Group classification of a general bond–option pricing equation of mathematical finance

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Research Article

Group Classification of a General Bond-Option

Pricing Equation of Mathematical Finance

Tanki Motsepa,

1

Chaudry Masood Khalique,

1

and Motlatsi Molati

1,2

1_{International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences,}

North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa

2_{Department of Mathematics and Computer Science, University of Lesotho, Roma 180, Lesotho}

Correspondence should be addressed to Chaudry Masood Khalique; masood.khalique@nwu.ac.za Received 30 January 2014; Accepted 24 March 2014; Published 13 April 2014

Academic Editor: Imran Naeem

Copyright © 2014 Tanki Motsepa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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.

1. Introduction

The theory of option pricing began in 1900 when the French mathematician Bachelier [1] deduced an option pricing for-mula based on the assumption that stock prices follow a Brownian motion. The Black-Scholes equation

𝑢_𝑡+1₂𝑥2𝜎2𝑢_𝑥𝑥+ 𝑟 (𝑥𝑢_𝑥− 𝑢) = 0 (1) was introduced by Black and Scholes [2] as the general equilibrium theory of option pricing which is particularly attractive because the final formula is a function of observable variables. Merton [3] 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 𝑢_𝑡+1

2𝜎2𝑢𝑥𝑥+ 𝜅 (𝜃 − 𝑥) 𝑢𝑥− 𝑥𝑢 = 0 (2)

was introduced by Vasicek [4] 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 mathe-matics equations, involve parameters, arbitrary elements, or functions, which need to be determined. Usually, these arbi-trary 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 [21] 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 indepen-dent variables. Suppose a differential equation contains an arbitrary element𝑓(𝑢). The main idea of group classification of this differential equation is to find the Lie point symmetries of the differential equation with arbitrary element𝑓(𝑢) and then find all possible forms of𝑓(𝑢) for which the principal Lie algebra can be extended.

Semi-invariants for the(1 + 1) linear parabolic equations with two independent variables and one dependent variable were derived by Johnpillai and Mahomed [22]. In addition, joint invariant equation was obtained for the linear parabolic equation and the (1 + 1) linear parabolic equation was

Volume 2014, Article ID 709871, 10 pages http://dx.doi.org/10.1155/2014/709871

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reducible via a local equivalence transformation to the one-dimensional heat equation. In [23], 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 [22].

Goard [24] 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, 𝑃(𝑟, 𝑇) = 1, where 𝑃 is the price of the bond. Here𝑟 is the short-term interest rate which is governed by the stochastic differential equation and𝑇 is time to maturity.

In [25], 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 trans-formations were used to construct the fundamental solutions for zero-coupon bond pricing equations.

Sinkala et al. [26] computed the zero-coupon bonds (group invariant solutions satisfying the terminal condition 𝑢(𝑇, 𝑇) = 1) using symmetry analysis for the Vasicek and Cox-Ingersoll-Ross (CIR) equations, respectively. In [27] an optimal system of one-dimensional subalgebras was derived and used to construct distinct families of special closed-form solutions of CIR equation. In [20], group classification of the linear second-order parabolic partial differential equation

𝑢𝑡+1₂𝜌2𝑥2𝛾𝑢𝑥𝑥+ (𝛼 + 𝛽𝑥 − 𝜆𝜌𝑥𝛾) 𝑢𝑥− 𝑥𝑢 = 0, (3)

where𝛼, 𝛽, 𝜆, 𝜌, and 𝛾 are constants, was carried out. Lie point symmetries and group invariant solutions were found for certain values of𝛾. 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. [28] used the invariant conditions developed in [23] to carry out group classification of [20] and some new cases were discovered.

Dimas et al. [29] 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

𝑢𝑡+ 𝛼𝑥𝑝𝑢𝑥𝑥+ 𝜆 (𝛽 − 𝑥) 𝑢𝑥+ 𝛾𝑥𝑞 𝑢 = 0, (4)

where 𝑡 is time, 𝑥 is the stock (share or equity) price or instantaneous short-term interest rate at current time𝑡, and 𝑢(𝑡, 𝑥) is the current value of the option or bond. Here 𝑝 ≥ 0, 𝑞 ≥ 0, and 𝛼, 𝛾, 𝜆, and 𝛽 are constants with 𝛼, 𝜆, 𝛾 ̸= 0. When 𝑞 = 0, (4) is the option pricing equation and it is the bond pricing PDE when𝑞 = 1.

This paper is structured as follows. InSection 2, we find two classifying equations on which group classification of (4) depends, one for 𝑝 ̸= 2 and the other for 𝑝 = 2. Then we use the two equations to find possible values for arbitrary constants for which (4) admits nontrivial Lie point symmetry algebras. InSection 3, we obtain symmetry reductions and construct group-invariant solutions for Case 2.1(1) and finally inSection 4we give conclusions.

2. Determination of Classifying

Equations of

(4)

The Lie point symmetries for (4) are given by the vector field: 𝑋 = 𝜏 (𝑡, 𝑥, 𝑢)_𝜕𝑡𝜕 + 𝜉 (𝑡, 𝑥, 𝑢)_𝜕𝑥𝜕 + 𝜂 (𝑡, 𝑥, 𝑢)_𝜕𝑢𝜕 , (5) if and only if

𝑋[2](𝑢_𝑡+ 𝛼𝑥𝑝𝑢_𝑥𝑥+ 𝜆 (𝛽 − 𝑥) 𝑢_𝑥+ 𝛾𝑥𝑞_{𝑢)󵄨󵄨󵄨󵄨󵄨}₍₄₎= 0, (6) where𝑋[2]is the second prolongation of𝑋 defined as

𝑋[2]= 𝑋 + 𝜁₁_𝜕𝑢𝜕

𝑡+ 𝜁2

𝜕

𝜕𝑢_𝑥+ 𝜁22_𝜕𝑢𝜕

𝑥𝑥. (7)

Here𝜁_𝑖’s are given by

𝜁₁= 𝐷_𝑡(𝜂) − 𝑢_𝑡𝐷_𝑡(𝜏) − 𝑢_𝑥𝐷_𝑡(𝜉) , 𝜁₂= 𝐷_𝑥(𝜂) − 𝑢_𝑡𝐷_𝑥(𝜏) − 𝑢_𝑥𝐷_𝑥(𝜉) , 𝜁₂₂= 𝐷_𝑥(𝜁₂) − 𝑢_𝑡𝑥𝐷_𝑥(𝜏) − 𝑢𝑥𝑥𝐷𝑥(𝜉) ,

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where the total derivatives𝐷_𝑡and𝐷_𝑥are defined as 𝐷_𝑡= 𝜕 𝜕𝑡+ 𝑢𝑡 𝜕 𝜕𝑢+ 𝑢𝑡𝑥 𝜕 𝜕𝑢_𝑥+ 𝑢𝑡𝑡 𝜕 𝜕𝑢_𝑡+ ⋅ ⋅ ⋅ , 𝐷_𝑥= 𝜕 𝜕𝑥+ 𝑢𝑥 𝜕 𝜕𝑢+ 𝑢𝑥𝑥 𝜕 𝜕𝑢_𝑥 + 𝑢𝑡𝑥 𝜕 𝜕𝑢_𝑡+ ⋅ ⋅ ⋅ . (9)

To perform the group classification of (4) it turns out that we need to consider two cases of𝑝 separately: 𝑝 ̸= 2 and 𝑝 = 2.

