Group Quantization of Quadratic Hamiltonians in Finance

Date: February 2 2021 Version 1.0

Santiago Garc´ıa1

Abstract The Group Quantization formalism is a scheme for constructing a functional space that is an irreducible infinite dimensional representation of the Lie algebra belonging to a dynamical symmetry group. We apply this formalism to the construction of functional space and operators for quadratic potentials- gaussian pricing kernels in finance. We describe the Black-Scholes theory, the Ho-Lee interest rate model and the Euclidean repulsive and attractive oscillators. The symmetry group used in this work has the structure of a principal bundle with base (dynamical) group a + semi-direct extension of the Heisenberg-Weyl group by SL(2,R), and structure group (fiber) R . + By using a R central extension, we obtain the appropriate commutator between the momentum and coordinate operators [pˆ,xˆ] = 1 from the beginning, rather than the quantum-mechanical [pˆ,xˆ] = −i}. The integral transformation between momentum and coordinate representations is the bilateral Laplace transform, an integral transform associated to the symmetry group. Keywords Black-Scholes – Lie Groups – Quadratic Hamiltonians – Central Extensions – Oscillator – Linear Canonical Transformations – Mellin Transform – Laplace Transform

[email protected]

Contents Introduction

1 Galilean Transformations as Numeraire Change3 The Group Quantization formalism ([1], [2], [3], [6]) is a 2 The Group Quantization Formalism3 scheme for constructing a functional space that is an irreducible infinite dimensional representation of the Lie algebra 3 The WSp(2,R) Group6 belonging to a dynamical symmetry group. 4 Black-Scholes9 This formalism utilizes Cartan geometries in a framework 5 Linear Potential. Ho-Lee Model 12 similar to the Hamiltonian framework, where, in finance, the 6 Harmonic Oscillator 15 coordinates represent prices, rates ... and the conjugate mo- 7 Repulsive Oscillator 18 menta operators are the corresponding deltas. 8 Conclusions 19 We apply the Group Quantization formalism to a modified WSp(2,R) group. WSp(2,R) is the semi direct product of the Appendices 20 1 two-dimensional real symplectic group Sp(2,R) ≈ SL(2,R) A Definitions 20 by the Heisenberg-Weyl group. Our modification consists in B Instrument Prices in Momentum Space 21 that the embedded 2-dimensional translation subgroup in the Heisenberg-Weyl group has been extended by +, rather than arXiv:2102.05338v2 [q-fin.MF] 17 Feb 2021 C Lagrangian Formulation 21 R U(1), which is the usual extension group in the physics and References 23 mathematical literature. Our interest in the WSp(2,R) group (called Group of Inho- mogeneous Linear Transformations in [32]) is that this group is the symmetry group of the second-order parabolic differential equations. Using the WSp(2,R) symmetry, one can find coordinate systems and operators that map the equations of motion and the corresponding solutions ([13], [32]). The use in finance of mathematical methods previously de-

