Evaluating the financial flows of Bessel processes by using spectral Analysis Текст научной статьи по специальности «Математика»

UDC 336.71

EVALUATING THE FiNANCiAL FLOWS OF BESSEL PROCESSES BY USiNG SPECTRAL ANALYSiS

© 2017 BURTNYAK I. V., MALYTSKA H. P.

UDC 336.71

Burtnyak I. V., Malytska H. P. Evaluating the Financial Flows of Bessel Processes by Using Spectral Analysis

The article solves the two-parameter task of evaluating the intensity of diffuse Bessel processes by the methods of spectral theory. In particular, barriers for cost of options, where the derivative of financial flows turns into zero, have been considered, and a task for the two-barrier option has been solved, which corresponds to Bessel process. A Green's function has been built for the diffusion Bessel process of the two-barrier option, decomposed according to the first-type system of Bessel functions. The barriers are taken in such a way that the derivative of financial flow in terms of price is turned to zero, i.e. there are the points where flow can acquire extreme values. On the basis of Green's function, the value of securities has been calculated. It is handier to use similar barriers when monitoring a stock market. The Green's function for this task, which represents the probability of spreading the option price, is represented through the Fourier series. This provides an opportunity to evaluate the intensity of financial flows in stock markets.

Keywords: spectral theory, barrier option, financial flows, Bessel functions, Green's function, singular parabolic operator, infinitesimal operator. Fig.: 1. Formulae: 12. Bibl.: 9.

Burtnyak Ivan V. - PhD (Economics), Associate Professor of the Department of Economic Cybernetics, Precarpathian National University named after V. Stefanyk (57 Shevchenka Str., Ivano-Frankivsk, 76018, Ukraine) E-mail: bvanya@meta.ua

Malytska Hanna P. - PhD (Physics and Mathematics), Associate Professor of the Department of Mathematical and Functional Analysis, Precarpathian National University named after V. Stefanyk (57 Shevchenka Str., Ivano-Frankivsk, 76018, Ukraine)

УДК 336.71

БуртнякI. В., Малицька Г. П. Оцнка фнансовихпотошв Бесел'юських процеав за допомогою спектрального анал'зу

У cmammi розв'язано двопараметричну задачу о^нювання 'нтенсив-Hocmi Бесел'вських дифузйних процеав методами спектральноi тео-рИ Зокрема, розглянуто бар'ери для вартостi опцюшв, в яких похiдна ф'тансових потошв перетворюеться в нуль, розв'язано задачу для двобар'ерного опцону, що вiдповiдае процесу Бесселя. Зд'шснено побу-дову функцИ Гр'на для дифузiйного процесу Бесселя двобар'ерного опцi-ону, яка розкладена по сисmемi функцй Бесселя першого роду. Бар'ери взятi таким чином, щоб у них похiдна ф'нансового потоку за цною перетворювалася в нуль, тобто це точки, де пот'ш може набувати екстремальних значень. На основi функцИГр'на проведено обчислення вартостi похiдних цнних папер'в. Для проведення мошторингу фондового ринку зручно використовувати саме так бар'ери. Функцю Гр'на цei задач'!, яка репрезентуе ймовiрнiсmь поширення цни опцону, представлено через ряди Фур'е. Це дае можливкть о^нити нтенсившсть ф'нансових потошв фондових риншв.

Ключов слова:спектральна теор'т, бар'ерний опцон, ф'нансов'1 потоки, функцИБесселя, функщя Гр'на, сингулярний парабол'мний оператор, 'нф'нтземальний оператор. Рис.: 1. Формул: 12. Б'бл.: 9.

Буртняк 1ван Володимирович - кандидат економiчних наук, доцент кафедри економiчноf юбернетики, Прикарпатський нацональний уш-верситет !м. В. Стефаника (вул. Шевченка, 57, iвано-Франкiвськ, 76018, Украна) E-mail: bvanya@meta.ua

Малицька Ганна Петрiвна - кандидат ф'вико-математичних наук, доцент кафедри математичного та функцюнального анал'ву, Прикарпатський нацональний ушверситет !м. В. Стефаника (вул. Шевченка, 57, iвано-Франкiвськ, 76018, Украна)

УДК 336.71

Буртняк И. В., Малицкая А. П. Оценка финансовых потоков процессов Бесселя с помощью спектрального анализа