2.1. Classifying Equation of (4) for 𝑝 ̸= 2. Expanding the

determining equation (6), we obtain

𝛾𝜉𝑞𝑢𝑥𝑞−1− 𝛾𝑢𝑥𝑞𝜂_𝑢+ 𝛾𝑢𝑥𝑞𝜉_𝑢𝑢_𝑥+ 𝛾𝜂𝑥𝑞+ 𝜂_𝑡− 𝜉_𝑡𝑢_𝑥 − 𝜆𝜉𝑢_𝑥+ 𝛽𝜆𝜂_𝑥− 𝜆𝑥𝜂_𝑥+ 𝛾𝑢𝑥𝑞𝜏_𝑡− 𝛾2𝑢2𝑥2𝑞𝜏_𝑢 + 𝛾𝜆𝑢𝑥𝑞+1_𝜏

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+ 𝜆𝑥𝑢_𝑥𝜉_𝑥− 𝛾𝜆𝑢𝑥𝑞+1𝜏_𝑥− 𝛽𝛾𝜆𝑢𝑥𝑞𝜏_𝑢𝑢_𝑥+ 𝛽𝛾𝜆𝑢𝑥𝑞𝜏_𝑥 + 𝜆2𝑥2𝑢𝑥𝜏𝑥+ 𝛽2𝜆2𝑢𝑥𝜏𝑥− 2𝛽𝜆2𝑥𝑢𝑥𝜏𝑥 − 2𝛼𝑥𝑝𝜏_𝑢𝑢_𝑥𝑢_𝑡𝑥− 2𝛼𝑥𝑝𝜏_𝑥𝑢_𝑡𝑥+ 𝛼𝜉𝑝𝑥𝑝−1𝑢_𝑥𝑥 − 2𝛼𝑥𝑝𝜉_𝑢𝑢_𝑥𝑢_𝑥𝑥− 2𝛼𝑥𝑝𝜉_𝑥𝑢_𝑥𝑥+ 𝑥𝑝 × ( 𝛼𝜏_𝑡𝑢_𝑥𝑥− 𝛼𝛾𝑢𝜏_𝑢𝑢_𝑥𝑥𝑥𝑞− 𝛼𝜆𝑥𝜏_𝑥𝑢_𝑥𝑥 +𝛼𝛽𝜆𝜏_𝑥𝑢_𝑥𝑥+ 𝛼𝑢2_𝑥𝜂_𝑢𝑢+ 2𝛼𝑢_𝑥𝜂_𝑥𝑢) + 𝛼𝛾𝑢𝑢2_𝑥𝜏_𝑢𝑢𝑥𝑝+𝑞− 𝛼𝜆𝑥𝑝+1𝑢3_𝑥𝜏_𝑢𝑢− 𝛼𝑥𝑝𝑢_𝑥3𝜉_𝑢𝑢 − 2𝛼𝑥𝑝_𝑢2 𝑥𝜉𝑥𝑢− 𝛼𝑥𝑝𝑢𝑥𝜉𝑥𝑥+ 𝛼𝑥𝑝𝜂𝑥𝑥 + 2𝛼𝛾𝑢𝑢_𝑥𝑥𝑝+𝑞𝜏_𝑥𝑢+ 𝛼2𝑥2𝑝𝑢_𝑥2𝜏_𝑢𝑢𝑢_𝑥𝑥− 2𝛼𝜆𝑥𝑝+1𝑢2_𝑥𝜏_𝑥𝑢 + 𝛼𝛽𝜆𝑥𝑝(𝑢3_𝑥𝜏_𝑢𝑢+ 2𝑢2_𝑥𝜏_𝑥𝑢) + 𝛼𝛾𝑢𝜏_𝑥𝑥𝑥𝑝+𝑞 + 𝛼2𝑥2𝑝𝑢𝑥𝑥(2𝑢𝑥𝜏𝑥𝑢+ 𝜏𝑥𝑥) − 𝛼𝜆𝑥𝑝+1𝑢𝑥𝜏𝑥𝑥 + 𝛼𝛽𝜆𝑥𝑝𝑢_𝑥𝜏_𝑥𝑥= 0. (10) Separating (10) with respect to the derivatives of𝑢, since the functions𝜏, 𝜉, and 𝜂 do not depend on them, leads to the following linear PDEs:

𝜏_𝑢= 0, (11) 𝜉_𝑢= 0, (12) 𝜂_𝑢𝑢= 0, (13) 𝜏_𝑥= 0, (14) 𝑝𝜉 + 𝑥 (𝜏_𝑡− 2𝜉_𝑥) = 0, (15) 𝑢𝛾𝑥𝑞𝜂_𝑢+ 𝑥𝜆𝜂_𝑥− 𝑞𝑢𝛾𝑥𝑞−1𝜉 − 𝛾𝑥𝑞𝜂 − 𝛼𝑥𝑝𝜂_𝑥𝑥 − 𝑢𝛾𝑥𝑞𝜏_𝑡− 𝜂_𝑡− 𝛽𝜆𝜂_𝑥= 0, (16) 𝜆𝜉 + 𝛼𝑥𝑝_𝜉 𝑥𝑥+ 𝑥𝜆𝜏𝑡+ 𝜉𝑡+ 𝜆 (𝛽 − 𝑥) 𝜉𝑥 − 2𝛼𝑥𝑝𝜂𝑥𝑢− 𝛽𝜆𝜏𝑡= 0. (17)

To solve the above system of equations, we first observe from (11) and (14) that𝜏 does not depend on 𝑥 and 𝑢, which means that𝜏 is a function of 𝑡. Thus

𝜏 = 𝜏 (𝑡) . (18)

Equation (12) implies that𝜉 depends on both 𝑡 and 𝑥 but not on𝑢. Hence

𝜉 = 𝜉 (𝑡, 𝑥) . (19)

Integration of (13) with respect to𝑢 twice gives

𝜂 (𝑡, 𝑥, 𝑢) = 𝐴 (𝑡, 𝑥) 𝑢 + 𝐵 (𝑡, 𝑥) , (20)

where𝐴(𝑡, 𝑥) and 𝐵(𝑡, 𝑥) are arbitrary functions of 𝑡 and 𝑥. Using the expressions for𝜏 and 𝜉 in (15) and integrating with respect to𝑥 lead to

𝜉 (𝑡, 𝑥) = 𝑐 (𝑡) 𝑥𝑝/2−𝑥𝜏_{𝑝 − 2}󸀠(𝑡), 𝑝 ̸= 2, (21) where𝑐(𝑡) is an arbitrary function of 𝑡. Using (18), (20), and (21) in (17) yields 8𝛼𝑥𝑝+1𝐴𝑥(𝑡, 𝑥) (2 − 𝑝) + 𝑐 (𝑡) 𝛼𝑝𝑥−1+3𝑝/2(𝑝 − 2)2 − 2𝑐 (𝑡) 𝜆𝑥1+𝑝/2(𝑝 − 2)2− 4𝑥2𝜏󸀠󸀠(𝑡) + 2𝛽𝜆𝑝𝐶 (𝑡) 𝑥𝑝/2(𝑝 − 2) + 4𝑥𝑝/2𝑥𝑐󸀠(𝑡) (𝑝 − 2) + 𝜆 ( 4𝑥2(𝑝 − 2) + 4𝛽𝑥 (1 − 𝑝)) 𝜏󸀠(𝑡) = 0. (22)

