Lie theory: applications to problems in Mathematical Finance and Economics

Lie Theory: Applications to Problems in Mathematical Finance and Economics Isabel Hernández1, Consuelo Mateos1, Juan Núñez1and Ángel F. Tenorio2 1Departamento de Geometría y Topología. Facultad de Matemáticas. Universidad de Sevilla. 2Dpto. Economía, Métodos Cuantitativos e Historia Económica. Escuela Politécnica Superior. Universidad Pablo de Olavide. [email protected] [email protected] jnv[email protected] [email protected] Abstract This paper is devoted to show and explain some applications of Lie theory to solve some problems in Economics and Mathematical Finance. So we put forward and discuss mathematical aspects and approaches for several economic problems which have been previously considered in the literature. Besides we also show our advances on this topic, mentioning some open problems for future research. Keywords and phrases: Mathematical Finance; Multidimensional screening problem; Technical progress; Holotheticity; Lie groups; Lie algebras. Introduction It is well-known how to use Lie Theory for solving problems related to other sciences different from Mathematics (like Physics or Engineering). However, most of them are usually experimental or technical; being at present quite more unknown the applications of this theory to other non-experimental, non-technical sciences. In this sense, this paper is devoted to show some applications of Lie Theory to Mathematical Finance and Economics. Let us note that such applications are only known in some very specific financial and economic topics. Regarding the relation between Lie Theory and some financial and economic problems, we would like to point out the existence of several recent works (all of them within this new century) which consistently represent the 1 application of well-known properties of Lie groups and algebras to several financial and economic problems and concepts. For example, Lo and Hui presented in [13] and [14] (at the beginnings of 2000s) different techniques based on Lie algebras to deal with the valuation of financial derivatives, in particular, multi-asset derivatives. They also deal with the PDE with timedependent coefficients, CEV models and moving barriers using Lie Theory (see [11] and [12]). Independently and using Lie algebras, Björk and Landén (2002) studied in [6] some models of interest-rate, previously introduced by Björk himself in [4]. Later, Polidoro (2003) studied a financial problem by using nilpotent Lie groups in [17]. Another interesting paper about the application of Lie Theory to Economics is due to Basov (2004), who described some methods based on Lie groups in order to solve the multidimensional screening problem in [3]. We also want to recall to Gaspar [8], who studied a general model for the structure of forward prices by using the methodology of Lie algebras given by Björk. Finally, Björk [4, 5] studied some applications of Lie algebras to certain economic concepts like constant volatility and constant direction volatility (i.e. considering that the volatility follows a constant vector field as its “direction”). The interested reader can consult the two previous references for an extensive and complete explanation about the concept of volatility and its different types. So this paper is devoted first to comment briefly on the basic aspects in some of the works previously cited, reaching an overall view for the current status in the application of Lie Theory to Economics, which constitutes a recent an innovative research. Secondly, we expound our advances on the concept of holotheticity (introduced by Sato in [18]) by endowing it with more mathematical explanations, in a similar way as one of the authors already did in [7]. More concretely, the mathematical groundwork for the concept of holotheticity is explained and checked in order to prove that the definition and some properties given by Sato have a correct mathematical foundation, and some problems relative to notation and formulation are also solved. 2 1 Lie groups of PDEs applied to multidimensional screening problems One of the main applications of Lie Theory to Economics is the use of Lie groups of partial differential equations to multidimensional screening problems, as Basov showed in [3]. In that paper, he described the use of group-theoretic methods for analyzing boundary problems arising when the Hamiltonian method is applied to solve the relaxed problem for the multidimensional screening problem. According to Basov, this technique provides a useful piece of information about the possible structure of the solutions and sometimes some particular solutions (but not general) can be obtained. In many industries the relationship between price paid by the customers and quantity purchased is not strictly