. In addition,

the difference is not remarkable. Therefore, this pa-

per mainly focuses on the option price under the Geo-

metric Brownian motion process. However, because

the classical assumption is too idealistic, it has obvi-

ous limitations, that is, the volatility and yield in the

actual market do not meet the constant assumption.

For example, Figure 1 & Figure 2 show the trend of

the Dow Jones Industrial Index and the United States

Federal Fund Interest Rate (1955-2023). It is obvious

Fig.1: Dow Jones Industrial Index from 2018-

2023

that the volatility and yield are not constant, and they

will change under various influences. If we consider

the case of random volatility, in most cases, we can-

not give a closed-form solution and must define the

appropriate distribution for the volatility. For exam-

ple, [25] considering the pricing of derivative options

with random volatility, only provided the pricing for-

mula, and did not give a closed-form solution. Hence,

this paper take the broken drift and singular volatility

into consideration.

The first hitting time is a mathematical term that

Fig. 2: The United States Federal Fund Interest

Rate from 1955-2023

represents the moment when a stochastic process first

reaches a certain set of states. In terms of application,

the first hitting time has important practical signifi-

cance, for example, in fields like financial risk man-

agement, communication network optimization, and

transportation scheduling. In addition, the first hitting

time can also be used for the optimization of predic-

tion models and algorithms, for example in fields like

machine learning and data mining. By understand-

ing the nature and calculation methods of the hitting

time, we can better understand and optimize the be-

havior of stochastic processes, thereby providing bet-

ter solutions for practical applications. On one hand,

the first hitting time problem is very crucial as a clas-

sic subject. On the other hand, the study of the first

hitting time problem of Geometric Brownian motion

with broken drift and singular volatility is very few at

present. We also view this problem as our target s-

tudy. Around the literature about the first hitting time

problems, the authors may consult [26] under skew

CIR process, [27] under regime switching Geometric

Brownian motion, [28] under sticky skew Brownian

motion, and [29] under sticky skew CIR process. A

most recent paper in 2019, [30] considered the opti-

mal stopping and first hitting time problems of Brow-

nian motion with broken drift. However, there ex-

ists few papers concerning on hitting time and op-

tion pricing problems by introducing singularity coef-

ficients. In addition, traditional methods when solv-

ing the pricing problem needs complex calculations

or numerical scheme. Complex calculations may re-

sult in inaccurate or even non-closed-form solution-

s. Numerical computations have high requirements

for accuracy and real-time performance of the results.

Therefore, this paper tries to solve the first hitting time

and option pricing problem under our setting mod-

el, respectively. In probability scheme, we only need

to employee the variable substitution and probability

distribution of random variable function for our tar-

get under risk-neutral measure. Now let us introduce

the Geometric Brownian motion with broken drift and

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singular volatility by

dSt

=µ(St)dt +σ(St)dWt,(1)

where

µ(St) = {µ1, St≥a,

µ2, St< a, σ(St) = {σ1, St≥a,

σ2, St< a.

The notations µ1and µ2are the drifts, σ1and σ2are

the volatilities, ais the singularity boundary, Stis the

stock price at time tand Wtis a standard Brownian

motion. When we consider the option pricing prob-

lem, in traditional option pricing theory, we always

assume the drift term to be the risk-free interest rate

r, this model (1) reduces to

dSt

=rdt +σ(St)dWt.(2)

Before proceeding the study, please let us retel-

l our target problem. In this paper, we consider the

Laplace transform of the first hitting time and op-

tion pricing problem under Geometric Brownian mo-

tion with singular volatility, respectively. To get the

Laplace transform of first hitting time under the above

model, we should solve the Sturm-Liouville equation,

and when facing the option pricing problem, we ap-

ply the probability scheme to calculate this object. All

the results we get are closed-form. At last, we provide

the numerical results to explain some interesting phe-

nomena under different coefficients.

The rest of this paper is organized as follows. Sec-

tion 2 analyzes the option pricing problem under Ge-

ometric Brownian motion with singular volatility and

use Matlab to get numerical results. Under our mod-

el the Laplace transform of first hitting time and the

theoretical and numerical results are respectively cal-

culated in Section 3. Section 4 concludes this paper

and explains some future study.

2 Option Pricing

2.1 Theoretical results

Let us assume that (Ω,F,P)is a complete proba-

bility space, Fis a σ-algebra on Ωand Prepresents a

probability measure of (Ω,F). In this paper, we con-

sider that the underlying asset satisfies (2).

