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Commun Nonlinear Sci Numer Simulat 16 (2011) 1752–1759
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Commun Nonlinear Sci Numer Simulat
journal homepage: www.elsevier .com/locate /cnsns
Algebraic solution of the Stein–Stein model for stochastic volatility
C. Sophocleous a, J.G. O’Hara b, P.G.L. Leach a,c,⇑a Department of Mathematics and Statistics, University of Cyprus, Lefkosia 1678, Cyprusb CCFEA, University of Essex, Wivenhoe Park, CO4 3SQ England, UKc School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa
a r t i c l e i n f o
Article history:Received 8 April 2010Accepted 6 August 2010Available online 12 August 2010
Keywords:Symmetry analysisSimilarity solutionsStochastic processesNonlinear evolution equations
1007-5704/$ - see front matter � 2010 Elsevier B.Vdoi:10.1016/j.cnsns.2010.08.008
⇑ Corresponding author at: School of Mathematic260 1008; fax: +27 31 260 2632.
E-mail addresses: [email protected] (C. Sopho(P.G.L. Leach).
a b s t r a c t
We provide an algebraic approach to the solution of the Stein–Stein model for stochasticvolatility which arises in the determination of the Radon–Nikodym density of the minimalentropy of the martingale measure. We extend our investigation to the case in which theparameters of the model are time-dependent. Our algorithmic approach obviates the needfor Ansätze for the structure of the solution.
� 2010 Elsevier B.V. All rights reserved.
1. Introduction
The Black–Scholes model for the pricing of derivatives and its use in risk management has enjoyed considerable popular-ity in the financial markets. However, one of the weaknesses of this classical model is the treatment of the volatility of theunderlying asset as a constant over the lifetime of the derivative contract. Precise measurement of volatility is central to thepricing of options. Many models have been developed to overcome this restrictive and unrealistic assumption, in particularthe idea of stochastic volatility has proved to be popular and in many cases gives valuable insight. The most established mod-els are those of Hull and White [9], Heston [8] and Stein and Stein [19]. Each model has its advantages and disadvantages.
In this article we focus upon the model of Stein and Stein. In this paradigm we assume that the volatility follows an arith-metic Ornstein–Uhlenbeck process. The volatility reverts to a long-run mean and the supporting Brownian motions, whichdrive the randomness of the volatility and the underlying security, are assumed to be independent. The advantage to thisapproach is that, although the calculations are awkward, they require only modest processing power.
Benth and Karlsen [3] demonstrated that the Radon–Nikodym density of the minimal entropy of the martingale measurecould be given in terms of the solution of a nonlinear (1 + 1) evolution partial differential equation. Both existence anduniqueness of the solution were established. A particular instance, known as the Stein–Stein model for stochastic volatility[19], of the equation,
2ut þ b2uxx � b2 1� q2� �
u2x þ 2 m� aþ nbqð Þxð Þux þ n2x2 ¼ 0; ð1:1Þ
where a, b, q and n are constants, was shown to have the solution
uðt; xÞ ¼ aðtÞx2 þ bðtÞxþ cðtÞ; ð1:2Þ
. All rights reserved.
al Sciences, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa. Tel.: +27 31
cleous), [email protected] (J.G. O’Hara), [email protected], [email protected], [email protected]
C. Sophocleous et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 1752–1759 1753
where the time-dependent functions satisfy some rather complicated relations, for the problem of a terminal condition givenby u(T,x) = 0.
Subsequently the problem was revisited by Kufakunesu [11] who added an interesting development to the model repre-sented by (1.1) in the sense that the parameters in the equation, apart from m, were taken to be explicitly time-dependent.Kufakunesu was able to construct a solution rather similar in form to that in (1.2) for the same terminal condition. In thegeneral treatment of Benth and Karlsen the parameters in (1.1), apart from b and q, could depend upon x so that the ap-proach of Kufakunesu marked a different philosophy in the approach to the nature of the variable dependence of the param-eters of the model.