В статье решена двухпараметрическая задача оценивания интенсивности диффузных процессов Бесселя методами спектральной теории. В частности, рассмотрены барьеры для стоимости опционов, где производная финансовых потоков превращается в ноль, решена задача для двухбарьерного опциона, что соответствует процессу Бесселя. Построена функция Грина для диффузионного процесса Бесселя двухбарьерного опциона, разложенная по системе функций Бесселя первого рода. Барьеры взяты таким образом, чтобы в них производная финансового потока по цене превращалась в ноль, то есть это точки, где поток может приобретать экстремальные значения. На основе функции Грина проведено вычисление стоимости ценных бумаг. Для проведения мониторинга фондового рынка удобно использовать именно такие барьеры. Функция Грина этой задачи, которая репрезентирует вероятность распространения цены опциона, представлена через ряды Фурье. Это дает возможность оценить интенсивность финансовых потоков фондовых рынков. Ключевые слова: спектральная теория, барьерный опцион, финансовые потоки, функции Бесселя, функция Грина, сингулярный параболический оператор, инфинитиземальный оператор. Рис.: 1. Формул: 12. Библ.: 9.

Буртняк Иван Владимирович - кандидат экономических наук, доцент кафедры экономической кибернетики, Прикарпатский национальный университет им. В. Стефаника (ул. Шевченко, 57, Ивано-Франковск, 76018, Украина) E-mail: bvanya@meta.ua

Малицкая Анна Петровна - кандидат физико-математических наук, доцент кафедры математического и функционального анализа, Прикарпатский национальный университет им. В. Стефаника (ул. Шевченко, 57, Ивано-Франковск, 76018, Украина)

Bessel processes play an important role in financial mathematics, since they are inherently closely related to models of geometric Brownian motion and Cox-Ingresol-Ross processes [1]. We are interested in considering those Bessel processes, which present generalization of the Ornstein-Uhlenbeck process for barrier options [2]. At certain characteristics, the diffusion process with the Bessel operator never hits zero, and a number of papers [3] are dedi-

cated to these cases. We consider those cases when the derivative of the financial flow of the Bessel process can hit zero. These conditions are used to determine the excessive growth rate of the stock portfolio as well as explain how exceeding the growth rate of the market portfolio provides a measure of internal volatility in the market at any given time [4].

A significant part of the problems of financial mathematics are described by diffusion processes or stochastic

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differential equations. In 1998, Kaufman showed that the Bessel diffusion {Z, t > 0} with constant negative drift and an infinitesimal generator has the form [5]:

(Of)(z) = 1 a 2 f" (z) + | d + cj f (z).

Problem statement. The spectral method is applied to derivative financial instruments, in particular there presented the price for the derivative u(t, x) through a function that is neutral to the risk of expecting the future value of the real-valued process X, that is as

u(t, x) = Ex[H(Xt)] = JH(y)p(t, x, y)dy, where p(t, x, y) - transition density ofX with the probability p. If the infinitesimal generator L of the real-valued process is self-ajoint in the Hilbert space with the increment of the measure m(x) dx, and the L - spectrum is discrete, then the transition density of X is developing with respect to its own functions [6]:

P(t, x, y) = m(y)> ne~Xnt<pn (y)<p„ (x), where {A,B} - eigenvalues, L{pn} - eigenfunctions: therefore,

= K

Let us consider the process for which the operator L has the form:

L = d2xx + x ldx - x 2p2,

w"+-

-w' + w = 0.

The solution of the resulting equation is a power series that is absolutely convergent for all x e (— oo; oo) and has the form:

(x /2)

r( p+1) +

(— 1)m (x /2) p+2m

(1)

where p - constant value called an index, x > 0.

We should note that L is a singular parabolic operator, an infinitesimal one, to which a number of operators where ff2 = 2x2 is reduced, L is called the Bessel operator.

Let us study L for eigenvalues and eigenfunctions using the Sturm - Liouville theory, thus Lv = -l2v, we obtain

v' p2 2

the equation v" +----r- v = —A v, after multiplying by

x x 2

we have

~ 2...m(p + 1)...(p + m)r(p + 1)

where r - gamma function. By transforming (3) on the basis of properties of the gamma function, we obtain a Bessel function of the first kind of the p-th order:

w , V (—!)m (x /2)p+2m

Jp (x) = > -.

pW m=0 r( m +1)r( p + m +1)

Note. Since equation (2) contains p2, the substitution of p for (-p) does not influence the solution of the equation, thus there exists a solution for any value of p.

Ifp is not an integer, then the Bessel functions cannot be linearly dependent and the general integral of equation (2) has the form:

J = cJp (x) + C J— p (x).