1 The infinite-dimensional representation theory of SL(2,R) was used by Bargmann ([40], [41]) for the description of the free non relativistic quantum mechanical particle. Contents 2/ 24 veloped in in physics is often based in the formal similarities and pricing Kernel both in the momentum space and in co- between the Black-Scholes equation and the quantum me- ordinate space. This Heisenberg-Weyl commutator makes chanical Schrodinger equation. These formal similarities have the bilateral Laplace transform the integral transform map- been explored both from the point of view of Lie algebra in- ping momentum and coordinate spaces. We give examples of variance ([12], [53],[36], [17] ) and global symmetries of the pricing in momentum space in appendixB. Black-Scholes Hamiltonian operator( [20], [23],[22],[21]). Sections5 to7 discuss the application of Group Quantization However, the mathematical properties of a wave equation solu- to theories with deformed Black-Scholes equations, i.e., Black- tion such as the Schrodinger equation are totally different than Scholes with a (at most quadratic) potential. the properties exhibited by a parabolic differential equation. The Group Quantization of the linear potential, that in finance Moreover, hermiticity and unitarity do not play a prominent represents the Hee-Lo interest rate theory, is discussed in role in finance: unlike the case of quantum mechanics, in section5. We obtain the polarized functions and pricing finance there is no probabilistic interpretation of solutions of equations both in momentum space and in coordinate space. the pricing equation, and time evolution is irreversible and We compare some of our results with the similarity methods non-unitary. developed in [13] and [32]. The Group Quantization formalism makes the most of the The harmonic and repulsive oscillators are studied in sec- principal fiber bundle structure linked to the central extension tions6 and7. The Heisenberg-Weyl commutator [pˆ,qˆ] = 1 of a Lie group. Although this formalism shares some features implies that the harmonic oscillator is generated by a hyper- with the Geometric Quantization scheme ([43], [44], [45]), bolic SL(2,R) subgroup, a change of scale operator, not a contrary to Geometric Quantization, the Group Quantization rotation. We discuss in detail the polarization constraints in formalism does not require the previous existence of a Poisson phase space and in a real analog of the Fock space. In spite algebra. In both formalisms, the word quantization signifies of our non-standard [pˆ,qˆ] = 1, we recover the usual ladder irreducible representation. operators when we construct higher order polarized functions. As an example of the applicability of this formalism to finance, The Mehler kernel is obtained first by building the Kernel we will obtain the Black-Scholes theory, the Ho-Lee model from the (Hermite) polarized functions, and later by applying and the Euclidean attractive and repulsive oscillators. the Linear Canonical Transformation technique described in [32]. 0.1 Outline From the standpoint of the Group Quantization formalism, the Section1 shows that a Galilean transformation on the space repulsive oscillator can be obtained from the harmonic and of Black-Scholes solutions constitutes a numeraire change. oscillator by making the frequency parameter pure imaginary. Strict Galilean invariance plays the role of phase invariance in Both oscillators, which as an interest rate theory represent quantum mechanics. quadratic interest rates, have also been recently proposed as We describe the main features of Group Quantization in sec- models for stock returns ( [61], [33]). tion2. Particularly important are the definition of the connec- AppendixA contains some general definitions in order to tion form, the polarization algebra and the concept of higher make this paper more self-contained and to establish notation. polarization. Group Quantization considers the action of the AppendixC shows examples of the Lagrangian formalism and group on itself, as opposed to the group acting on an exter- the relationship of the connection form in the Group Quanti- nal manifold. This guarantees the existence of two sets of zation formalism and the Poincare-Cartan´ form in classical commuting generators, the right invariant fields and the left mechanics. invariant fields. Left invariant fields provide naturally a set of We have favored clarity over mathematical rigor, and impor- polarization constraints that result in pricing equations, while tant topics such as group cohomology are just glossed over. the right invariant fields provide operators compatible with For simplicity, we use coordinates as much as possible in- these constraints. stead of a more compact notation. We refer the reader to the Section3 gives a brief survey of the WSp(2,R) group, the publications in the bibliography for a detailed and rigorous SL(2,R) group and the Linear Canonical transformations. treatment of the mathematical concepts relevant to this article. + By using a R central extension to create the embedded Heisenberg-Weyl group, Group Quantization provides the appropriate commutator between the momentum and coordinate operators [pˆ,xˆ] = 1, compatible with pˆ representing a delta, rather than the quantum-mechanical [pˆ,xˆ] = −i}, without resorting to analogies with any quantum theory or rotating to a fictitious Euclidean time. Section4 applies the Group Quantization formalism to a parabolic SL(2,R) subgroup in order to construct the Black- Scholes theory. We obtain polarization constraints, operators 2 The Group Quantization Formalism 3/ 24