The mathematical apparatus used by Benth and Karlsen on the one hand and Kufakunesu on the other is rooted in func-tional analysis. Our approach is quite different. We seek the conditions under which an equation of the form (1.1) possesses asufficient number of Lie point symmetries to be able to accommodate the terminal condition and then to construct the solu-tion of (1.1) subject to u(T,x) = 0 in the usual approach of reduction in the Lie theory. This approach has been successfullyused for a variety of equations arising in Financial Mathematics as it seems to be part of the nature of the modelling processto develop evolution partial differential equations rather redolent with symmetry. Some examples the application of the Lietheory to equations which arise in Financial Mathematics are to be found in [10,7,13,1,14–18,6]. The calculation of Lie sym-metries is a tedious task even for elementary equations and we make use of one of the packages available to enable the com-puter to deal with the tedium. Because of the number of parameters, subsequently unspecified functions of time, aninteractive approach is to be preferred. We make use of the Mathematica-based package Sym [4,5,2].
In a successful application of the Lie theory to evolution partial differential equations, such as of the type being consideredin this paper, one finds for a (1 + 1) equation a sufficient number of Lie point symmetries to accommodate the additionalconstraints imposed by the terminal condition.1 In the case of linear (1 + 1) evolution partial differential equations thereare always two Lie point symmetries, namely u@u and f(t,x)@u, where the latter is a solution of the equation and gives rise toan infinite-dimensional abelian subalgebra. Obviously the former reflects the homogeneity of the equation in the dependentvariable u(t,x). Any additional symmetries are determined by the coefficients of the various terms involving the dependent var-iable and its derivatives. When one considers nonlinear (1 + 1) evolution partial differential equations, there are two possibil-ities. In one of them the infinite-dimensional subalgebra of solution symmetries disappears completely. In the other the infinite-dimensional subalgebra persists, but the coefficient functions, f(t,x), are solutions of a linear (1 + 1) evolution partial differentialequation and one finds a function of u multiplied by f in the symmetry. That function gives the clue for the linearising trans-formation. In other words the second class of equation is a linear equation in disguise.
In our investigation of the symmetries of (1.1) and its generalisation with time-dependent coefficients we find that bothpossibilities occur. Eq. (1.1), as it stands, is essentially Burgers equation in potential form and so is transformable to the stan-dard form of the classical heat equation. When we consider the generalisation to time-dependent coefficients, the situation isnot so simple as we see below. One should emphasise that the simple fact that the differential equation can be transformedto a standard equation is not sufficient to dismiss the current problem, whether it be this one or some other, since we havethe associated terminal condition. In the present form of the problem the terminal condition is quite simple. After a series oftransformations one may find that the penalty is to deal with a very complex terminal condition.
In the next Section we examine (1.1) for its Lie point symmetries on the assumption that the parameters are constants.We see that there are two symmetries which are compatible with the terminal condition. It is necessary only to use one ofthese symmetries to determine the solution of the stated problem. Fortunately we are assured that the solution is unique dueto the Fokker–Planck Theorem. In Section 3, we follow the generalisation of Kufakunesu in admitting the possibility that allof the parameters in (1.1) are functions of time. We find that there are two possibilities and discuss them in Sections 3.1 and3.2. We conclude the paper with some observations in Section 4.
2. Symmetry analysis of (1.1)
Due to the number of parameters in (1.1) it is preferable to use Sym in interactive mode. We find that (1.1) possesses a Liepoint symmetry of the form
1 Onewhen tdeterm
C ¼ aðtÞ@t þ12
_axþ bðtÞ� �
@x þ Gðt; xÞ þ 11� q2 exp 1� q2� �
u� �
Fðt; xÞ�
@u; ð2:1Þ
where
2Ft þ b2Fxx þ 2 m� ðaþ bnqÞx½ �Fx � 1� q2� �n2x2F ¼ 0; ð2:2Þ
and provides an infinite number of symmetries for (1.1) which indicates that the equation is linearisable. The function G(t,x)is given by
must bear in mind that the statement ‘subject to u(T,x) = 0’ is actually two conditions. The first condition is that t = T. The second condition is that u = 0= T, where one is reminded that in the symmetry analysis the dependent variable, u(t,x), is treated as an independent variable when one is analysing theining equations.
