Mathematical Methods and Models in the Natural to the Life SciencesView this Special Issue
Group Classification of a General Bond-Option Pricing Equation of Mathematical Finance
We carry out group classification of a general bond-option pricing equation. We show that the equation admits a three-dimensional equivalence Lie algebra. We also show that some of the values of the constants which result from group classification give us well-known models in mathematics of finance such as Black-Scholes, Vasicek, and Cox-Ingersoll-Ross. For all such values of these arbitrary constants we obtain Lie point symmetries. Symmetry reductions are then obtained and group invariant solutions are constructed for some cases.
The theory of option pricing began in 1900 when the French mathematician Bachelier  deduced an option pricing formula based on the assumption that stock prices follow a Brownian motion. The Black-Scholes equation was introduced by Black and Scholes  as the general equilibrium theory of option pricing which is particularly attractive because the final formula is a function of observable variables. Merton  extended the Black-Scholes theory of option pricing by introducing more assumptions and found new explicit formulas for pricing both the call and put options as well as the warrants and the down-and-out options. The equation is mainly used to find the fair price of a financial instrument (option or derivative) and to find the implied volatility.
The first bond pricing equation was introduced by Vasicek  and thereafter many other researchers (see, e.g., [5–12]) came up with other one-factor models which modelled the term structure of interest rates.
Many differential equations, including financial mathematics equations, involve parameters, arbitrary elements, or functions, which need to be determined. Usually, these arbitrary parameters are determined experimentally. However, the Lie symmetry approach through the method of group classification has proven to be a versatile tool in specifying the forms of these parameters systematically (see, e.g., [13–20]).
In 1881, Lie  was the first one to investigate the problem of group classification. In this regard, he studied a linear second-order partial differential equation with two independent variables. Suppose a differential equation contains an arbitrary element . The main idea of group classification of this differential equation is to find the Lie point symmetries of the differential equation with arbitrary element and then find all possible forms of for which the principal Lie algebra can be extended.
Semi-invariants for the linear parabolic equations with two independent variables and one dependent variable were derived by Johnpillai and Mahomed . In addition, joint invariant equation was obtained for the linear parabolic equation and the linear parabolic equation was reducible via a local equivalence transformation to the one-dimensional heat equation. In , a necessary and sufficient condition for the parabolic equation to be reducible to the classical heat equation under the equivalence group was provided which improved the work done in .
Goard  found group invariant solutions of the bond pricing equation by the use of classical Lie method. The solutions obtained were shown to satisfy the condition for the bond price, that is, , where is the price of the bond. Here is the short-term interest rate which is governed by the stochastic differential equation and is time to maturity.
In , the fundamental solutions were obtained for a number of zero-coupon bond models by transforming the one-factor bond pricing equations corresponding to the bond models to the one-dimensional heat equation whose fundamental solution is well known. Subsequently, the transformations were used to construct the fundamental solutions for zero-coupon bond pricing equations.
Sinkala et al.  computed the zero-coupon bonds (group invariant solutions satisfying the terminal condition ) using symmetry analysis for the Vasicek and Cox-Ingersoll-Ross (CIR) equations, respectively. In  an optimal system of one-dimensional subalgebras was derived and used to construct distinct families of special closed-form solutions of CIR equation. In , group classification of the linear second-order parabolic partial differential equation where , , , , and are constants, was carried out. Lie point symmetries and group invariant solutions were found for certain values of . Also the forms where equation (3) admitted the maximal seven Lie point symmetry algebra were transformed into the heat equation. Vasicek, CIR, and Longstaff models were recovered from group classification and some other equations were derived which had not been considered before in the literature. Furthermore, Mahomed et al.  used the invariant conditions developed in  to carry out group classification of  and some new cases were discovered.
Dimas et al.  investigated some of the well-known equations that arise in mathematics of finance, such as Black-Scholes, Longtsaff, Vasicek, CIR, and heat equations. Lie point symmetries of these equations were found and their algebras were compared with those of the heat equation. The equations with seven symmetries were transformed to the heat equation.
In this paper, we study a general bond-option pricing equation. The partial differential equation which will be investigated is a generalisation of (1) and (2) and is given by where is time, is the stock (share or equity) price or instantaneous short-term interest rate at current time , and is the current value of the option or bond. Here , , and , , , and are constants with , , . When , (4) is the option pricing equation and it is the bond pricing PDE when .
This paper is structured as follows. In Section 2, we find two classifying equations on which group classification of (4) depends, one for and the other for . Then we use the two equations to find possible values for arbitrary constants for which (4) admits nontrivial Lie point symmetry algebras. In Section 3, we obtain symmetry reductions and construct group-invariant solutions for Case 2.1(1) and finally in Section 4 we give conclusions.
2. Determination of Classifying Equations of (4)
The Lie point symmetries for (4) are given by the vector field: if and only if where is the second prolongation of defined as Here ’s are given by where the total derivatives and are defined as To perform the group classification of (4) it turns out that we need to consider two cases of separately: and .
2.1. Classifying Equation of (4) for
Expanding the determining equation (6), we obtain Separating (10) with respect to the derivatives of , since the functions , , and do not depend on them, leads to the following linear PDEs: To solve the above system of equations, we first observe from (11) and (14) that does not depend on and , which means that is a function of . Thus Equation (12) implies that depends on both and but not on . Hence Integration of (13) with respect to twice gives where and are arbitrary functions of and . Using the expressions for and in (15) and integrating with respect to lead to where is an arbitrary function of . Using (18), (20), and (21) in (17) yields Integrating (22) with respect to and solving for we obtain where is an arbitrary function of .