Integrating (22) with respect to𝑥 and solving for 𝐴 we obtain 𝐴 (𝑡, 𝑥) = 𝑑 (𝑡) + 𝑥−𝑝−1 4𝛼(𝑝 − 2)2 × {(𝑝 − 2)2𝑥𝑝/2𝑐 (𝑡) (𝑝𝛼𝑥𝑝+ 2𝜆𝑥 (𝑥 − 𝛽)) −2𝑥3𝜏_𝑡𝑡} − 𝑥−𝑝−1 4𝛼(𝑝 − 2)2 × {2𝑥2( 2 (𝑝 − 2) 𝑥𝑝/2𝑐󸀠(𝑡) + (2 − 𝑝) 𝜆𝜏󸀠(𝑡) (𝛽 − 𝑥))} , (23)

where𝑑(𝑡) is an arbitrary function of 𝑡. Using (18)–(21) and (23) in (16) gives

16𝛼(𝑝 − 2)2𝑥2+𝑝/2(𝛼𝑥𝑝𝐵_𝑥𝑥+ 𝐵𝛾𝑥𝑞+ 𝐵_𝑡+ 𝛽𝜆𝐵_𝑥− 𝜆𝑥𝐵_𝑥) + 8𝑢𝛼 (𝑝 − 2) 𝑥2+𝑝/2(2𝛾 (𝑝 − 𝑞 − 2) 𝑥𝑞 −𝜆 (𝑝 − 2) (𝑝 − 1)) 𝜏_𝑡 − 8𝑢𝜆2_{(𝑝 − 2)}2_𝑥4−𝑝/2_𝜏 𝑡+ 8𝛽𝜆𝑢 (𝑝 − 2) × {𝛼 (𝑝 − 1) 𝑝𝑥1+𝑝/2𝜏𝑡− 𝛽𝜆 (𝑝 − 1) 𝑥2−𝑝/2𝜏𝑡 +𝜆 (2𝑝 − 3) 𝑥3−𝑝/2𝜏_𝑡} + 𝑢 {8 (𝑝 − 2) (2𝛼 (𝑝 − 2) 𝑑_𝑡𝑥2+𝑝/2+𝛼 (𝑝 − 1) 𝑥2+𝑝/2𝜏_𝑡𝑡 −2𝑥3𝑐𝑡𝑡) + 8𝑥4−𝑝/2𝜏𝑡𝑡𝑡} + 𝛼(𝑝 − 2)2𝑐 (𝑡) 𝑥𝑝−1𝑢 {𝛼𝑝3𝑥𝑝− 6𝛼𝑝2𝑥𝑝+ 8𝛼𝑝𝑥𝑝 −8𝛽𝜆𝑝𝑥 + 16𝛾𝑞𝑥𝑞+2} + 4𝜆2(𝑝 − 2)2𝑥𝑢𝑐 (𝑡) (𝑥 − 𝛽) (−𝛽𝑝 + 𝑝𝑥 − 2𝑥) = 0. (24)

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Since the functions𝐵, 𝑐, 𝑑, and 𝜏 do not depend on 𝑢, we split (24) with respect to𝑢 and get

1 : 𝐵_𝑡+ 𝛼𝑥𝑝𝐵_𝑥𝑥+ 𝜆 (𝛽 − 𝑥) 𝐵_𝑥+ 𝛾𝑥𝑞𝐵 = 0, (25) 𝑢 : 8𝛼 (𝑝 − 2) 𝑥2+𝑝/2(2𝛾 (𝑝 − 𝑞 − 2) 𝑥𝑞 −𝜆 (𝑝 − 2) (𝑝 − 1)) 𝜏_𝑡 − 8𝜆2(𝑝 − 2)2𝑥4−𝑝/2𝜏𝑡+ 8𝛽𝜆 (𝑝 − 2) × ( 𝛼 (𝑝 − 1) 𝑝𝑥1+𝑝/2𝜏_𝑡− 𝛽𝜆 (𝑝 − 1) 𝑥2−𝑝/2𝜏_𝑡 +𝜆 (2𝑝 − 3) 𝑥3−𝑝/2𝜏_𝑡) + (8 (𝑝 − 2) (2𝛼 (𝑝 − 2) 𝑑_𝑡𝑥2+𝑝/2+ 𝛼 (𝑝 − 1) 𝑥2+𝑝/2𝜏_𝑡𝑡 −2𝑥3𝑐𝑡𝑡) + 8𝑥4−𝑝/2𝜏𝑡𝑡𝑡) + 𝛼(𝑝 − 2)2𝑐 (𝑡) 𝑥𝑝−1(𝛼𝑝3𝑥𝑝− 6𝛼𝑝2𝑥𝑝+ 8𝛼𝑝𝑥𝑝 −8𝛽𝜆𝑝𝑥 + 16𝛾𝑞𝑥𝑞+2) + 4𝜆2(𝑝 − 2)2𝑥𝑐 (𝑡) (𝑥 − 𝛽) (−𝛽𝑝 + 𝑝𝑥 − 2𝑥) = 0. (26) It is clear that 𝐵 satisfies the original equation of (4). Rewriting (26) yields our classifying equation as

ℎ0(𝑡) 𝑥5𝑝/2+ ℎ1(𝑡) 𝑥𝑞+2+3𝑝/2+ ℎ2(𝑡) 𝑥1+3𝑝/2+ ℎ3(𝑡) 𝑥4 + ℎ₄(𝑡) 𝑥3+ ℎ₅(𝑡) 𝑥𝑝+2+ ℎ₆(𝑡) 𝑥𝑝+𝑞+3+ ℎ₇(𝑡) 𝑥𝑝+3 + ℎ₈(𝑡) 𝑥3+𝑝/2+ ℎ₉(𝑡) 𝑥2+𝑝/2+ ℎ₁₀(𝑡) 𝑥4+𝑝/2+ ℎ₁₁(𝑡) 𝑥5 = 0, ₍₂₇₎ where ℎ0(𝑡) = 𝛼2(𝑝 − 4) (𝑝 − 2)3𝑝𝑐 (𝑡) , ℎ₁(𝑡) = 16𝛼𝛾(𝑝 − 2)2𝑞𝑐 (𝑡) , ℎ2(𝑡) = −8𝛼𝛽𝜆(𝑝 − 2)2𝑝𝑐 (𝑡) , ℎ₃(𝑡) = 8𝛽𝜆2(𝑝 − 2) (2𝑝 − 3) 𝜏󸀠(𝑡) , ℎ₄(𝑡) = −8𝛽2𝜆2(𝑝 − 2) (𝑝 − 1) 𝜏󸀠(𝑡) , ℎ5(𝑡) = 8𝛼𝛽𝜆 (𝑝 − 2) (𝑝 − 1) 𝑝𝜏󸀠(𝑡) , ℎ₆(𝑡) = 16𝛼𝛾 (𝑝 − 2) (𝑝 − 𝑞 − 2) 𝜏󸀠(𝑡) , ℎ7(𝑡) = 8𝛼 (𝑝 − 2) ((𝑝 − 1) 𝜏󸀠󸀠(𝑡) − 𝜆 (𝑝 − 2) × (𝑝 − 1) 𝜏󸀠(𝑡) + 2 (𝑝 − 2) 𝑑󸀠(𝑡)) , ℎ₈(𝑡) = −8𝛽𝜆2(𝑝 − 2)2(𝑝 − 1) 𝑐 (𝑡) , ℎ₉(𝑡) = 4𝛽2𝜆2(𝑝 − 2)2𝑝𝑐 (𝑡) , ℎ₁₀(𝑡) = 4 (𝑝 − 2) ( 𝜆2𝑝2𝑐 (𝑡) − 4𝜆2𝑝𝑐 (𝑡) +4𝜆2𝑐 (𝑡) − 4𝑐󸀠󸀠(𝑡)) , ℎ₁₁(𝑡) = −8 ( 𝜆2𝑝2𝜏󸀠(𝑡) − 4𝜆2𝑝𝜏󸀠(𝑡) +4𝜆2𝜏󸀠(𝑡) − 𝜏󸀠󸀠󸀠(𝑡)) . (28)