1754 C. Sophocleous et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 1752–1759
Gðt; xÞ ¼ g½t� � 14b2 1� q2ð Þ
€ax2 � 2ðm� ðaþ bnqÞxÞx _aþ 4x _bþ 4ðaþ bnqÞxbh i
; ð2:3Þ
and the functions a(t), b(t) and g(t) satisfy the system of equations
av�4K2 _a ¼ 0; ð2:4Þ
€b� K2b ¼ �32ðaþ bnqÞm _a; ð2:5Þ
_g ¼ 14b2 1� q2ð Þ
b2€a� 2 m2 � b2ðaþ bnqÞ� �
_aþ 4m _bþ 4mðaþ bnqÞbh i
; ð2:6Þ
where K2 = a2 + 2abnq + b2n2.The solutions of (2.4), (2.5) and (2.6) are
aðtÞ ¼ N1 þ N2 cosh 2Kt þ N3 sinh 2Kt; ð2:7Þ
bðtÞ ¼ N4 cosh Kt þ N5 sinh Kt �mK
aþ bnqð ÞðN3 cosh 2Kt þ N2 sinh 2KtÞ; ð2:8Þ
gðtÞ ¼ N6 þm
b2K 1� q2ð ÞKN4 þ ðaþ bnqÞN5½ � cosh Kt þ KN5 þ ðaþ bnqÞN4½ � sinh Ktf g
þ 12b2K2 1� q2ð Þ
f½ðK2ðab2 þ b3nq� 2m2Þ þ b2m2n2ð1� q2ÞÞN2 þ Kðb2K2 � 2m2ðaþ bnqÞÞN3� cosh 2Kt
þ ½ðK2ðab2 þ b3nq� 2m2Þ þ b2m2n2ð1� q2ÞÞN3 þ Kðb2K2 � 2m2ðaþ bnqÞÞN2� sinh 2Ktg: ð2:9Þ
The algebra of the symmetries (1.1) is {sl(2,R) � sW} � s1A1 which is the usual algebra of an (1 + 1) evolution partial dif-ferential equation of maximal symmetry comprising three subalgebras. These are sl(2,R) the elements of which are deter-mined by the solutions of (2.4) and their contributions to the particular solutions of (2.5) and (2.6), the Weyl–Heisenbergalgebra, W, the three elements of which are given by the two complementary solutions of (2.5) and their contributions tothe particular solution of (2.6) and the constant solution of (2.6) and finally the infinite-dimensional abelian subalgebra gen-erated by the solutions of (2.2). A feature of the algebraic structure of equations of this type is that the Lie Bracket of an ele-ment of the finite-dimensional subalgebra with the infinite-dimensional abelian subalgebra generates another solution of(2.2). The linearisation of (1.1) leads to (2.2).
In the case of the Stein–Stein problem we require the symmetry given in (2.1) to be consistent with the dual conditions,t = T and u(T,x) = 0. Given the mess of the explicit expressions for the coefficient functions in (2.7), (2.8) and (2.9), it is some-thing of a relief to see that the dual conditions simply require that
aðTÞ ¼ 0 and GðT; xÞ ¼ 0; ð2:10Þ
since u does not appear in the finite-dimensional subalgebra. Since G is a quadratic expression in the free variable, x, (2.10)imposes four constraints upon the six constants in the expression for the general form of the symmetry, Ni, i = 1,6, whichmeans that we obtain two symmetries which are consistent with the terminal condition. The additional constraints fromthe separation of G(T,x) by coefficients of separate powers of x are
12
€aþ ðaþ bnqÞ _a ¼ 0; ð2:11Þ
_bþ ðaþ bnqÞb� 12
m _a ¼ 0; ð2:12Þ
g ¼ 0; ð2:13Þ
all evaluated at t = T. After what is unfortunately not a little algebra we find that the symmetries are
C1 ¼ K2 cosh½KðT � tÞ� þ Kðaþ bnqÞ sinh½KðT � tÞ�h i
@x
þ n2 �mþm cosh½KðT � tÞ� þ Kx sinh½KðT � tÞ�½ �@u; and ð2:14Þ
C2¼ 1�cosh½2KðT� tÞ�þsinh½2KðT� tÞ�½ �@tþ �ðb2K2þm2ðaþbnqÞÞcosh½KðT� tÞ�
2mðaþbnqÞ þ �mþ K2xaþbnq
!cosh½2KðT� tÞ�
þðb2K2þm2ðaþbnqÞÞsinh½KðT� tÞ�
2mðaþbnRÞ þ m� K2xaþbnq
!sinh½2KðT� tÞ�
!@x�
nðb2K2�am2�bnm2qþ2K2mxÞ2bK2q
�ðcosh½KðT� tÞ��cosh½2KðT� tÞ��sinh½KðT� tÞ�þsinh½2KðT� tÞ�Þ@u; ð2:15Þ
in which the two independent parameters are taken to be N6 and N1, respectively. Since C2 is constructed from elements ofthe sl(2,R) subalgebra and C1 from the elements of the Weyl–Heisenberg subalgebra and the algebraic structure of the finite
C. Sophocleous et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 1752–1759 1755
subalgebra of (1.1) is sl(2,R) � sW, it follows that [C1,C2]LB = constC1 and so we use C1 to reduce (1.1) to a first-order ordin-ary differential equation which we then solve to give the solution of the problem of (1.1) coupled with the terminalcondition.