1. Galilean Transformations as Numeraire Numeraire invariance is analogous to Phase Invariance in Change quantum mechanics, However, the relevant Black-Scholes gauge group is not the quantum mechanical U(1), but rather + As a motivation for this work, we show that the action of the multiplicative positive real line R . the Galilei group on the space of Black-Scholes solutions Exponential gauge factors (cocycles) analog to the one in constitutes a numeraire change equation (6) play a pivotal role in the Group Quantization The Black-Scholes equation for a stock that pays no dividends formalism. is ∂V 1 ∂ 2V ∂V = − σ 2S2 − rS + rV (1) 2. The Group Quantization Formalism ∂t 2 ∂S2 ∂S where σ is the stock volatility, r is the risk free rate, S is We describe the main features of the Group Quantization the stock price, t is time, and V = V(S,t) is the price of a formalism ([1], [2], [3], [6]). financial instrument. For simplicity, let’s assume that r and σ are constant. Under the change of variables 2.1 The Quantization Group 0 0 S → S0 = Sev t t → t0 (2) A Quantization Group G˜ is a connected Lie group 3 G˜ which is a central extension of a Lie group G (called the dynamical 0 where v is constant, the Black-Scholes equation becomes group) by another Lie group U called the structural group. ∂V 0 1 ∂ 2V 0 ∂V 0 The central extension defines a principal bundle (G˜,G,π,U). = − σ 2S02 − (r + v0)S0 + rV 0 (3) ∂t0 2 ∂S02 ∂S0 The structure group acts on the fiber by left multiplication. G is called base manifold of the bundle and U is called the 0 0 0 where V = V(S ,t ). fiber. The projection π is a continuous and surjective map The transformations (5) can be expressed in log-stock coordi- π : G˜ → G nates as x0 = x + v0, t v0 = v t0 = t (4) were x ≡ ln(S). We recognize the familiar Galilean transformations, with the position x representing the logarithm of the stock price, and the velocity v the stock’s growth rate. Given that the action of the Galileo group is linear in the log-stock cooordinates, it is convenient to express the Black-Scholes equation (1) in terms of the logarithm of the stock price ∂V 1 ∂ 2V ∂V = − σ 2 − µ + rV ∂t 2 ∂x2 ∂x 1 with µ = r − σ 2. Equation (3) becomes 2 Figure 1. One can visualize G˜ by imagining that through ∂V 0 1 ∂ 2V 0 ∂V 0 = − σ 2 − (µ + v0) + rV 0 (5) each point in the base manifold G there is a line (fiber) of 0 02 0 ∂t 2 ∂x ∂x points with different values of the fiber coordinate ζ ∈ U. where V 0 = V(x0,t0) The Black-Scholes equation is only covariant under the action A Quantization Group G˜ with structural group U is a Cartan of the Galilei group. Strict Galilean invariance can be restored geometry. The Cartan geometry is the geometry of spaces that 0 by multiplying the solution V by an exponential factor are locally (infinitesimally) like quotient spaces G = G˜/U. 0 0 Although G is not a subgroup of G˜ , G˜/U ' G as topo- V¯ (x0,t0) = eε(x ,t ) V 0(x0,t0) logical spaces any g˜ ∈ G˜ can be decomposed in two parts, v0 1 ˜ ε(x0,t0) = v0 t0 − x0 − µt0 (6) g˜ = (g,u), g ∈ G,u ∈ U. This means that, locally, G looks σ 2 2 like the Cartesian product of G and U. 2 In this paper, we will only consider Quantization Groups with One has + structural group U = R . ¯ 2 ¯ ¯ ∂V 1 2 ∂ V ∂V = − σ − µ + rV¯ 3 ∂t0 2 ∂x02 ∂x0 A Lie group is a differential manifold G endowed with a group composition law F : G × G → G such as F and its inverse are smooth, differentiable 2Note that Φ(x,t) ≡ exp(rt0 + ε(x0,t0)) is a solution of equation (5) applications. 2 The Group Quantization Formalism 4/ 24

2.2 Invariant Vector Fields 2.3 Cocycles The Group Quantization method considers the action of the The function ε in equation (7) is called a cocycle. Cocycles group on itself, as opposed to considering the group acting are restricted by the group law properties. on an external manifold. This guarantees the existence of two Associativity of the group law implies that ε must satisfy the sets of commuting group operators, the right invariant fields following functional equation and the left invariant fields. 00 0 00 0 00 0 0 The left and right translations, Lg and Rg are defined as ε(g ,g ) + ε(g g ,g) = ε(g ,g g) + ε(g ,g) L : G → G / L (g0) = gg0 g g and existence of an inverse element 0 0 Rg : G → G / Rg(g ) = g g