2.2. Classifying Equation of (4) for𝑝 = 2. In the case when

𝑝 = 2 in (4), we proceed as above to obtain the determining equation as 𝑞𝑢𝛾𝜉𝑥𝑞−1+ 𝛾𝜂𝑥𝑞+ 𝑢𝛾𝜏_𝑡𝑥𝑞− 𝑢𝛽𝛾𝜆𝑢_𝑥𝜏_𝑢𝑥𝑞+ 𝑢𝛽𝛾𝜆𝜏_𝑥𝑥𝑞 + 𝑢𝛾𝑢_𝑥𝜉_𝑢𝑥𝑞− 𝑢𝛾𝜂_𝑢𝑥𝑞− 𝑢2𝛾2𝜏_𝑢𝑥2𝑞+ 𝑢𝛾𝜆𝑢_𝑥𝜏_𝑢𝑥𝑞+1 − 𝑢𝛾𝜆𝜏_𝑥𝑥𝑞+1_{− 𝑢𝛼𝛾𝜏} 𝑢𝑢𝑥𝑥𝑥𝑞+2+ 𝑢𝛼𝛾𝑢2𝑥𝜏𝑢,𝑢𝑥𝑞+2 + 2𝑢𝛼𝛾𝑢_𝑥𝜏_𝑥𝑢𝑥𝑞+2+ 𝑢𝛼𝛾𝜏_𝑥𝑥𝑥𝑞+2+ 𝛼2𝑢2_𝑥𝑢_𝑥𝑥𝜏_𝑢𝑢𝑥4 + 2𝛼2𝑢_𝑥𝑢_𝑥𝑥𝜏_𝑥𝑢𝑥4+ 𝛼2𝑢_𝑥𝑥𝜏_𝑥𝑥𝑥4− 𝛼𝜆𝜏_𝑥𝑢_𝑥𝑥𝑥3 − 𝛼𝜆𝑢3_𝑥𝜏𝑢𝑢𝑥3− 2𝛼𝜆𝑢2𝑥𝜏𝑥𝑢𝑥3− 𝛼𝜆𝑢𝑥𝜏𝑥𝑥𝑥3 + 𝜆2𝑢_𝑥𝜏_𝑥𝑥2− 2𝛼𝑢_𝑥𝜏_𝑢𝑢_𝑡𝑥𝑥2− 2𝛼𝜏_𝑥𝑢_𝑡𝑥𝑥2+ 𝛼𝜏_𝑡𝑢_𝑥𝑥𝑥2 + 𝛼𝛽𝜆𝜏_𝑥𝑢_𝑥𝑥𝑥2− 2𝛼𝑢_𝑥𝜉_𝑢𝑢_𝑥𝑥𝑥2− 2𝛼𝜉_𝑥𝑢_𝑥𝑥𝑥2 + 𝛼𝛽𝜆𝑢3_𝑥𝜏_𝑢𝑢𝑥2+ 2𝛼𝛽𝜆𝑢2_𝑥𝜏_𝑥𝑢𝑥2+ 𝛼𝛽𝜆𝑢_𝑥𝜏_𝑥𝑥𝑥2 − 𝛼𝑢3_𝑥𝜉_𝑢𝑢𝑥2− 2𝛼𝑢2_𝑥𝜉_𝑥𝑢𝑥2− 𝛼𝑢_𝑥𝜉_𝑥𝑥𝑥2+ 𝛼𝑢2_𝑥𝜂_𝑢𝑢𝑥2 + 2𝛼𝑢𝑥𝜂𝑥𝑢𝑥2+ 𝛼𝜂𝑥𝑥𝑥2− 𝜆𝑢𝑥𝜏𝑡𝑥 − 2𝛽𝜆2𝑢𝑥𝜏𝑥𝑥 + 𝜆𝑢𝑥𝜉𝑥𝑥 − 𝜆𝜂𝑥𝑥 + 2𝛼𝜉𝑢𝑥𝑥𝑥 − 𝜆𝑢𝑥𝜉 + 𝛽𝜆𝑢𝑥𝜏𝑡 + 𝛽2𝜆2𝑢_𝑥𝜏_𝑥− 𝑢_𝑥𝜉_𝑡− 𝛽𝜆𝑢_𝑥𝜉_𝑥+ 𝜂_𝑡+ 𝛽𝜆𝜂_𝑥 = 0. (29) As before, splitting (29) on derivatives of𝑢 and simplifying lead to 𝜏_𝑢= 0, (30) 𝜉_𝑢= 0, (31) 𝜂_𝑢𝑢= 0, (32) 𝜏_𝑥= 0, (33) 2𝛼𝜂_𝑥𝑢𝑥3− 𝛼𝜉_𝑥𝑥𝑥3− 𝜆𝜉_𝑥𝑥2+ 𝜆𝜉𝑥 − 𝜉_𝑡𝑥 + 𝛽𝜆𝜉_𝑥𝑥 − 2𝛽𝜆𝜉 = 0, (34) 2𝑥𝜉_𝑥− 2𝜉 − 𝑥𝜏_𝑡= 0, (35)

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𝑥𝑞(𝑢𝛾𝜉 (𝑞 − 2) + 𝑥 (𝛾𝜂 − 𝑢𝛾𝜂_𝑢+ 2𝑢𝛾𝜉_𝑥))

+ 𝑥2𝛼𝜂_𝑥𝑥𝑥 + 𝑥𝜆𝜂_𝑥(𝛽 − 𝑥) + 𝑥𝜂_𝑡= 0. (36) Equations (30) and (33) imply that

𝜏 = 𝜏 (𝑡) , (37)

whereas (31) means that𝜉 does not depend on 𝑢; that is,

𝜉 = 𝜉 (𝑡, 𝑥) , (38)

and (32) gives

𝜂 (𝑡, 𝑥, 𝑢) = 𝐴 (𝑡, 𝑥) 𝑢 + 𝐵 (𝑡, 𝑥) , (39) after integrating twice by𝑢 and for some arbitrary functions 𝐴 and𝐵. Substituting (37) and (38) into (35), we get a linear first-order ODE in𝜉 which can be easily integrated with respect to 𝑥 to give

𝜉 (𝑡, 𝑥) = 𝑥𝑒 (𝑡) +1₂𝑥𝜏󸀠(𝑡) ln 𝑥, (40) where𝑒(𝑡) is an arbitrary function of 𝑡. If we substitute (37), (40), and (39) into (34) and solve the resulting differential equation for𝐴, we get

𝐴 (𝑡, 𝑥) = _8𝛼𝑥1 [{𝑥 [4𝑒󸀠(𝑡) + ln 𝑥𝜏󸀠󸀠(𝑡)]