The associated Lagrange’s system for (2.14) is
dt0¼ dx� K
n2mK cosh KðT � tÞ þ ðaþ bnqÞ sinh KðT � tÞ½ �
¼ du1� cosh KðT � tÞ � x K
m sinh KðT � tÞ; ð2:16Þ
and this gives the invariants to be
v ¼ t; and ð2:17Þ
w ¼ uþn2m 2ð1� cosh KðT � tÞÞx� x2 K
m sinh KðT � tÞ� �
2K K cosh KðT � tÞ þ ðaþ bnqÞ sinh KðT � tÞ½ � : ð2:18Þ
We write
u ¼ f ðtÞ �n2m 2ð1� cosh KðT � tÞÞx� x2 K
m sinh KðT � tÞ� �
2K K cosh KðT � tÞ þ ðaþ bnqÞ sinh KðT � tÞ½ � ; ð2:19Þ
so that (1.1) becomes
2 _f þ 1
pðtÞ22mpðtÞqðtÞ þ b2 pðtÞsðtÞ � ð1� q2ÞqðtÞ2
�h i¼ 0; ð2:20Þ
and the solution follows from a simple quadrature. The actual expression is distinguished by is length rather than any intrin-sic complexity and is not written here. The constant of integration is determined by the requirement that f(T) = 0.
Since the solution of the problem (1.1) subject to this terminal condition is unique as a consequence of the Fokker–PlanckTheorem, the solution which would be obtained using the more complicated symmetry, C2, is the same as derived above.
In [3], a solution of the terminal problem for (1.1) as treated here is assumed to have the form
uðt; xÞ ¼ aðtÞx2 þ bðtÞxþ cðtÞ; ð2:21Þ
and the coefficients are determined by requiring that the equation and terminal condition be satisfied. Without an a prioriknowledge of the x dependence in the solution such a process of assumption has every chance of being quite unsuccessful.What we have demonstrated here is the fact that an analysis of the differential equation in question, in this case (1.1), for itsLie point symmetries provides an algorithmic route to the determination of the structure of the solution and its precise res-olution by using the standard method of reduction of order of the Lie theory.
As a side result to our calculation of the solution of the terminal problem associated with (1.1) we have seen an interest-ing split of the algebra. The finite subalgebras of (1.1), sl(2,R) and W, both contribute to the symmetries which are compatiblewith the terminal condition. The Weyl–Heisenberg subalgebra leads to C1 and the special linear group in two dimensions toC2. Both symmetries lead to the unique solution. The path provided by the former is the easier route to take. One is remindedof the manner in which symmetry is lost when more complicated terms enter into (1 + 1) evolution partial differential equa-tions. The first symmetries to be lost are two of the elements of W. These two elements are peculiar to the equation underdiscussion. Oftentimes they are called ‘solution symmetries’ since they correspond to the solution symmetries of the Noethersymmetries for the corresponding Lagrangian of a one-degree-of-freedom system. Basically, their presence means that thedifferential equation is simpler and it would appear that the greater ease of solution is one of the benefits of that augmentedsimplicity.