We can write the definition of the option price un-

der risk-neutral measure by:

f(t, St) = e−r(T−t)EQ[max(ST−K)+],(3)

where Kis the exercise price and Tis the maturity

date.

Next, we naturally draw the theorem of option

pricing:

Theorem 1. If the stock price satisfies the Geometric

Brownian motion with risk-free interest rate and sin-

gular volatility in (2), then the option price defined by

(3) takes the form of:

f(t, St) =











StΦ(ln St

K+(r+1

2σ2

1)(T−t)

σ1√T−t)−

Ke−r(T−t)Φ(ln St

K+(r−1

2σ2

1)(T−t)

σ1√T−t),

St≥a,

StΦ(ln St

K+(r+1

2σ2

2)(T−t)

σ2√T−t)−

Ke−r(T−t)Φ(ln St

K+(r−1

2σ2

2)(T−t)

σ2√T−t),

St< a.

Proof. According to Itˆo's formula:

df(t, St) =(∂f

∂t +∂f

∂St

µSt+1

∂2f

∂t2σ2St2)dt (4)

+∂f

∂St

σStdWt.(5)

Let lnSt=Gt, then we have ∂Gt

∂St=1

St,∂2Gt

∂St2=−1

St2

and ∂G

∂t = 0.

We recall (5), getting

dGt=dlnSt= (µ−σ2

2)dt +σdWt.

Further, we get

lnST∼N(lnSt+ (µ−σ2

2)(T−t), σ2(T−t)).

Letting m=lnSt+ (µ−σ2

2)(T−t),s=σ√T−t

and applying the probability scheme, we have

EQ[max(ST−K)+]

=∫∞

(St−K)f(St)dSt

=∫∞

lnK

(elnSt−K)g(lnSt)d(lnSt)

=∫∞

lnK−m

[e(sw+m)−K]h(w)dw

=∫∞

lnK−m

e(sw+m)(2π)−1

2e(−w2

2)dw −KΦ(m−lnK

=e(s2

2+m)∫∞

lnK−m

(2π)−1

−(w−s)2

2dw −KΦ(m−lnK

=e(s2

2+m)Φ(s−lnK −m

s)−KΦ(m−lnK

s),

where f(St)is the density function of St,g(lnSt)is

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the density function of lnSt,h(·)is the standard nor-

mal density function and lnK obeys normal distribu-

tion that the mean is mand the standard deviation is

s, that is lnK ∼N(m, s2).

Therefore, we apply the result to (3), acquiring

f(t, St) =e−r(T−t)[es2

2+mΦ(s−lnK −m

−KΦ(m−lnK

s)],

where

s−lnK −m

s=σ2(T−t) + lnSt/K+ (r+1

2σ2)(T−t)

σ√T−t

=ln(St

K)+(r+1

2σ2)(T−t)

σ√T−t=d1,

m−lnK

s=lnSt+ (r−1

2σ2)(T−t)−lnK

σ√T−t

=ln(St

K)+(r−1

2σ2)(T−t)

σ√T−t=d2,

and

m+s2

2=lnSt+ (r−0.5r2)(T−t) + σ2(T−t)

=lnSt+r(T−t).

Finally, we replace µand σby rand σ(·), which fin-

ishes our proof.

2.2 Numerical results

We use Matlab for numerical simulation to realize

the trend and comparison of option prices and stock

prices. In our setting, we choose ato be close to the

median value of 7 and r= 0.1. When the stock price

Stis greater than or equal to 7, we take volatility σ

as 1.8, and when Stis less than 7, we take volatili-

ty σas 1.2. Because risk in markets is usually high,

implying that the greater the risk is, the greater the

potential gain gets, but the greater the potential loss

happens. Therefore, the higher the stock price is, the

more severe the fall is likely to be; while the lower

the stock price is, the less room for further decline the

stock price is. More precisely, we set

a= 7, r = 0.1, σ(t) = {1.8, St≥a,

1.2. St< a.

Figure 3 is the results of our numerical simulation.

It can be seen from Figure 3 that the cumulative val-

ue of derivatives has existed for a long time due to

the continuous compounding of stocks, so the overall

trend is upward.