+2𝜏󸀠(𝑡) [𝑥 (𝛼 + 𝜆) − 𝛽𝜆]} ln 𝑥 − 4𝛽𝜆𝑒 (𝑡)] + 𝑓 (𝑡) ,

(41) where𝑓(𝑡) is an arbitrary function of 𝑡. Substituting (41) and (40) into (36), we obtain 8𝛼2𝑥4𝐵_𝑥𝑥− 8𝛼𝜆𝑥3𝐵_𝑥+ 8𝛼𝛽𝜆𝑥2𝐵_𝑥+ 8𝛼𝑥2𝐵_𝑡+ 8𝛼𝛾𝑥𝑞+2𝐵 + 𝑢 ( − 4𝛼𝑥2𝑒󸀠(𝑡) − 4𝜆𝑥2𝑒󸀠(𝑡) + 4𝑒 (𝑡) × ( 2𝛼𝛾𝑞𝑥𝑞+2+ 𝛽𝜆 (𝛽𝜆 − 𝑥 (2𝛼 + 𝜆))) + 4𝑥2𝑒󸀠󸀠(𝑡) ln 𝑥 + 8𝛼𝑥2𝑓󸀠(𝑡) + 8𝛼𝛾𝑥𝑞+2𝜏󸀠(𝑡) + 4𝛼𝛾𝑞𝑥𝑞+2𝜏󸀠(𝑡) ln 𝑥 − 2𝛽2𝜆2𝜏󸀠(𝑡) − 2𝛼2𝑥2𝜏󸀠(𝑡) − 4𝛼𝜆𝑥2𝜏󸀠(𝑡) + 2𝛼𝑥2𝜏󸀠󸀠(𝑡) − 2𝜆2𝑥2𝜏󸀠(𝑡) + 𝑥2𝜏(3)(𝑡) ln2𝑥 + 4𝛽𝜆2𝑥𝜏󸀠(𝑡) + 8𝛼𝛽𝜆𝑥𝜏󸀠(𝑡) − 4𝛼𝛽𝜆𝑥𝜏󸀠(𝑡) ln 𝑥 +2𝛽2𝜆2𝜏󸀠(𝑡) ln 𝑥 − 2𝛽𝜆2𝑥𝜏󸀠(𝑡) ln 𝑥) = 0. (42) Splitting (42) on𝑢 yields 1 : 𝐵_𝑡+ 𝛼𝑥2𝐵_𝑥𝑥+ 𝜆 (𝛽 − 𝑥) 𝐵_𝑥+ 𝛾𝑥𝑞𝐵 = 0, (43) 𝑢 : (4𝑥2𝑒󸀠󸀠(𝑡) + 4𝛼𝛾𝑞𝑥𝑞+2𝜏󸀠(𝑡) + 2𝛽2𝜆2𝜏󸀠(𝑡) − 4𝛼𝛽𝜆𝑥𝜏󸀠(𝑡) −2𝛽𝜆2𝑥𝜏󸀠(𝑡)) ln 𝑥 − 4𝛼𝑥2𝑒󸀠(𝑡) − 4𝜆𝑥2𝑒󸀠(𝑡) + 4𝑒 (𝑡) × ( 2𝛼𝛾𝑞𝑥𝑞+2+ 𝛽𝜆 (𝛽𝜆 − 𝑥 (2𝛼 + 𝜆))) + 8𝛼𝑥2𝑓󸀠(𝑡) + 8𝛼𝛾𝑥𝑞+2𝜏󸀠(𝑡) − 2𝛽2𝜆2𝜏󸀠(𝑡) − 2𝛼2𝑥2𝜏󸀠(𝑡) − 4𝛼𝜆𝑥2𝜏󸀠(𝑡) + 2𝛼𝑥2𝜏󸀠󸀠(𝑡) − 2𝜆2𝑥2𝜏󸀠(𝑡) + 𝑥2𝜏󸀠󸀠󸀠(𝑡) ln2𝑥 + 8𝛼𝛽𝜆𝑥𝜏󸀠(𝑡) + 4𝛽𝜆2𝑥𝜏󸀠(𝑡) = 0. (44) Rewriting (44) we get our classifying equation as

𝑏₀(𝑡) + 𝑏₁(𝑡) ln 𝑥 + 𝑥 (𝑏₂(𝑡) + 𝑏₃(𝑡) ln 𝑥) + 𝑥2(𝑏₄(𝑡) + 𝑏₅(𝑡) ln 𝑥 + 𝑏₆(𝑡) ln2𝑥) + 𝑥𝑞+2(𝑏₇(𝑡) + 𝑏₈(𝑡) ln 𝑥) = 0, (45) where 𝑏0(𝑡) = 4𝛽2𝜆2𝑒 (𝑡) − 2𝛽2𝜆2𝜏󸀠(𝑡) , 𝑏 (1) = 2𝛽2𝜆2𝜏󸀠(𝑡) , 𝑏₂(𝑡) = −4𝛽𝜆 (2𝛼 + 𝜆) (𝑒 (𝑡) − 𝜏󸀠(𝑡)) , 𝑏3(𝑡) = −2𝛽𝜆 (2𝛼 + 𝜆) 𝜏󸀠(𝑡) , 𝑏₄(𝑡) = 8𝛼𝑓󸀠(𝑡) − 4 (𝛼 + 𝜆) 𝑒󸀠(𝑡) − 2(𝛼 + 𝜆)2𝜏󸀠(𝑡) + 2𝛼𝜏󸀠󸀠(𝑡) , 𝑏₅(𝑡) = 4𝑒󸀠󸀠(𝑡) , 𝑏6(𝑡) = 𝜏󸀠󸀠󸀠(𝑡) , 𝑏₇(𝑡) = 4𝛼𝛾 (2𝑞𝑒 (𝑡) + 2𝜏󸀠(𝑡)) , 𝑏₈(𝑡) = 4𝛼𝛾𝑞𝜏󸀠(𝑡) . (46)

3. Results of Group Classification

We note that our classifying equations (27) and (45) are satisfied if we choose

𝑐 (𝑡) = 𝑒 (𝑡) = 0, 𝑑 (𝑡) = 𝑓 (𝑡) = 𝑐2, 𝜏 (𝑡) = 𝑐1, (47)

for some constants𝑐₁and𝑐₂. Thus using these values, for both cases, the coefficients of the infinitesimal operator are

𝜏 = 𝑐1, 𝜉 = 0, 𝜂 = 𝑐2𝑢 + 𝐵 (𝑡, 𝑥) , (48)

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Case 0(𝛼, 𝛾, 𝜆, 𝛽, 𝑝, 𝑞 arbitrary). We obtain the following Lie

point symmetries:

𝑋₁=_𝜕𝑡𝜕, 𝑋₂= 𝑢_𝜕𝑢𝜕 , 𝑋_𝐵= 𝐵 (𝑡, 𝑥)_𝜕𝑢𝜕 , (49) where the symmetry associated with𝐵 is the solution symme-try. Lie symmetries (49) generate what is called the principal Lie algebra.

By equating the powers of𝑥 in (27) and solving for𝑝 we infer that possible extensions of the principal Lie algebra are possible for the following values of𝑝:

0, 1,3 2, 4 3, 6 5, 8 5, 8 3, 3, 4, 6. (50) In this paper, we consider𝑝 = 0, 1, and 2, as these values of 𝑝 provide us with very important equations in mathematics of finance. For example, when𝑝 = 2 and 𝑞 = 0, we have the Black-Scholes equation. We obtain the Vasicek equation when𝑝 = 0 and 𝑞 = 1 and CIR equation when 𝑝 = 1 and 𝑞 = 1.

We show only the different combinations of parameters which extend the principal Lie algebra.

Case 1 (𝑝 = 0)

Case 1.1 (𝑞 = 0). In this case, the principal Lie algebra

extends by the following Lie point symmetries:

𝑋₃= 𝑒2𝜆𝑡 2𝜆 𝜕 𝜕𝑡+ 𝑒2𝜆𝑡( 𝑥 2 − 𝛽 2) 𝜕 𝜕𝑥 + 𝑢𝑒2𝜆𝑡(𝛽2𝜆 2𝛼 − 𝛾 2𝜆+ 𝜆𝑥2 2𝛼 − 𝛽𝜆𝑥 𝛼 − 1 2) 𝜕 𝜕𝑢, 𝑋₄=𝑒−2𝜆𝑡_2𝜆 [−_𝜕𝑡𝜕 + 𝜆 (𝑥 − 𝛽)_𝜕𝑥𝜕 + 𝛾𝑢_𝜕𝑢𝜕 ] , 𝑋₅= 𝑒𝜆𝑡[𝜕 𝜕𝑥+ 𝜆 𝛼𝑢 (𝑥 − 𝛽) 𝜕 𝜕𝑢] , 𝑋₆= 𝑒−𝜆𝑡_𝜕𝑥𝜕 . (51)

Case 1.2 (𝑞 = 1). The principal Lie algebra extends by

𝑋₃= 𝑒2𝜆𝑡 2𝜆 𝜕 𝜕𝑡+ 𝑒2𝜆𝑡(− 𝛼𝛾 𝜆2 − 𝛽 2 + 𝑥 2) 𝜕 𝜕𝑥 + 𝑢𝑒2𝜆𝑡(𝛽_2𝛼2𝜆+_2𝜆𝛼𝛾₃2 +𝛽𝛾_𝜆 +𝜆𝑥_2𝛼2 −𝛽𝜆𝑥_𝛼 −3𝛾𝑥_2𝜆 −1₂)_𝜕𝑢𝜕 , 𝑋₄= 𝑒−2𝜆𝑡 2𝜆 [− 𝜕 𝜕𝑡+ 1 𝜆(𝜆2(𝑥 − 𝛽) − 2𝛼𝛾) 𝜕 𝜕𝑥 +𝛾𝑢 𝜆2 (𝜆2𝑥 − 𝛼𝛾)_𝜕𝑢𝜕 ] , 𝑋₅= 𝑒𝜆𝑡[_𝜕𝑥𝜕 +_𝛼𝜆𝑢 (−𝛼𝛾 − 𝛽𝜆2+ 𝜆2𝑥)_𝜕𝑢𝜕 ] , 𝑋₆= 𝑒−𝜆𝑡[ 𝜕 𝜕𝑥+ 𝛾𝑢 𝜆 𝜕 𝜕𝑢] . (52) It should be noted that this case results in the Vasicek equation [26].