proportional. Moreover, for nonlinear tariffs, the payment is often determined as a function of several characteristics, which can be differently valued by each customer. Consequently, a one-dimensional characteristic is not enough to capture the typology of customers in general. This leads to a multi-dimensional screening problem. This problem is described as follows: let us consider a multi-product monopoly producing ngoods (or, in a more simplified model, a good with n quality dimensions), being convex its cost function. An arbitrary consumer’s preferences over these goods are parameterizable by a m-dimensional vector and the typology of consumers follows a distribution with density function fover a convex open bounded set Ω⊂Rm. Let also suppose that fis continuously differentiable on Ωand extending by continuity to its closure Ω. For the monopolist, maximizing its profits is always one of its main goals. An interesting way to achieve this is the suitable choice of a tariff, bearing in mind that a tariff is simply a real function on the set of bundles of goods. Therefore, the tariff allows us to determine a consumer’s payment for a particular bundle of goods. Consequently, finding a solution of this problem often involves solving a boundary problem involving a system of nonlinear PDEs. We must take into consideration that there do not exist any general methods for solving the previous problem. However, when symmetries are present, the problem can be considerably simplified and even solved explicitly. In this sense, Basov applied the theory of Lie groups of PDEs to multidimensional 3 screening problems for the first time in the literature. Besides, Basov obtained particular explicit solutions for some types of these problems using his own methods. For example, he could compute explicit solutions for the monopolist problem. Even when explicit solutions cannot be obtained, the corresponding problem can be simplified and non-explicit solutions are obtained. Next, an example of Basov’s method and its application are shown according to the assumptions considered in this section. For a multi-product monopoly producing ngoods, consumers’ preferences over the bundles of these goods are parameterized by m-dimensional column vectors and the typology of consumers follows the distribution given by a density function fdefined on Ω∈Rm, which is open, bounded, and convex. Indeed, Basov assumed that: •Ω = Πm i=1 (ai, bi). •fis continuous and strictly positive on a convex open subset of Ω. When a consumer of type α∈Ωconsumes a bundle x∈X⊂R+nand pays the tariff t, the utility for him/her is given by: U(α, x, t) = m X i=1 αivi(x)−t(x), where each function vi(which represents the marginal rate of the superplus for a consumer of type α) is increasing, continuously differentiable and satisfying the Lipschitz condition in x∈X. Let us note that the notation U(α, x, t)for the utility of consumers of type αwas introduced by Basov himself to define a function simplify the classical notation given in [2, 21]. Hence, given the tariff t:X→R, the profits of the monopolist can be computed as: π=Z[t(x(α)) −c(x(α))] f(α)dα, where cis the production cost and x(α)is the bundle purchased by every consumer of type α. So the monopolist is interested in determining a tariff t which maximizes the utility on profits: s(α) = max x∈XU(α, x, t(x)). This is equivalent to solving a system of PDEs, whose solution can be obtained explicitly by Lie Theory. 4 To obtain the solution of the problem using symmetries, the following steps were considered by Basov: 1. Find an invariance group of the problem. 2. Find all the independent invariants of the group computed in the previous step. 3. Express the problem in terms of invariants of the group. 4. Attempt to solve it. To do this, it is necessary to introduce some notions related to Lie Theory. Next, we recall the definition of a one-parameter Lie group (which was incorrectly defined by Basov but appropriately applied): Definition 1.1. Let us consider an open set E⊂Rmand a function F: Rm×R→Esmooth at α∈Rmand analytic at τ∈R. If the set Gof coordinate transformations: gτ:Rm→E:α7→ F(α, τ) has a group structure by considering the inner product operation m:G×G→ Gdefined by: m(gτ1, gτ2) = gτ2◦gτ1=gτ1+τ2:α→F(F(α, τ1), τ2) = F(α, τ1+τ2), the pair (G, m)is called a one-parameter Lie group, where τis the parameter. Remark 1.2. In Definition 1.1, the function Fhas two inputs, a vector α∈Rmand a real number τ1. The unique output of Fis a vector F(α, τ1) belonging to the open set E⊂Rm. Consequently, F(α, τ1)is a vector in Rmand hence (F(α, τ1), τ2)belongs to the domain of the function F, for all τ2∈R. For screening problems, the arisen PDEs are of first and second-order and its expression is the following: Φ(α, u, ∇(u), D2u) = 0, where Φis a continuously differentiable function; α∈Rm;u:Rm→Ris twice continuously differentiable; ∇(u)is the gradient of uand D2(u)is the symmetric Hessian matrix. 