10 20 30 40 50 60 70 80 90 100 110 120

time/day

price

The option price

Call

Put

10 20 30 40 50 60 70 80 90 100 110 120

time/day

price

The stock price

Price

Fig.3: The trend and comparison of option prices

and stock prices

We observe the changes of option prices and s-

tock prices by different parameters. First, fix broken

boundary aand volatility σ, and change the value of

r. The specific values are as follows:

a= 7, r = 0.2, σ(t) = {1.8, St≥a,

1.2, St< a.

a= 7, r = 0.05, σ(t) = {1.8, St≥a,

1.2, St< a.

It can be seen from Figure 4 & Figure 5 that blue

curve is bigger than the red (green) curve. With the

change of rvalue, both stock price and option price

will change, in which the price of stock price and call

option increases relative with the increase of value,

which reflects the fact that the great rpushes the high

level for stock price and the corresponding call option

price.

0 20 40 60 80 100 120 140

time/day

price

The stock price

Price1

Price2

Fig. 4: The stock price and call option price with

different r. The blue line relies on r= 0.2and the

red line relies on r= 0.05, respectively.

Moreover, we get the intuition that the change of

σhas an effect on the price of the stock and the rela-

tive size of the impact of the yield on the price trend.

Therefore, by changing the value of σ, the influences

of different parameters on the stock price and option

price are compared. Hence we fix broken boundary a

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0 20 40 60 80 100 120 140

time/day

price

Call option price

Call1

Call2

Fig. 5: The stock price and call option price with

different r. The blue line relies on r= 0.2and the

green line relies on r= 0.05, respectively.

and r, and change the volatility value of σ. The spe-

cific values are as follows:

a= 7, r = 0.05, σ(t) = {1.8, St≥a,

1.2, St< a.

a= 7, r = 0.05, σ(t) = {2, St≥a,

1.2, St< a.

It can be seen from the Figure 6 & Figure 7 that blue

curve is bigger than the red (green) color curve. When

the value changes, the price of stock and option will

also change. The price of stock and call option in-

creases with the increase of σvalue, which may ex-

plain that the great σ, leading high reward, pushes the

high level for stock price and the corresponding call

option price. We can also observe that the impact of

rvalue on option prices is smaller than that of σval-

ue, that is, the impact of volatility on asset prices is

greater than that of yield. In our opinion, managing

risk is far more important than guaranteeing expecta-

tion.

0 20 40 60 80 100 120 140

time/day

3.5

4.5

5.5

6.5

7.5

price

The stock price

Price1

Price2

Fig. 6: The stock price under GBM with risk-free

interest rate. The red line relies on σ1= 1.8and σ2=

1.2and the blue line relies on σ1= 2 and σ2= 1.2,

respectively.

0 20 40 60 80 100 120 140

time/day

0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

price

Call option price

Call1

Call2

Fig. 7: Call option price under GBM with risk-free

interest rate. The green line relies on σ1= 1.8and

σ2= 1.2and the blue line relies on σ1= 2 and σ2=

1.2, respectively.

3 Laplace transform of first hitting

time

3.1 Main results regarding first hitting time

First hitting time is the moment when a random

variable passes a given threshold for the first time.

For example, random walks leave a given space for

the first time. It can also be diffusion limited reac-

tions, polymer ring formation, stock market price dy-

namics, search problems, etc. The problem of lead-

time is an important part of stochastic process theory.

Stochastic process is the fundamental theory in finan-

cial mathematics, so the problem of first hitting time

also plays an important role in financial mathematics.

In the study of financial markets, especially the

pricing and risk management of path-dependent fi-

nancial derivatives, the problem of first hitting time is

almost always encountered. It is necessary to under-

stand the distribution of first hitting time. However, it

is not easy to obtain the distribution of the first hitting

time directly, so some scholars obtained the distribu-

tion of the first hitting time by studying the Laplace

transform of the first hitting time. More precisely, if

we set the density function of the first hitting time to

be f(t), then the Laplace transform of the first hitting

time is defined by

L(λ;x;l) = Ex[exp(−λτl)] = ∫+∞

f(t)e−λtdt,

. By inverse Laplace transform,

f(t) = 1

2πj ∫λ+j∞

λ−j∞

L(λ;x;l)eλtdt,

we get the density function of the first hitting time.