With an integer p we find one more partial solution:

Jp (x) cos p ft + J—p (x)

Yp(x) = : , F sin p ft

which is expressed by a Bessel function of the second kind that is undefined at x = 0. Using L'Hopital's rule we find the boundary for x ^ 0 and by this value define the function at zero:

x

Yo =- Jo(x)^ln^ + C I —

^^ 2 (—1)m(x/2)

2m

K

m=1

1+2+1+..+11

2 3 m

x2 v" + xv' + (A2 x2 — p 2 )v = 0. (2)

Equation (2) is a Bessel equation with the parameter A.

The solution of equation (2), except for the partial values of p, is not expressed in terms of elementary functions (in the finite form), these non-elementary functions are called Bessel functions, they are widely used in economics, technology and physics. Since the Euler-Bes-sel equation is a linear one, its total integral can be put in the form

v = qvi + 02^

where v1, v2 are any two linearly independent partial solutions of the Euler - Bessel equation, and C1, C2 are arbitrary constants.

In the case ofp > 0 we make a substitution v = xpw and obtain for the function w the following equation: 2 p + 1 ,

For any value of p the following formulas can be used: d (xpJp (x)) = xpJp—1( x),

dx (x—PJp (x)) = x—pJp+i (x).

The Bessel functions J,(Ax), Jp(^x) where I and |i are the roots of the equation Jp(x) = 0 are orthogonal on the interval [0, 1] with the weight x, thus

/0 xJp (Xx)Jp (¡ax)dx = 0, A ^ and if A = yU, the two cases are possible:

/ xJp(Ax)dx = 0

1Jp2( A), Jp (A):

2\ 1—^ A2

J2p (A), Jp (A) = 0.

For all a, ß > 0, a + ß > 0 there exists a countable set of positive roots

av'k (¡) + ßjuvk (¡) = 0,

whose boundary point is at infinity.

If v(x) is the solution of (2), then the function v(Ax) will also be the solution of the equation of the following form:

x2 v" + xv' + (A2 x2 — p2 )v = 0. (4)

Equation (4) is a Bessel equation with the parameter A. Any solution of equation (2) expressed by a Bessel func-

1

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tion has an infinite set of positive roots that are close to the roots of the function sin (x + a), which has the form: kn = nn - a, a = const, n - an integer (it is similar for negative roots, because they are symmetrical relative to the origin of coordinates), if kn ? 0 they are simple roots and form a countable set [7].

Since Bessel functions are alternating series, the calculation of values can be performed using the Leibniz lemma, which makes it possible to determine the accuracy of the approximation.

To find the eigenfunctions and eigenvalues, let us consider the following boundary value problem:

(5)

x2vk + xvk + (A2kx2 - P>k = 0:

kl=o <+œ'

avk (x0 ) + Pvk (xo ) =

(6) (7)

-P 2)vk

:0,

KU <+œ'

v'k ( Xo ) = °.

(8)

From (6) it follows that p > 0. Let us consider the case with p > 0, since (2) has as its integral v = C1Jp(x) + C2Y (x), then, based on the properties of the Bessel function, problem (8) has the following solution:

v = C1 Jp (Ax) + C2Yp (Ax).

Taking into account the boundary conditions we will have C2 = 0 and C1Jp(Xkx0) = 0, thus Jp(Xkx0) = 0, therefore

Xkx0 = jk, where Ak = —, 0 < j1 < j2 < ... < jk< ..., where x0

jkare the roots of J'p (Ak) = 0. The norm of vk(x) will have the form:

Il r J2 1

IK (x)|| = ^

2 Vk

(Jp (Vk ))2, k = 1, 2,

Let us consider the case with p = 0. Thus we have the problem:

( xv') + A2 xv = 0,

Hx=0 <+^

v'( X0 ) = 0.

(9)

With X = 0 the solution is v(x) = 1, then it foll ows

that X = 0 is the eigenvalue and v0 (x) = 1 is the eigenfunc-tion.

Let us consider X > 0. As in the previous case we will

find Vk(x) = J0

/ \

V \X0 )

k = 1, 2,..., Ak = , where ¡i

are positive roots of the equation J'p (a) = 0, in this case

the norm vk (x) will be equal to:

2 x 2

v (x)|| = -2-(J0(a))2, k = 1,2,...