Lg and Rg are diffeomorphisms of G. Note that Lg and Rg ε(g,e) = ε(e,g) = ε(e,e) = 0 commute. Let g,g0,g00 ∈ G. In an abuse of notation, we write g00 = g0g where e is the identity element of G. for the base group G composition law. We denote an element + g˜ ∈ G˜ by g˜ = (g,ζ), with ζ in the structural group R . The 2.4 Coboundaries ˜ composition law for G is A coordinate change in the fiber U generates a mathematically g˜00 ≡ (g00,ζ 00) = (g0g,ζ 0ζ exp(ε(g0,g)) (7) trivial cocycle. If the fiber elements ζ undergo the coordinate change ζ → ζ e f (g) in the group law (7), the extension cocycle where G ∈ G, ζ ∈ + and ε(g0,g) ∈ is the extension cocy- R R ε gets an extra additive factor δc( f ) cle. 0 0 0 A vector field X is called a left invariant vector field (LIVF) if ε(g,g ) → ε(g,g ) + δc( f )(g,g ) T L Xg0 = Xgg0 g˜ ˜ ˜ ˜ with and a right invariant vector field (RIVF) if 0 0 0 T δc( f )(g,g ) = f (gg ) − f (g) − f (g ) Rg˜ Xg˜0 = Xg˜0g˜ In local coordinates, the LIVF are given by Trivial cocycles are called coboundaries4. The coboundary 0 δc( f )(g,g ) can can be undone by a change of coordinates. L L ∂ Xi = ∑ Xi,k + λi Ξ Although group extensions whose cocycles differ in a cobound- k ∂gk ary mathematically define the same group representation, the with dynamical effects generated by coboundaries are not neces- 0 ∂ ∂ε(g ,g) sarily trivial. Ξ = ζ λi ≡ ∂ζ ∂gi g=g0,g0=e One useful analogy is a change of variables in a differen- 0 canonical forms L ∂(g g)k tial equation: differential equations have to Xi,k ≡ which they can be reduced, however it is often convenient to ∂gi g=g0,g0=e work in non-canonical coordinates. For instance, even if the and the RIVFs Black-Scholes equation can be reduced to the simpler heat equation, in practice its solution is calculated using variables R R ∂ Xi = ∑ Xi,k + γi Ξ which allow to implement naturally the instrument’s pricing ∂gk k boundary conditions. where 0 ∂ ∂ε(g ,g) 2.5 Connection. Vertical Form Ξ = ζ γi ≡ ∂ζ ∂g 0 i g =g,g=e A connection on G˜ is a smooth choice of horizontal subspace ∂(g0g) ˜ XL ≡ k H and vertical subspace V such that G can be decomposed as i,k 0 ˜ ∂gi g0=g,g=e a direct sum, G = H ⊕V. L 5 The i-th component of a LIVF Xi is calculated as the deriva- The Group Quantization formalism uses a Cartan connection, tive respect to unprimed coordinates evaluated at the identity: where the geodesics coincide with the flows of an integrable first set g0 = g in the derivative, then set g = e. The i-th com- LIVF algebra. Concretely, the connection form in the Group L ponent of a LIVF Xi is calculated as the derivative respect to primed coordinates evaluated at the identity: first set g = g0 in 4Not all coboundaries are generated by a coordinate change. 5 the derivative, then set g0 = e. The notion of an affine connection on a differential manifold ( Poincare-´ Cartan form ) was introduced by the French mathematician Ellie´ Cartan in Note that, by construction, RIVFs and RIVFs commute. two papers published in 1923 and 1924 [42]. 2 The Group Quantization Formalism 5/ 24

fields in classical mechanics, and the integral flows of the CΘ elements are the geodesics of the Cartan connection. CΘ is defined as the set of LIVFs that leave the connection form (vertical form) Θ strictly invariant 7