5 For such a PDE, let us consider the following transformations of the independent and dependent variables: ½αi=Fi(α, u, τ), u=G(α, u, τ), where functions Fiand Gare infinitely differentiable at αand uand analytic at τ∈R, with Fi(α, u, 0) = αiand G(α, u, 0) = u. The set of transformations previously defined also has a structure of one-parameter Lie group when defining the product of two transformations as the inner product operation defined in Definition 1.1. Definition 1.3. Asymmetry group of the equation Φ(α, u, ∇(u), D2u) = 0 is any subgroup of the Lie group of transformations leaving invariant the equation. Therefore, solving a PDE is necessary to solve the problem presented in this section. This can be done by computing and using symmetry groups of such a PDE. 2 Non-linear PDEs in Mathematical Finance Another application of PDEs and Lie Theory to Economics is related to Mathematical Finance. This section is devoted to explaining Polidoro’s work [17] about this application. He studied a non-linear PDE arisen from solving a problem in Mathematical Finance. To solve such a PDE, he used some properties of both Lie commutator and Lie groups. More concretely, Polidoro studied a non-linear degenerate Cauchy problem arising from Mathematical Finance, proving the existence of a locally strong solution. Besides, the regularity of this solution was also studied in the paper by considering the framework of sub-elliptic operators on nilpotent Lie groups. Moreover, he gave sufficient conditions for the existence of global solutions. Let us consider the following PDE with three variables z= (x, y, t)∈R3: Lu ≡∂xxu+u ∂yu−∂tu=f, (1) that first appeared in a financial problem studied by Antonelli, Barucci and Mancino [1]. Polidoro proved that there exists a unique utility function 6 for the previos cited financial problem. Moreover, this solution is also the unique viscosity solution of the Cauchy problem corresponding to the given PDE with a suitable initial datum, namely: u(x, y, 0) = g(x, y).(2) Besides, he proved that the solution of the Cauchy problem (1)-(2) is defined in a short time interval [0, T )satisfying the following condition: |u(x, y, t)−u(ξ, η, τ)|≤ C(|x−ξ|+|y−η|+|t−τ|)1 2.(3) for all (x, y, t),(ξ, η, τ)∈R2×[0, T )and assuming that fand gare uniformly Lipschitz continuous functions. Indeed, Polidoro was interested in the interior regularity of the solution. First, we must note that both Standard Regularity Theory for distributional solutions and regularity results for viscosity solutions cannot be applied to the problem studied by Polidoro. On the contrary, the operator Lcan be hyperbolic in some cases. However, under some suitable hypotheses (namely, uis a classical solution of Lu ≡fwith f∈C∞and ∂xu > 0), the solution was proved to be smooth by Polidoro. To obtain such results, he applied the framework of Lie groups analysis. In fact, the operator defined in (1) is a second-order operator and the matrix of coefficients for second-order derivatives is semi-definite positive. Therefore, the solution of the equation Lu ≡f∈C∞is smooth in the directions in which the matrix is non-degenerate, but not necessarily in other directions. As an example of this problem, Polidoro considered the Kohn- Laplace operator in R3, defined as follows: L≡(∂x+ 2y∂t)2+ (∂y−2x∂t)2. This operator is defined as the addition of the directional derivatives X= ∂x+ 2y∂tand Y=∂y−2x∂t. So every solution of Lu = 0 is smooth in both directions. However, these solutions are also smooth in the direction of the commutator of both derivatives: [X, Y ] = −4∂t. The Kohn-Laplace operator is the simplest meaningful example studied by Hörmander [10]. In this way, Hörmander considered a set {X0, . . . , Xp} of directional derivatives (i.e. smooth vector fields on Ω) defined as: Xj= n X i=1 aij(x)∂xi,∀j= 0, . . . , p 7 where aij ∈C∞(Ω) for some open set Ω∈Rn. If f∈C∞(Ω), Hörmander proved the following result: if uis a solution on Ωof the equation X i,j XiXju+X0u=f, (4) and Rnis the linear span of the vector fields X0, . . . , Xpand their respective commutators, then u∈C∞(Ω). This result has been the starting point in the research of regularity properties for the operators involved in (4), as well as their relationship with some Lie group structures on Rn. For the non-linear equation (1), Polidoro applied the previously cited linear theory to study the corresponding regularity problem. In this way, Polidoro considered the linearized operator Lu=∂2 x+u ∂y−∂t, where uis used as a coefficient. He checked that the smoothness of the coefficients aij was an essential hypothesis which cannot be removed. Obviously, this assumption could not be done for uas a