Results of the research. Let us consider the Bessel process described by the equation:

dv(t, x) d2v(t, x) -1 dv(t, x) _-;--h x

dt

dx

dx

(10)

2 —2

-p x v(t, x), 0 < x < x0,

Thus, we are considering a Sturm - Liouville problem. The given problem has a unique solution. We impose condition (6) because x = 0 is a special point of equation (5) and the operator L. x0 is a regular point of equation (5). The values of Xk with which the boundary value problem (5) - (7) has a non-trivial solution vk are called eigenvalues, and vk - eigenfunctions of the problem. It is known that under conditions (6) the operator L has a countable number of eigenvalues, they are simple and not negative [8]. The multiplication of L by x2 does not change either the eigenvalues, the eigenfunctions, or their quantity.

Let us consider the following problem:

x2 v'k + xvk + (A2 x2-

and the boundary condition

v(0, x) = K(ex — 1)+, vx (t, x0) = 0, (11) where K is a strike value. The process is homogeneous, therefore, v(t, x) = p(t) v(x).

From the Sturm - Liouville theory we have:

v(t, x) =2

n=1

2

Vn, —t

c e x0 J u np e u p

Vn

+ c0, p > 0, Vn > 0,

where jn - positive roots of the equation are calculated by the formula with p > 0, c0p = 0 ifp = 0 then

K1 /0x° x(ex — 1) dx K1

K11 x0e'

ex0 +1-

0 p

/0x0 xdx

: K1(2x—1ex0 — 2x—2ex0 + 2x-2 —1), K1 = e-

K /0x0 x( ex — 1) J.

Vn

dx

cnp

/00 xJp

Vn

dx

The financial flows have the following form: u(t, x) = > Kc

ln K I

(T—t )

n=1

np

Jp I Vnlnk |.

In the case when the process is completed at time T,

when XT = K:

u(t, x) =2 Kcne

n=0

\2

Vn

ln t

(T—t )

Jp

Vn| ln

ln

R

' L

where L < x < R, L, R - barriers, K - strike value, and cnp are calculated as follows:

np

cnp = 2K

/01 ( eKt —1) Jp ( Vnt ) dt

Jp+1(Vn )

We have calculated the expansion of the financial flow in terms of the system of Bessel functions Jp of the first kind, while the distribution of the flows is set by the Green function of the corresponding problem. Therefore, for the cal-

X

0

2

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culations it is convenient to expand the Green function in terms of the system of Bessel functions. The process that we consider can be expressed by a correspondent inhomoge-neous boundary value problem:

du d2u _i du p2u (t, x) „ ,

— = — + x V - 2 + f(t, x), x > 0, (12)

dt dx2 dx x2

where f(t, x) is twice continuously differentiable in x and continuously differentiable in t, absolutely integrable with

the derivatives, (t, x) e [0, + and is expressed as follows:

/ \

Pnx

we have:

*(t, x) = 2 /(

t „_K(t_ß

7oxo Sf(S, t) Jp

i

p.n S

n=0

/ \/

X

V xo

Xd ßd S J p

Pnx

\ x0 I

/

/ox0 ^J 2

Pnx

_i

dy

\x0 I I

f (t, x) =2 fn (t) Jp

n=0

0 < x < x0 <+œ, 0 < t < T,

= f x0 ft 2 f 0 f0¿

n_0

Pn S

yJ

P.y

\ \_2

V x0 I I

0I

XS Jp

therefore,

Jp

\ x0 I

Pnx

\ x0 I

dy

f (S, t ) d Sd ß,

e_X»(t_ß) X

where « are the roots of the equation J (u ) = 0.

tion:

The problem can be solved using the following equa- G(t — S, x, £) = 2 J

u (t, x) = 2 t. (t) Jp

n=0

/ \ Pnx

By substituting (12) we obtain:

2 T(t) Jpl^^ 1=2

Pnx

Pnx

_¿2 Jpl^lT (t) +2 fn (t) Jp!^

then

I x0 I , ,2, |Pnx --2 +AnJp

Pnx

2 [Tn'(t) + (t) _ fn (t)] Jp

/ \ Pnx

n= 0

= 0,

Pn

therefore, u(t, x) =2 /0e_A2(t_ß) fn(ß)dß J

/ \ Pnx

n_0

Taking into account that

fn (t) = Lx° Sfn (S, t) Jp

I

Pn S

d S

/0° xJ;

Pnx

\_1

dx

n=0

Pn S

Jp

/ /

X

2V

\ v

x2 x0 2 Pi

V x0 / _i

Pnx

_i (t_ß)

? x0 X

V x0 /

( Jp ( Pi ))2

therefore, T; (t) +(t) — / (t) = 0, Xn =LJL, n e W

x0

with the initial condition Tn(0) = 0.