3. Time-dependent models
Kufakunesu [11] introduced a model based upon (1.1) in which all of the parameters apart from m were explicitly time-dependent. He was able to show that there existed a solution to the problem with the terminal condition as above withoutany constraint upon the nature of the time-dependence in the functions. Indeed, the structure of the solution is the same asfor the autonomous problem discussed by Benth and Karlsen [3] ((2.21) above) and our solution. In our analysis, we find thatthere is a natural bifurcation in the process of the calculation of the Lie point symmetries and we treat them separately. Thecritical difference is whether or not q(t) is a constant or not. When it is a constant, the equation is still linearisible by meansof a point transformation. If q(t) is not a constant, this linearisation is no longer possible.
3.1. The case q a constant
The equation we analyse is
2ut þ bðtÞ2uxx � bðtÞ2 1� q2� �
u2x þ 2 m� aðtÞ þ nðtÞbðtÞqð Þxð Þux þ nðtÞ2x2 ¼ 0: ð3:1Þ
In the course of our analysis using Sym in interactive mode we find that the general form of the symmetry of (3.1) is
1756 C. Sophocleous et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 1752–1759
C ¼ aðtÞ@t þ bðtÞ þ x
2b2
ddt
aðtÞbðtÞ2 �� �
@x þ1
1� q2 Fðt; xÞ exp 1� q2� �u
� �þ Gðt; xÞ
� �@u; ð3:2Þ
where
avþ4UðtÞ _aþ 2 _UðtÞa ¼ 0; ð3:3Þ
UðtÞ ¼ �a2 � 2abnq� b2n2 þ _a� _bnqþ b _nq� 2a _bb� 2 _b2
b2 þ€bb;
€bþ b2 ddt
1b2
� �_bþ UðtÞ � b
d2
dt2
1b
� � !¼ WðtÞ; ð3:4Þ
WðtÞ ¼ 32
m _a aþ bnqþ 2_bb
!þmab
ddt
ab
� �þ d2
dt2
1b
� � !;
_g ¼ 11� q2
14
€aþ 12
aþ bnqþ_bb�m2
b2
!_aþ 1
2ddt
aþ bnq�m2
b2 þ_bb
!aþm _b
b2 þm
b2 ðaþ bnqÞb( )
; ð3:5Þ
2Ft þ bðtÞ2Fxx þ 2 m� ðaðtÞ þ bðtÞnðtÞqÞx½ �Fx � 1� q2� �n2x2F ¼ 0; and ð3:6Þ
Gðt; xÞ ¼ gðtÞ � 12 1� q2ð Þ
1b
2 ddt
12
_aþ aðaþ bnqÞ þ a _bb
!" #x2 þ 2b
aþ bnqb2 þ 2 _b
b2 �ma
b2
" #x
( ): ð3:7Þ
One can make some observations on the mathematical aspects of the equations defining the coefficient functions of thesymmetries. Eq. (3.3) is a third-order equation of maximal symmetry and as such its solution may be written as
aðtÞ ¼ C1pðtÞ2 þ 2C2pðtÞqðtÞ þ C3qðtÞ2; ð3:8Þ
where p(t) and q(t) any two linearly independent solutions of
€zþUz ¼ 0: ð3:9Þ
Eq. (3.4) is not in normal form. It is rendered into normal form by means of the transformation bðtÞ ! hðtÞbðtÞ. Then it is
€hþUðtÞh ¼ WðtÞbðtÞ ; ð3:10Þ
i.e., the complementary solution can be written in the same form as the solution of (3.9).As in the case of (1.1) the application of the general symmetry to the terminal conditions, t = T and u(T,x) = 0, leads to four
equations in the six parameters in the symmetry. The nature of the conditions is still such that the elements of the sl(2,R)subalgebra and those of the Weyl–Heisenberg subalgebra are separated. The general symmetry of the latter may be writtenas
CW ¼ bðtÞ@x þ gðtÞ � aþ bnq1� q2ð Þb2 bðtÞx
" #@u; ð3:11Þ
where the functions comprising g[t] do not include and part of a(t), from which it is quite apparent that the process of solu-tion can in principle proceed along the same lines as that of (1.1). The important thing to note is that the structure of thesolution remains a quadratic in x.