As the first hitting time problem is an important

subject in optimal stopping, the first hitting time prob-

lem of Geometric Brownian motion with broken drift

term is studied next. Then, set a constant level l≥0

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and define the first hitting time of Xfor touching lby

τl=inf{t≥0; Xt=l}. The Laplace transform of

τlis L(λ;x;l) = Ex[exp(−λτl)], where X0=xis

the initial point of Xand λis a parameter. Next lem-

ma provides the main result under Geometric Browni-

an motion about the Laplace transform of the first hit-

ting time, the proof can be seen in Borodin and Salmi-

nen.

Theorem 2. (Lemma in [31]) If Xtis the Geometric

Brownian motion, then the Laplace transform of first

hitting time of Xfor touching lis expressed by

L(λ;x;l) = Ex[exp(−λτl)] = f(x)

f(l),

where

f(x) = xr1+xr2

and

r1=(1 −2µ

σ2) + √(2µ

σ2−1)2+8λ

σ2

r2=(1 −2µ

σ2)−√(2µ

σ2−1)2+8λ

σ2

Further we consider the first hitting level lunder

Geometric Brownian motion with broken drift and

singular volatility, this immediately leads to the fol-

lowing theorem.

Theorem 3. If Xtis the Geometric Brownian motion

with broken drift and singular volatility, then the so-

lution to the S-L equation is expressed by

Case 1: if x≥l,

{f(x) = xr12 , l > x ≥a,

c1xr11 +c2xr12 , a < x < l. (6)

Case 2:if x < l,

{f(x) = xr21 , x > l ≥a,

b1xr12 +b2xr22 , l < a < x. (7)

where

c1=ar12 (r21 −r12)

ar11 (r21 −r11), c2=ar12 (r12 −r11)

ar21 (r21 −r11),

b1=ar21 (r22 −r21)

ar12 (r22 −r12), b2=ar21 (r21 −r12)

ar22 (r22 −r12),

and

r11 =

(1 −2µ1

σ2

1) + √(2µ1

σ2

1−1)2+8λ

σ2

r12 =

(1 −2µ2

σ2

2) + √(2µ2

σ2

2−1)2+8λ

σ2

r21 =

(1 −2µ1

σ2

1)−√(2µ1

σ2

1−1)2+8λ

σ2

r22 =

(1 −2µ2

σ2

2)−√(2µ2

σ2

2−1)2+8λ

σ2

Proof. First of all, for case 1 (X≥l), we have the fact

that the function and its derivative are continuous at

point of a, namely

ar12 =c1ar11 +c2ar21 ,(8)

r12a(r12−1) =c1r11a(r11−1) +c2r21a(r12 −1).(9)

By solving equations (8) and (9), we get

c1=ar12 (r21 −r12)

ar11 (r21 −r11),

c2=ar12 (r12 −r11)

ar21 (r21 −r11).

Recalling Lemma 2, under our model, we have the

above result.

On the other hand, for case 2 (X < l), we have the

fact that the function and its derivative are continuous

at point of a, namely

ar21 =b1ar12 +b2ar22 ,

r21a(r21−1) =b1r12a(r12−1) +b2r22a(r22 −1).

In a similar way, we have

b1=ar21 (r22 −r21)

ar12 (r22 −r12),

b2=ar21 (r21 −r12)

ar22 (r22 −r12).

Thus the proof is finished.

Theorem 4. The expectation of the first hitting time of

Geometric Brownian motion with singular volatility

term is infinite, i.e. Ex[τl] = ∞.

Proof. Recall the Laplace transform expression

Ex[e−λτl] = g(λ).

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Taking the derivative of both sides of this equation

with respect to λhave

Ex[−τle−λτl] = g′(λ).

Then, as λapproaches 0+, we have

Ex[τl] = −lim

λ→0+

g(λ)−g(0)

λ=−lim

λ→0+

f(l)−f(x)

λf(l).

Using the L'Hopital's rule implies the result.

3.2 Numerical results

At first, we observe the change of underlying mod-

el and the value of Laplace transform with different l.

For numerical results, the common coefficients are

a= 7, µ(t) = {0.125, Xt≥a,

0.15, Xt< a,

σ(t) = {1.1, Xt≥a,

1.5, Xt< a, X0= 8, λ = 0.8.

The boundary lare supposed to be l= 4.5and

l= 6.5, respectively. The Figure 8 & Figure 9 below

show the numerical simulation results. In this way,

the larger lis, the larger Laplace transform is, the s-

maller the first hitting time is.