Case 1.3 (𝑞 = 2). (1) Consider 𝛼 ̸= 𝜆2/4𝛾. The additional Lie

point symmetries are given by

𝑋₃= 𝑒_2𝜅2𝜅𝑡_𝜕𝑡𝜕 + 𝑒2𝜅𝑡(1₂𝑥 − 𝛽𝜆_2𝜅2)_𝜕𝑥𝜕 +𝑢𝑒2𝜅𝑡 4𝛼𝜅3(−2𝛼𝛽2𝛾𝜆2+ (𝜅 + 𝜆) × ( (𝜅𝑥 − 𝛽𝜆) ( 𝜅2𝑥 − 𝛽𝜆2) − 𝛼𝜅2)) 𝜕 𝜕𝑢, 𝑋₄= −𝑒−2𝜅𝑡 2𝜅 𝜕 𝜕𝑡+ 𝑒−2𝜅𝑡 2𝜅2 (𝜅2𝑥 − 𝛽𝜆2) 𝜕 𝜕𝑥 −𝑢𝑒−2𝑡𝜅 4𝛼𝜅3 (𝛼𝜆 (2𝛽2𝛾𝜆 + 𝜅 (𝜅 + 𝜆)) − 4𝛼2𝛾𝜅 − 𝛽2𝜆3(𝜅 + 𝜆) + 𝜅3𝑥2(𝜅 − 𝜆) +𝛽𝜅𝜆𝑥(𝜅 − 𝜆)2) 𝜕 𝜕𝑢, 𝑋₅= 𝑒𝜅𝑡[ 𝜕 𝜕𝑥+ 1 2𝛼𝜅𝑢 (𝜅 + 𝜆) (𝑥𝜅 + 𝛽𝜆) 𝜕 𝜕𝑢] , 𝑋₆= 𝑒−𝜅𝑡[ 𝜕 𝜕𝑥− 1 2𝛼𝜅𝑢 (𝜅 − 𝜆) (𝑥𝜅 − 𝛽𝜆) 𝜕 𝜕𝑢] , (53) where𝜅 = √𝜆2− 4𝛼𝛾.

(2) Consider 𝛼 = 𝜆2/4𝛾. The additional Lie point symmetries are 𝑋₃= 𝑡_𝜕𝑡𝜕 + (𝑥₂ −3₄𝛽𝜆2𝑡2)_𝜕𝑥𝜕 + 𝑢 (1 2𝛽2𝛾𝜆2𝑡3+ 3 2𝛽2𝛾𝜆𝑡2− 3 2𝛽𝛾𝜆𝑡2𝑥 +𝛽2𝛾𝑡 −𝜆𝑡 2 − 3𝛽𝛾𝑡𝑥 + 𝛾𝑥2 𝜆 − 𝛽𝛾𝑥 𝜆 ) 𝜕 𝜕𝑢,

(7)

𝑋4= 𝑡2_𝜕𝑡𝜕 + (𝑡𝑥 −1₂𝛽𝜆2𝑡3)_𝜕𝑥𝜕 + 𝑢 (1 4𝛽2𝛾𝜆2𝑡4+ 𝛽2𝛾𝜆𝑡3− 𝛽𝛾𝜆𝑡3𝑥 + 𝛽2𝛾𝑡2− 𝜆𝑡2 2 −3𝛽𝛾𝑡2𝑥 + 2𝛾𝑡𝑥2 𝜆 − 2𝛽𝛾𝑡𝑥 𝜆 − 𝑡 2+ 𝛾𝑥2 𝜆2 )_𝜕𝑢𝜕 , 𝑋₅= 𝜕 𝜕𝑥+ 𝑢 ( 2𝛾𝑥 𝜆 − 2𝛽𝛾𝑡) 𝜕 𝜕𝑢, 𝑋6= 𝑡_𝜕𝑥𝜕 + 𝑢 (−𝛽𝛾𝑡2−2𝛽𝛾𝑡_𝜆 +2𝛾𝑡𝑥_𝜆 +2𝛾𝑥_𝜆₂ )_𝜕𝑢𝜕 . (54) Case 2 (𝑝 = 1)

Case 2.1 (𝑞 = 0). (1) Consider 𝛽 ̸= 𝛼/2𝜆 and 𝛽 ̸= 3𝛼/2𝜆. The

principal Lie algebra is extended by

𝑋₃= 𝑒𝜆𝑡[_𝜆1_𝜕𝑡𝜕 + 𝑥_𝜕𝑥𝜕 + 𝑢 (𝜆𝑥_𝛼 −𝛽𝜆_𝛼 −_𝜆𝛾)_𝜕𝑢𝜕 ] , 𝑋₄=𝑒−𝜆𝑡 𝜆 [− 𝜕 𝜕𝑡+ 𝑥𝜆 𝜕 𝜕𝑥+ 𝛾𝑢 𝜕 𝜕𝑢] . (55)

(2) Consider 𝛽 = 𝛼/2𝜆. The additional Lie point symmetries are 𝑋₃= 𝑒𝜆𝑡[1_𝜆_𝜕𝑡𝜕 + 𝑥_𝜕𝑥𝜕 + 𝑢 (𝜆𝑥_𝛼 −𝛾_𝜆−1₂)_𝜕𝑢𝜕 ] , 𝑋₄= 𝑒−𝜆𝑡 𝜆 [− 𝜕 𝜕𝑡+ 𝑥𝜆 𝜕 𝜕𝑥+ 𝛾𝑢 𝜕 𝜕𝑢] , 𝑋₅= √𝑥𝑒𝜆𝑡/2 𝜕 𝜕𝑥+ 𝜆𝑢√𝑥𝑒𝜆𝑡/2 𝛼 𝜕 𝜕𝑢, 𝑋₆= √𝑥𝑒−𝜆𝑡/2 𝜕 𝜕𝑥. (56)

(3) Consider 𝛽 = 3𝛼/2𝜆. The Lie point symmetries that extend the principal Lie algebra are

𝑋₃= 𝑒𝜆𝑡 𝜆 𝜕 𝜕𝑡+ 𝑥𝑒𝜆𝑡 𝜕 𝜕𝑥+ 𝑒𝜆𝑡( 𝜆𝑥 𝛼 − 𝛾 𝜆− 3 2) 𝑢 𝜕 𝜕𝑢, 𝑋₄= −𝑒−𝜆𝑡 𝜆 𝜕 𝜕𝑡+ 𝑥𝑒−𝜆𝑡 𝜕 𝜕𝑥+ 𝛾𝑢𝑒−𝜆𝑡 𝜆 𝜕 𝜕𝑢, 𝑋₅= √𝑥𝑒𝜆𝑡/2_𝜕𝑥𝜕 + 𝑒𝜆𝑡/2(𝜆√𝑥_𝛼 − 1 2√𝑥) 𝑢 𝜕 𝜕𝑢, 𝑋₆= √𝑥𝑒−𝜆𝑡/2_𝜕𝑥𝜕 −𝑢𝑒−𝜆𝑡/2 2√𝑥 𝜕 𝜕𝑢. (57)

Case 2.2 (𝑞 = 1/2). This has two subcases.