3.2. The case for q(t) being an unspecified function of t
Preliminary calculations with the parameters in (1.1) replaced by arbitrary functions of time without the constancy of theratio between the coefficients of uxx and u2
x indicate that the expressions obtained using Mathematica become essentiallymeaningless. Consequently we rewrite (1.1) in the simpler, albeit not in a structure closely related to the financial model,form
2ut þ pðtÞuxx þ qðtÞu2x þ 2ðm� rðtÞxÞux þ sðtÞx2 ¼ 0: ð3:12Þ
After the first round, as it were, of computation we find that a symmetry has the form
C ¼ aðtÞ@t þ bðtÞ þ dapdt
x2p
� @x þ ua
_pp�
_qq
� � exp �u
qp
� Fðt; xÞ þ Gðt; xÞ
� @u; ð3:13Þ
where F(t,x) and G(t,x) are to be determined from the remaining equations. We note that the form of (3.13) differs little fromthat of (2.1). The major difference is the term in the coefficient of @u which is linear in u. The effect of this term is to providethe choice F(t,x) = 0 or q(t) / p(t). The latter case was treated in the previous subsection.
The second, so far unspecified, function, G(t,x), is given by
C. Sophocleous et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 1752–1759 1757
Gðt; xÞ ¼ gðtÞ þ x_bþ br
q�mp
2qddt
ap
� �" #þ x2 1
2qddtðarÞ þ 1
4qddt
a _p4q
� �þ
€ap
� ; ð3:14Þ
where g(t) is a function of integration and consequently is so far arbitrary.At the next level of equations we separate initially by the coefficient of u and not-u. Integration of the former with respect
to time gives that
aðtÞ ¼ M1pqq _p� p _q
; ð3:15Þ
where M1 is a constant of integration. The latter can be further separated by coefficients of powers of x. The coefficient of x2 isa third-order equation for a(t), that of x a second-order equation for b(t) and the term independent of x a first-order equationfor g(t). This is the same structure as we saw in the previous subsection, but now we have a(t) already specified. Conse-quently the third-order equation becomes a constraint on the four parametric functions of time in (3.12) unless M1 is setto zero in which case the third-order equation is satisfied identically. The trade-off is a reduction in the number of symme-tries to three for no constraint on the parametric functions of time or four symmetries and a constraint. We consider theimplications below. In the presence of a nonzero a(t) the effect of (3.15) is to make the equation a third-order equationfor, say p(t), given q(t), r(t) and s(t), a second-order equation for r(t) given that the other three functions or a first-order equa-tion for s(t). We take the last and solve it to find that the constraint is
sðtÞ ¼ �M2
a2qþ r2
q� d
dtrq
� �� 2a€a� _a2
4a2q� 2q€q� 3 _q2
4q3 ; ð3:16Þ
where M2 is the second constant of integration.Without an explicit knowledge of the functions which constitute the coefficient functions of the symmetries of (3.12) we
can still proceed in a symbolic way. We write the four symmetries as
C1 ¼ a@t þ ax@x þ au@u;
C2 ¼ b1@x þ b10u þ xb11uð Þ@u;
C3 ¼ b2@x þ b20u þ xb21uð Þ@u;
C4 ¼ @u;
where ax and au are the coefficient functions for @x and @u engendered by a(t) and the same applies for the two functions b(t)except that we keep the x dependence explicit. The algebra of the four symmetries is A1 � sW, i.e., the loss of symmetry hasbeen from the sl(2,R) subalgebra.