10 20 30 40 50 60 70 80 90 100 110 120

time

0.015

0.02

0.025

0.03

0.035 The Laplace transform of first hitting time

10 20 30 40 50 60 70 80 90 100 110 120

time

The underlying model

Fig. 8: The figures are the Laplace transform of

first hitting time and the underlying model, respec-

tively. The figures rely on l= 4.5.

Then, we want to know the change of the under-

lying model and the value of Laplace transform by

changing the value of X0by using Matlab. For nu-

merical simulation, let

a= 7, µ(t) = {0.125, Xt≥a,

0.15, Xt< a,

σ(t) = {1.1, Xt≥a,

1.5, Xt< a, l= 6.5, λ = 0.8.

X0= 8 and X0= 13 are taken respectively. The Fig-

ure 10 and Figure 11 below show the numerical simu-

lation results. In this way, the larger X0is, the smaller

Laplace transform is, the larger the first hitting time

is.

10 20 30 40 50 60 70 80 90 100 110 120

time

0.02

0.025

0.03

0.035 The Laplace transform of first hitting time

10 20 30 40 50 60 70 80 90 100 110 120

time

11 The underlying model

Fig. 9: The figures are the Laplace transform of

first hitting time and the underlying model, respec-

tively. The figures rely on l= 6.5.

10 20 30 40 50 60 70 80 90 100 110 120

time

0.015

0.02

0.025

0.03

0.035 The Laplace transform of first hitting time

10 20 30 40 50 60 70 80 90 100 110 120

time

The underlying model

Fig.10: The figures are the Laplace transform of

first hitting time and the underlying model, respec-

tively. The figures rely on X0= 8.

10 20 30 40 50 60 70 80 90 100 110 120

time

0.02

0.025

0.03

0.035 The Laplace transform of first hitting time

10 20 30 40 50 60 70 80 90 100 110 120

time

11 The underlying model

Fig. 11: The figures are the Laplace transform of

first hitting time and the underlying model, respec-

tively. The figures rely on X0= 13.

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4 Conclusion

In this paper, the option pricing and Laplace trans-

form of first hitting time problem under Geometric

Brownian motion with singular volatility are studied

and the numerical simulations are carried out to ver-

ify the accuracy of the conclusions. The following

conclusions can be drawn:

(i)A rich body of literature on option pricing prob-

lem contain complex calculations and numerical re-

sults under complex models. Unlike their method-

s, we adopt probability scheme to directly get the

closed-form solution instead of previous methods. In

contemporary financial problems, especially in the

pricing of path-dependent financial derivatives.

(ii)We use the Sturm-Liouville equation and Eu-

ler's formula to get the Laplace transform of the first

hitting time. Furthermore, the density of the first hit-

ting time can be deduced by inverse Laplace trans-

form. In order to better depict the financial market,

Geometric Brownian motion model has been widely

used in the financial field, and there is a large room

for improvement. But most of the model are too ideal

to make the pricing problem difficult. Hence in our

paper we discuss Geometric Brownian motion with

singular volatility.

In summary, in the process of deducing op-

tion pricing we use probability scheme. In solving

Laplace transform of the first hitting time we use the

Sturm-Liouville equation and Euler's formula to de-

rive the closed-form solution, and then give some

main results. Compared with other numerical results,

our solutions are closed-form and accurate.

In the future, we will focus on introducing jump

or random volatility and extending our model to the

sticky OU or sticky CIR model to reflect more real-

istic phenomena, but need the density function of the

underlying models. This seems to be a big challenge

for future study.

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Contribution of Individual Authors to

the Creation of a Scientific Article

(Ghostwriting Policy)

Haoyan Zhang proposed the idea of the method

and checked the correctness of the manuscript. Yece

Zhou gave the method and calculated the option price.

Xuan Li wrote the article and provided the numeri-

cal results. Yinyin Wu computed the first hitting time

problem and polish the language.

Sources of Funding for Research

Presented in a Scientific Article or

Scientific Article Itself

This work is supported by the National Natural

Science Foundation of China (No.12101602).

Conflict of Interests

The authors declare that there is no conflict of in-

terests regarding the publication of this paper.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

_US

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.95

Haoyan Zhang, Yece Zhou, Xuan Li, Yinyin Wu

E-ISSN: 2224-2880

883

Volume 22, 2023