(1) Consider 𝛽 = 𝛼/2𝜆. This case results in the following extra Lie point symmetries:

𝑋₃= 𝑒𝜆𝑡 𝜆 𝜕 𝜕𝑡+ 𝑒𝜆𝑡(𝑥 − 2𝛼𝛾√𝑥 𝜆2 )_𝜕𝑥𝜕 + 𝑢𝑒𝜆𝑡(𝜆𝑥 𝛼 − 3𝛾√𝑥 𝜆 + 𝛼𝛾2 𝜆3 −1₂)_𝜕𝑢𝜕 , 𝑋₄= −𝑒−𝜆𝑡_𝜆 _𝜕𝑡𝜕 + 𝑒−𝜆𝑡(𝑥 −2𝛼𝛾√𝑥_𝜆₂ )_𝜕𝑥𝜕 + 𝑢𝑒−𝜆𝑡(𝛾√𝑥 𝜆 − 𝛼𝛾2 𝜆3 ) 𝜕 𝜕𝑢, 𝑋₅= √𝑥𝑒𝜆𝑡/2 𝜕 𝜕𝑥+ 𝑢𝑒𝜆𝑡/2( 𝜆√𝑥 𝛼 − 𝛾 𝜆) 𝜕 𝜕𝑢, 𝑋₆= √𝑥𝑒−𝜆𝑡/2 𝜕 𝜕𝑥+ 𝛾𝑢𝑒−𝜆𝑡/2 𝜆 𝜕 𝜕𝑢. (58)

(2) Consider𝛽 = 3𝛼/2𝜆. The extra Lie point symmetries are 𝑋₃= 𝑒𝜆𝑡 𝜆 𝜕 𝜕𝑡+ 𝑒𝜆𝑡(𝑥 − 2𝛼𝛾√𝑥 𝜆2 ) 𝜕 𝜕𝑥 + 𝑒𝜆𝑡(𝛼𝛾2 𝜆3 − 3 2+ 𝛼𝛾 𝜆2_√𝑥+ 𝜆𝑥 𝛼 − 3𝛾√𝑥 𝜆 ) 𝑢 𝜕 𝜕𝑢, 𝑋4= −𝑒 −𝜆𝑡 𝜆 𝜕 𝜕𝑡+ 𝑒−𝜆𝑡(𝑥 − 2𝛼𝛾√𝑥 𝜆2 ) 𝜕 𝜕𝑥 + 𝑒−𝜆𝑡(−𝛼𝛾_𝜆₃2 +_𝜆₂𝛼𝛾 √𝑥+ 𝛾√𝑥 𝜆 ) 𝑢 𝜕 𝜕𝑢, 𝑋₅= √𝑥𝑒− 𝜆𝑡 2 𝜕 𝜕𝑥+ 𝑒 −𝜆𝑡 2 ( 𝛾_𝜆 − 1 2√𝑥) 𝑢 𝜕 𝜕𝑢, 𝑋6= √𝑥𝑒 𝜆𝑡 2 𝜕 𝜕𝑥+ 𝑒 𝜆𝑡 2 (𝜆√𝑥_𝛼 −_𝜆𝛾 −_2√𝑥1 ) 𝑢_𝜕𝑢𝜕 . (59)

Case 2.3 (𝑞 = 1). This leads to six subcases.

(1) Consider 𝛽 ̸= 𝛼/2𝜆 and 𝛽 ̸= 3𝛼/2𝜆 and 𝛼 ̸= 𝜆2/4𝛾. The principal Lie algebra is extended by

𝑋₃= 𝑒_𝜅𝑡𝜅_𝜕𝑡𝜕 + 𝑥𝑒𝑡𝜅_𝜕𝑥𝜕 +𝑢𝑒_2𝛼𝑡𝜅(𝑥𝜅 − 𝛽𝜆_𝜅2 + 𝜆𝑥 − 𝛽𝜆)_𝜕𝑢𝜕 , 𝑋₄= −𝑒−𝑡𝜅 𝜅 𝜕 𝜕𝑡+ 𝑥𝑒−𝑡𝜅 𝜕 𝜕𝑥 +𝑢𝑒_2𝛼−𝑡𝜅(𝛽𝜆_𝜅2 − 𝛽𝜆 + 𝜆𝑥 − 𝑥𝜅)_𝜕𝑢𝜕 , (60)

(8)

where𝜅 = √𝜆2− 4𝛼𝛾. We note that this case gives us the CIR equation [26,27].

(2) Consider𝛼 = 𝜆2/4𝛾, 𝛽 ̸= 𝛼/2𝜆, and 𝛽 ̸= 3𝛼/2𝜆. The extra symmetries that extend the principal Lie algebra are given by

𝑋₃= 𝑡𝜕 𝜕𝑡 + 𝑥 𝜕 𝜕𝑥+ 𝛾𝑢 ( 2𝑥 𝜆 − 2𝛽𝑡) 𝜕 𝜕𝑢, 𝑋4= 𝑡2_𝜕𝑡𝜕 + 2𝑡𝑥_𝜕𝑥𝜕 + 𝛾𝑢 (4𝑡𝑥_𝜆 +4𝑥_𝜆₂ − 2𝛽𝑡2−4𝛽𝑡_𝜆 )_𝜕𝑢𝜕 . (61) (3) Consider 𝛼 = 𝜆2/4𝛾, 𝛽 = 𝛼/2𝜆. Additional Lie point symmetries are 𝑋₃= 𝑡𝜕 𝜕𝑡+ 𝑥 𝜕 𝜕𝑥+ 𝑢 ( 2𝛾𝑥 𝜆 − 1 4𝜆𝑡) 𝜕 𝜕𝑢, 𝑋₄= 𝑡2_𝜕𝑡𝜕 + 2𝑡𝑥_𝜕𝑥𝜕 + 𝑢 (4𝛾𝑡𝑥_𝜆 −₄1𝜆𝑡2−1₂𝑡 +4𝛾𝑥_𝜆₂ )_𝜕𝑢𝜕 , 𝑋₅= √𝑥𝜕 𝜕𝑥+ 2𝛾𝑢√𝑥 𝜆 𝜕 𝜕𝑢, 𝑋₆= 𝑡√𝑥_𝜕𝑥𝜕 + 2𝛾𝑢√𝑥 (_𝜆2₂ +_𝜆𝑡)_𝜕𝑢𝜕 . (62) (4) Consider 𝛼 = 𝜆2/4𝛾, 𝛽 = 3𝛼/2𝜆. Additional Lie point symmetries to the principal Lie algebra are given by

𝑋₃= 𝑡𝜕 𝜕𝑡+ 𝑥 𝜕 𝜕𝑥+ 𝑢 ( 2𝛾𝑥 𝜆 − 3 4𝜆𝑡) 𝜕 𝜕𝑢, 𝑋₄= 𝑡2_𝜕𝑡𝜕 + 2𝑡𝑥_𝜕𝑥𝜕 + 𝑢 (4𝛾𝑡𝑥_𝜆 −₂3𝑡 +4𝛾𝑥_𝜆₂ −3₄𝜆𝑡2)_𝜕𝑢𝜕 , 𝑋5= √𝑥_𝜕𝑥𝜕 + 𝑢 (2𝛾√𝑥_𝜆 −_2√𝑥1 )_𝜕𝑢𝜕 , 𝑋₆= 𝑡√𝑥 𝜕 𝜕𝑥+ 𝑢 ( 2𝛾𝑡√𝑥 𝜆 − 𝑡 2√𝑥+ 4𝛾√𝑥 𝜆2 ) 𝜕 𝜕𝑢. (63) (5) Consider 𝛽 = 𝛼/2𝜆, 𝛼 ̸= 𝜆2/4𝛾. The extra Lie point symmetries are 𝑋₃= 𝑒𝑡𝜅 𝜅 𝜕 𝜕𝑡+ 𝑥𝑒𝑡𝜅 𝜕 𝜕𝑥+ 𝑢𝑒𝑡𝜅( 𝑥𝜅 2𝛼− 1 4− 𝜆 4𝜅+ 𝜆𝑥 2𝛼) 𝜕 𝜕𝑢, 𝑋₄= −𝑒−𝑡𝜅 𝜅 𝜕 𝜕𝑡+ 𝑥𝑒−𝑡𝜅 𝜕 𝜕𝑥 + 𝑢𝑒−𝑡𝜅( 𝜆 4𝜅− 1 4+ 𝜆𝑥 2𝛼− 𝑥𝜅 2𝛼) 𝜕 𝜕𝑢, 𝑋₅= √𝑥𝑒(1/2)𝑡𝜅 𝜕 𝜕𝑥+ 𝑢√𝑥𝑒(1/2)𝑡𝜅 𝜆 + 𝜅 2𝛼 𝜕 𝜕𝑢, 𝑋6= √𝑥𝑒−(1/2)𝑡𝜅_𝜕𝑥𝜕 + 𝑢√𝑥𝑒−(1/2)𝑡𝜅𝜆 − 𝜅_2𝛼 _𝜕𝑢𝜕 , (64) where𝜅 = √𝜆2− 4𝛼𝛾.