We apply the general symmetry
C ¼X4
i¼1
aiCi;
to the terminal conditions, t = T and u(T,x) = 0 and obtain
a1 ¼ 0; anda2 b10u þ xb11uð Þ þ a3 b20u þ xb21uð Þ þ a4 ¼ 0;
respectively. Since a1 = 0, the existence or not of C1 is immaterial to the problem at hand. Consequently we may take a(t) = 0and so have no constraint upon the coefficient functions in (3.12). Since x is a free variable, the second equation gives twoconditions on the three parameters and we obtain the symmetry
Ctcp ¼1
b10ub21u � b11ub20uð ÞðTÞ �b21uðTÞb1ðtÞ þ b11uðTÞb2ðtÞ½ �@x þ �b21uðTÞ b10uðtÞ þ xb11uðtÞð Þ½f
þb11uðTÞ b20uðtÞ þ xb21uðtÞð Þ þ b10ub21u � b11ub20uð ÞðTÞ�@ug: ð3:17Þ
In (3.17), we see that (3.12) has a symmetry of the same structure as the symmetry which enabled us to reduce (1.1) to afirst-order ordinary differential equation. It is evident that the solution of (3.12) has the same form, namely
uðt; xÞ ¼ A0ðtÞ þ A1ðtÞxþ A2ðtÞx2; ð3:18Þ
as the previous forms of the equation considered in this paper. Our analysis justifies the Ansätze made in [3] and [11]. Whenone substitutes (3.18) into (3.12) and equates the coefficients of the three distinct powers of x to zero, three first-order equa-tions, namely
0 ¼ 2mA1ðtÞ þ 2A2ðtÞpðtÞ þ A21ðtÞqðtÞ þ 2 _A0ðtÞ; ð3:19Þ
0 ¼ 2mA2ðtÞ þ 2A1ðtÞA2ðtÞqðtÞ � A1ðtÞrðtÞ þ _A1ðtÞ; ð3:20Þ0 ¼ 4A2
2ðtÞqðtÞ � 4A2ðtÞrðtÞ þ sðtÞ þ 2 _A2ðtÞ: ð3:21Þ
1758 C. Sophocleous et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 1752–1759
The essential difficulty in the solution of these three equations is (3.21) which is a Riccati equation. Its second-order linearversion is
2 Thecan undsecond-end it i
€v þ€q
2q� 3 _q2
4q2 �r _qqþ _r � r2 þ qs
� �v ¼ 0;
where
A2 ¼r
2qþ
_q4q2 þ
_v2qv ;
the solution of which is not immediately obtainable for general parametric functions.
4. Concluding comments
We have demonstrated that the solution provided by Benth and Karlsen [3] to the determination of the Radon–Nikodymdensity of the minimal entropy martingale measure in the case of the Stein–Stein model of stochastic volatility can be ob-tained in an algorithmic way by means of the application of the Lie theory of continuous groups. The critical equation, (1.1),albeit nonlinear, possesses the same algebraic properties as the archetypal (1 + 1) evolution equation and the route to itsresolution in combination with the specific terminal condition for this problem is straightforward. The form of the modeltreated by Benth and Karlsen was autonomous. The generalisation to the nonautonomous case proposed by Kufakunesu[11] separated into two distinct cases. In the first case the coefficient of the nonlinear term was constantly proportionalto that of the second-order derivative and there was no essential difference by comparison with the treatment of the auton-omous problem. Naturally in terms of the details of solution the nonautonomous nature of the equations involved in thedetermination of the coefficients in the symmetries does interpose a technical problem. In the second case for which thisconstant proportionality did not exist there was a definite change in the algebraic structure of the equation. The numberof symmetries of the finite subgroup decreased to either three or four depending upon the existence of a constraint amongstthe time-dependent coefficients in the equation. As it so happened, the terminal condition removed the need for the con-straint on the coefficients in the equation and so it was possible to obtain, albeit again in a formal nature, an explicit solutionof the nonlinear evolution equation.2
It is interesting to note that the Ansätze proposed by Benth and Karlsen and Kufakunesu in their respective models areexactly in terms of their dependence upon x those predicted by the symmetry analysis. What does happen in the symmetryanalysis is that the nature of the time dependence is already largely determined from the solution of the differential equa-tions specifying the time-dependence in the coefficient functions of the symmetries of the differential equation itself.
A further development of the volatility model has been offered by Heston [8] in which the constant m in (1.1) is replacedby m/x. The complications introduced by this seemingly simple change in the equation are to be examined in a later work.
Acknowledgement
This work is part of a project funded by the Research Promotion Foundation of Cyprus, Grant No. PPOREKKYRH/PPOEM/0308/02, devoted to the algebraic resolution of problems in Financial Mathematics. PGLL thanks the University of KwaZulu-Natal and the National Research Foundation of South Africa for their continued support. The opinions expressed in this papershould not be construed as being those of either institution.
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