Using (18)–(21) and (23) in (16) gives Since the functions , , , and do not depend on , we split (24) with respect to and get It is clear that satisfies the original equation of (4). Rewriting (26) yields our classifying equation as where
2.2. Classifying Equation of (4) for
In the case when in (4), we proceed as above to obtain the determining equation as As before, splitting (29) on derivatives of and simplifying lead to Equations (30) and (33) imply that whereas (31) means that does not depend on ; that is, and (32) gives after integrating twice by and for some arbitrary functions and . Substituting (37) and (38) into (35), we get a linear first-order ODE in which can be easily integrated with respect to to give where is an arbitrary function of . If we substitute (37), (40), and (39) into (34) and solve the resulting differential equation for , we get where is an arbitrary function of . Substituting (41) and (40) into (36), we obtain Splitting (42) on yields Rewriting (44) we get our classifying equation as where
3. Results of Group Classification
We note that our classifying equations (27) and (45) are satisfied if we choose for some constants and . Thus using these values, for both cases, the coefficients of the infinitesimal operator are where is any solution of (4).
Case 0 (, , , , , arbitrary). We obtain the following Lie point symmetries: where the symmetry associated with is the solution symmetry. Lie symmetries (49) generate what is called the principal Lie algebra.
By equating the powers of in (27) and solving for we infer that possible extensions of the principal Lie algebra are possible for the following values of : In this paper, we consider , and , as these values of provide us with very important equations in mathematics of finance. For example, when and , we have the Black-Scholes equation. We obtain the Vasicek equation when and and CIR equation when and .
We show only the different combinations of parameters which extend the principal Lie algebra.
Case 1 ()
Case 1.1 (). In this case, the principal Lie algebra extends by the following Lie point symmetries:
Case 1.2 (). The principal Lie algebra extends by It should be noted that this case results in the Vasicek equation .
Case 1.3 (). (1) Consider . The additional Lie point symmetries are given by where .
(2) Consider . The additional Lie point symmetries are
Case 2 ()
Case 2.1 ()
(1) Consider and . The principal Lie algebra is extended by
(2) Consider . The additional Lie point symmetries are
(3) Consider . The Lie point symmetries that extend the principal Lie algebra are
Case 2.2 (). This has two subcases.
(1) Consider . This case results in the following extra Lie point symmetries:
(2) Consider . The extra Lie point symmetries are
Case 2.3 (). This leads to six subcases.
(2) Consider , , and . The extra symmetries that extend the principal Lie algebra are given by
(3) Consider , . Additional Lie point symmetries are
(4) Consider , . Additional Lie point symmetries to the principal Lie algebra are given by
(5) Consider , . The extra Lie point symmetries are where .
(6) Consider , . The principal Lie algebra is extended by where .
Case 3 (). We can conclude from (45) that can only take the value . Proceeding as before, we find that the principal Lie algebra extends for the case when by the following symmetry operators: This case gives us the Black-Scholes equation .
4. Symmetry Reductions and Group Invariant Solutions
Case 2.1 (). The operator results in two invariants and . The group invariant solution is given by , where solves
Case 2.2 (). The operator gives the following two invariants: Hence, the invariant solution of (67) under is given by where satisfies The solution of the above equation is where and are arbitrary constants. Hence, the group invariant solution under is
Case 2.3 (). The symmetry operator gives the invariants and . Thus, the group invariant solution in this case is given by where and are arbitrary constants.
It should be noted that the operators and do not provide invariant solutions.
In this paper we carried out group classification of the general bond-option pricing PDE (4) for , and . The principal Lie algebra was found to be three-dimensional. These values of resulted in cases, which extended the principal Lie algebra. We presented the Lie point symmetries for each case. Three cases gave us the option pricing equations, which were given by Cases 1.1, 2.1(1), and 3. In the last case, Black-Scholes equation was recovered. Seven bond pricing equations were obtained and these were Case 1.2 and Cases 2.3(1)–2.3(6). Cases 1.2 and 2.3(1) were found to be the Vasicek and CIR equations, respectively. Finally, symmetry reductions and construction of group invariant solutions for Case 2.1(1) were presented.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
L. Bachelier, “Théorie de la spéculation,” Annales Scientifiques de l'École Normale Supérieure, vol. 3, pp. 21–86, 1900.View at: Google Scholar
T. S. Y. Ho and S. Lee, “Term structure movements and pricing interest rate contingent claims,” The Journal of Finance, pp. 1011–1029, 1986, 41.View at: Google Scholar
N. H. Ibragimov, Ed., CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1–3, CRC Press, Boca Raton, Fla, USA, 1994–1996.
M. Molati and C. M. Khalique, “Group classification and exact solutions of a generalized Emden-Fowler equation,” International Journal of Non-Linear Mechanics, vol. 27, pp. 785–791, 2012.View at: Google Scholar
M. Molati and C. M. Khalique, “Lie symmetry analysis of the time-variable coefficient B-BBM equation,” Advances in Differential Equations, vol. 2012, no. 212, 2012.View at: Google Scholar
S. Lie, “Uber die integration durch bestimmte integrale von einer klasse linearer partieller differentialgleichungen,” Archiv for Mathematik og Naturvidenskab, vol. 6, pp. 328–368, 1881.View at: Google Scholar
F. M. Mahomed, K. S. Mahomed, R. Naz, and E. Momoniat, “Invariant approaches to equations of finance,” Mathematical and Computational Applications, vol. 18, pp. 244–250, 2013.View at: Google Scholar