coefficient, because the goal of his paper was precisely proved the smoothness of the solution u. Let us note that the reader can also consider the Stochastic Calculus of Variations, developed by Malliavin [16] in 1976. It was originally introduced to study the smoothness of the densities of solutions of stochastic differential equations, obtaining a probabilistic proof of a condition for a partial differential operator to be hypo-elliptic. Malliavin’s works involve the use of Lie brackets of vector fields for diffusion matrices satisfying a kind of Hörmander’s condition. Taking into consideration the nonexistence of a general theory for operators with non-smooth coefficients, Polidoro had to represent the operator L following the structure in (4): Lu =X2u+Y u, by considering X=∂xand Y=u∂y−∂t. In this way, for any ς= (ξ, η, τ)∈Ω he took the operator approximation as: Lςu=X2u+Yςu, where Yς= (u(ς)+(x−ξ)∂xu(ς)) ∂y−∂t. Lςis a linear Hörmander operator and a good approximation of Lbecause, under Condition (3), it holds: |Lu(z)−Lςu(z)|=|u(z)−u(ς)−(x−ξ)∂xu(ς)| · | ∂yu(z)|≤ C|z−ς|. 8 Under these conditions, results about regularity could be obtained using representations of the solution uin terms of the fundamental solution of Lς. Consequently, the regularity problem had to be studied using Lie groups analysis. Besides, it was necessary to consider and apply the Lie commutator of Xand Y(i.e. [X, Y ] = ∂xu·∂y) and take into consideration that Hörmander condition is satisfied if ∂xu(z)6= 0, for every z. With this example, we have shown another application of Lie Theory for dealing with and solving questions related to problems in Finance. In fact, there are several papers giving explicit or concrete examples about how to apply the Lie group analysis for solving problems in Mathematical Finance, like [14, 15, 20]. Besides, the reader can also consult Gazizov and Ibragimov [9] to consult how Lie group analysis can be applied in general to problems in Finance involving PDEs. 3 Technical Progress and Lie groups A praiseworthy application of Lie Theory to Economics is the corresponding to study changes in a technology over the time. These changes can be differentiated in two types: those due to a technical progress or those produced by scale effects (both notions will be discussed in some detail later). This distinction between the two types of effects leads to Solow-Stigler Controversy, which will be also explained later. At this point, we want to highlight that Sato [18, 19] introduced the concept of holotheticity for a production function in the 1980s, solving the previously cited controversy using Lie Theory. The solution given by Sato will be explained next. Any given technology is expressed according to two variables: the capital Kand the labor L. In this way, such a technology can be represented by a production function Y=f(K, L), where Kand Lare two real vectors (whose dimensions can be, in general, different). Definition 3.1. A production function Y=f(K, L)is called neoclassical if the following two conditions hold: 1. Yis homogeneous of degree 1 (constant returns to scale). 2. Yis smoothly diminishes returns to individual factors (law of diminishing returns). 9 [11] C.F. Lo, C.H. Hui and P.H. Yuen: Constant elasticity of variance option pricing model with time-dependent parameters. Int. J. Theor. Appl. Fin. 3:4 (2000), 661-674. [12] C.F. Lo, C.H. Hui and P.H. Yuen: Option risk measurement with timedependent parameters. Int. J. Theor. Appl. Fin. 3:3 (2000), 581-589. [13] C.F. Lo and C.H. Hui: Valuation of financial derivatives with time-dependent parameters. Quantitative Finance 1(2001), 73-78. [14] C.F. Lo and C.H. Lui: Pricing multi-asset financial derivatives with timedependent parameters - Lie algebraic approach. Int. J. Math. Math. Sci. 32:7 (2002), 401-410. [15] C.F. Lo and C.H. Lui: Lie algebraic approach for pricing moving barrier options with time-dependent parameters. J. Math. Anal. Appl. 323:2 (2006), 1455–1464. [16] P. Malliavin: Stochastic calculus of variations and hypoelliptic operators. In: K. Itô (ed.): Proc. Int. Symp. Stochastic Diff. Eq. (Kyoto, 1976) (1978), Wiley. [17] S. Polidoro: A Nonlinear PDE in Mathematical Finance. In: F. Brezzi, A. Buffa, S. Corsaro and A. Murli (eds.): Numerical Mathematics and Advanced Application (2003), Springer. [18] R. Sato: The impact of technical change on the holotheticity of production functions. Economic Studies 47 (1980), 767-776. [19] R. Sato: Theory of technical change and economic invariance. Application of Lie groups (1981), Edward Elgar Publishing. [20] W. Sinkala, P.G.L. Leach and J.G. O’Hara: An optimal system and groupinvariant solutions of the Cox-Ingersoll-Ross pricing equation. Appl. Math. Comp. 201:1–2 (2008), 95–107. [21] R.B. Wilson: Non Linear Pricing (1993), Oxford University Press. 16