The inhomogeneous differential equation of the first order is solved by the method of constant variation. Since

T^ (t) + l2nTn (t) = 0 has the first integral Tn (t) = Ce—^ (the solution of a inhomogeneous equation), then

Tn (t) = C(t) e—'^, thus

C'(t) =/«(t)e#, C(t) = /0eA«S/n(j3)dS + Cj.

Tn(t) = /0eS/n(S)d^e—Ani +Cle~x2nt with t = 0, Cj = 0,

Tn (t) = /0e—"n (t—S) /n ( S) d S,

«(t, x) = /0'G(i-r, x, t)/(r, t)d£.

Since the problem of evaluating and studying two-

dimensional barrier options is reduced to considering and

solving boundary value problem [9]

2 2 a«(t, x) d « (t, x) + -1 d«(t, x) p «(t, x)

at dx2 dx x '

x e [L,H], t e [0, T],

«x (t,L) = 0, «x(t, H) = 0,

«x (T, x) = max (± (x (T) - KX 0) [ (L<x(t)<H; t e[0,T]).

This problem is reduced to solving the boundary value problem for the singular parabolic equation:

a« a2« — a« p2« (t, x)

T7 = 2 + y ---2-,

at ay2 ay y2 y = lnx, y e [A, B], t e [0, T], A = lnL, B = lnH,

«x (t, A) = 0, «x (t, B) = 0, «x (0, y) = y(ey(T)) =

= max (±( x(T ) - K ), 0) [ (L<x (t )<h ; te[0,T ].

Taking into account all the considerations as to the solution of classical boundary value problems for the singular parabolic operator L, we have:

« (T, x) = /0n H (etL - K) | (L<x(t)<H; te[0, t]G(x, t) d 1 =

/00'L (e S L _ r )

(L<x(í)<ff ; í6[0, T]),

\2

Pn

2 2 e

n=0

J

Pn S

ln

H

\

J

Pn ln

ln

H

X

X

//

v

ln H\2 _ \

L

Pi

_i

( Jp ( Pi ))2

where is the [[(L < x(í) < H t e [0 T]) Heaviside step function.

<

OQ 2

o

ZT

o

o

<

o

U

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so

00

00

\

00

Note. Since the roots of the Bessel functions of the first kind are simple, then between two neighboring roots of the Bessel functions there is a derivative root and we can assume that the roots of the derivative are distributed similarly to the roots of the functions. Thus for the Green's function and its first derivative the correct evaluation is

C > c0 ln2 t <+v Vx e [L, H ], 0 < t < T, C > 0, c0 > 0.

n=1

Approximate calculations do not require a large number of coefficients in a row because of the rapid convergence.

0.006 -| 0.005 -0.004 -0.003 -0.002 -0.001 -0 -4

^ G(t, x, S)|t = , G(t, x, ^ = 3 ^ G(t, x, ^ = ,2

Fig. 1. Graph of the Green's function as the distribution density at L = 90, H = 120, f = 0,5

It should be noted that the Bessel diffusion is widely used in financial mathematics. Thus, the study considers the problem for the two-barrier option, which corresponds to the Bessel process.

CONCLUSIONS

Thus, as a result of this study, a Green's function for the Bessel diffusion process of a two-barrier option expanded in terms of Bessel functions of the first kind is built. As barriers there chosen the points at which the derivative of the financial flow by price is equal to zero, that is, the points where the flow can take extreme values. By means of the Green's function the value of derivative prices are calculated. Barriers of this kind are convenient for stock market monitoring. ■

7. Lebedev, N. N. Special Functions and Their Applications. New York: Dover, 1972.

8. Владимиров В. С. Уравнения математической физики. М.: Наука, 1981. -512 с.

9. Mendoza-Arriaga, R., Carr, P. and V. Linetsky. Time-changed markov processes in unified credit-equity modeling. Mathematical Finance. 2010. Vol. 20. Issue 4. P. 527-569.

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Cox, J. C., Ingersoll, J. E., and Ross, S. A. "A theory of the term structure of interest rates". Econometrica. Vol. 53 (1985): 385-407.

Linetsky, V. "The spectral representation of Bessel processes with constant drift: applications in queueing and finance". Journal of Applied Probability. Vol. 41, no. 2 (2004): 327-344.

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LITERATURE

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