(6) Consider𝛽 = 3𝛼/2𝜆, 𝛼 ̸= 𝜆2/4𝛾. The principal Lie algebra is extended by 𝑋₃= 𝑒𝜅𝑡 𝜅 𝜕 𝜕𝑡+ 𝑥𝑒𝜅𝑡 𝜕 𝜕𝑥+ 𝑒𝜅𝑡( 𝜅𝑥 2𝛼− 3𝜆 4𝜅− 3 4+ 𝜆𝑥 2𝛼) 𝑢 𝜕 𝜕𝑢, 𝑋₄= −𝑒−𝜅𝑡 𝜅 𝜕 𝜕𝑡+ 𝑥𝑒−𝜅𝑡 𝜕 𝜕𝑥 + 𝑒−𝜅𝑡(3𝜆 4𝜅 − 3 4+ 𝜆𝑥 2𝛼− 𝜅𝑥 2𝛼) 𝑢 𝜕 𝜕𝑢, 𝑋₅= √𝑥𝑒𝜅𝑡/2 𝜕 𝜕𝑥+ 𝑒𝜅𝑡/2( 𝜆√𝑥 2𝛼 + 𝜅√𝑥 2𝛼 − 1 2√𝑥) 𝑢 𝜕 𝜕𝑢, 𝑋₆= √𝑥𝑒−𝜅𝑡/2 𝜕 𝜕𝑥+ 𝑒−𝜅𝑡/2( 𝜆√𝑥 2𝛼 − 𝜅√𝑥 2𝛼 − 1 2√𝑥) 𝑢 𝜕 𝜕𝑢, (65) where𝜅 = √𝜆2− 4𝛼𝛾.

Case 3 (𝑝 = 2). We can conclude from (45) that 𝑞 can

only take the value0. Proceeding as before, we find that the principal Lie algebra extends for the case when𝛽 = 0 by the following symmetry operators:

𝑋₃= 𝑡𝜕 𝜕𝑡+ 1 2𝑥 ln 𝑥 𝜕 𝜕𝑥 + 𝑢 (𝜆2𝑡 4𝛼 + 1 4𝛼𝑡 − 𝛾𝑡 + 1 2𝜆𝑡 + 𝜆 ln 𝑥 4𝛼 + 1 4ln𝑥) 𝜕 𝜕𝑢, 𝑋₄= 𝑡2 𝜕 𝜕𝑡+ 𝑡𝑥 ln 𝑥 𝜕 𝜕𝑥 + 𝑢 (𝜆2𝑡2 4𝛼 + 1 4𝛼𝑡2− 𝛾𝑡2+ 1 2𝜆𝑡2− 1 2𝑡 +𝜆𝑡 ln 𝑥 2𝛼 + 1 2𝑡 ln 𝑥 + ln2𝑥 4𝛼 ) 𝜕 𝜕𝑢, 𝑋₅= 𝑥 𝜕 𝜕𝑥, 𝑋₆= 𝑡𝑥 𝜕 𝜕𝑥+ 𝑢 ( 𝜆𝑡 2𝛼+ 1 2𝑡 + ln𝑥 2𝛼) 𝜕 𝜕𝑢. (66) This case gives us the Black-Scholes equation [30].

4. Symmetry Reductions and Group

Invariant Solutions

We obtain symmetry reductions [31] and construct group invariant solutions of Case 2.1(1), that is, when𝑝 = 1, 𝑞 = 0, 𝛽 ̸= 𝛼/(2𝜆), and 𝛽 ̸= 3𝛼/(2𝜆), while all other constants in (4) are arbitrary. Equation (4) is then given by

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Case 2.1 (𝑋₁). The operator𝑋₁results in two invariants𝐽₁= 𝑥 and 𝐽₂ = 𝑢. The group invariant solution is given by 𝑢 = 𝑓(𝑥), where 𝑓 solves

𝛼𝑥𝑓󸀠󸀠(𝑥) + 𝜆 (𝛽 − 𝑥) 𝑓󸀠(𝑥) + 𝛾𝑓 (𝑥) = 0. (68)

Case 2.2 (𝑋₃). The operator 𝑋₃ gives the following two

invariants:

𝐽₁= 𝑥𝑒−𝜆𝑡, 𝐽₂= 𝑢 exp {𝛼𝛾𝑡 + 𝛽𝜆2𝑡 − 𝜆𝑥

𝛼 } . (69)

Hence, the invariant solution of (67) under𝑋₃is given by 𝑢 = 𝑓 ( 𝑥𝑒−𝜆𝑡) exp {𝜆𝑥 − 𝑡 (𝛼𝛾 + 𝛽𝜆

2₎

𝛼 } , (70)

where𝑓 satisfies

𝛼𝑧𝑓󸀠󸀠(𝑧) + 𝛽𝜆𝑓󸀠(𝑧) = 0, 𝑧 = 𝑥𝑒−𝜆𝑡. (71) The solution of the above equation is

𝑓 (𝑧) = 𝑐1𝛼𝑧(𝛼−𝛽𝜆)/𝛼

𝛼 − 𝛽𝜆 + 𝑐2, (72)

where 𝑐₁ and 𝑐₂ are arbitrary constants. Hence, the group invariant solution under𝑋₃is

𝑢 (𝑡, 𝑥) = [ [ 𝑐₁𝛼(𝑥𝑒−𝜆𝑡₎(𝛼−𝛽𝜆)/𝛼 𝛼 − 𝛽𝜆 + 𝑐2] ] × exp {𝜆𝑥 − 𝑡 (𝛼𝛾 + 𝛽𝜆 2₎ 𝛼 } . (73)

Case 2.3 (𝑋₄). The symmetry operator𝑋₄ gives the

invari-ants 𝐽₁ = 𝑥𝑒𝜆𝑡 and 𝐽₂ = 𝑢𝑒𝛾𝑡. Thus, the group invariant solution in this case is given by

𝑢 (𝑡, 𝑥) = [ [ 𝑐₁𝛼(𝑥𝑒𝜆𝑡₎(𝛼−𝛽𝜆)/𝛼 𝛼 − 𝛽𝜆 + 𝑐2] ] 𝑒−𝛾𝑡, (74) where𝑐₁and𝑐₂are arbitrary constants.

It should be noted that the operators𝑋₂ and𝑋_𝐵do not provide invariant solutions.

5. Conclusions

In this paper we carried out group classification of the general bond-option pricing PDE (4) for 𝑝 = 0, 1, and 2. The principal Lie algebra was found to be three-dimensional. These values of𝑝 resulted in 16 cases, which extended the principal Lie algebra. We presented the Lie point symme-tries 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.

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