X ∈ CΘ −→ d(iX Θ) = 0 iX (dΘ) = 0 (9)

The curvature form ω = dΘ can be written in local coordinates as

∂θi ω = ∑ dg j ∧ dgi = i, j ∂g j

∂θi ∂θ j ∑( − )dg j ⊗ dgi Figure 2. Transformation along the ﬁber U are vertical, i, j ∂g j ∂gi quantities deﬁned on the base manifold G ' G˜/U are horizontal. then, the second of equations (9) reads, using the results in section 2.5

Quantization formalism is the vertical part Θ of the Maurer- ∂θi ∂θ j 6 ∑( − )Xj = 0 Cartan form j ∂g j ∂gi + In this work, the vertical space is the structural group U = R ∂ Note that from the Maurer-Cartan equations, ω(X,Y) = 0 with a unique generator Ξ = ζ (see equation (7)). ∂ζ implies Θ([X,Y]) = 0. Θ provides a natural definition of horizontality by requiring that the horizontal fields belong to the kernel of Θ. 2.7 Polarization Algebra The Group Quantization formalism constructs an irreducible L iXL Θ ≡ Θ(X ) = 0 iΞ Θ ≡ Θ(Ξ) = 1 (8) representation of G˜ starting from the ring F of functions with domain G˜ and range R. where XL stands for all LIVFs (Left Invariant Vector Fields). The reduction is achieved by restricting the arguments of In coordinates, equations (8) read (see section 2.2) the functional space F using a set of horizontal operators 1 and building a polarized function space FP that provides a Θ = ∑ θi dgi + d Ξ d Ξ ≡ dζ representation of the group. i ζ The action of vertical fields (generators of the structural group V where the θi are solutions of a linear algebraic system U) X on FP is

L V V ∑ θk Xi,k + λi = 0 X Ψ = f (X )Ψ ∀Ψ ∈ FP k where f (XV ) is a function associated to the generator XV . We say that FP is a U- invariant functional space. In this 2.6 Characteristic Module + V document, U = R and f (X ) = 1. In the Group Quantization formalism, the elements of the We will see later in this document that, for the Black-Scholes characteristic module C are interpreted as evolution opera- Θ and the Ho-Lee groups, polarized functions have the meaning tors. C is an integrable system analogous to Hamiltonian Θ of ﬁnancial instrument prices. 6The Maurer-Cartan form. Γ : G˜ → G˜ takes values in the Lie algebra of G˜, G˜ and it is the unique left-invariant 1-form such that Γ|e is the identity 2.7.1 First Order Polarization map. L The Group Quantization formalism provides a natural choice Γ can be written in terms of the left invariant ﬁelds at the identity Xi ,i = 1,2...dim(G˜) and their dual forms θ for polarization by requiring invariance respect to a horizontal L i i L algebra P ( polarization algebra ) that imposes constraints Γ = ∑Xi ⊗ θ θ (Xi ) = δi, j i on F .

The Maurer-Cartan equations 7A vector ﬁeld X is a symmetry of a 1-form Γ if X leaves Γ semi-invariant,

i 1 i j k that is, the Lie derivative of Γ with respect to X is a total differential dθ = − ∑c jk θ ∧ θ . 2 X X jk LX Γ = d f −→ d(iX Γ) + iX (dΓ) = d f i X express the differential of the form in terms of the c jk, the structure constants where f is some function associated with the vector ﬁeld X. Then, CΘ = of the Lie algebra G˜ . kerΘ ∩ kerdΘ. 3 The WSp(2,R) Group 6/ 24

A first-order polarization (or just polarization) P is defined 2.10 Lagrangian as a maximal horizontal commutative left invariant algebra The classical Lagrangian L (x,x˙) is obtained by the projection containing the characteristic module CΘ, so that the con- onto the base manifold G of the connection form Θ along straints are compatible with the evolution operators. the CΘ flows (trajectories). See appendixC for a discussion The polarized functions FP will be characterized by condi- of the Poincare-Cartan´ form of classical mechanics and the tions of the form Lagrangian function.

F = {Ψ ∈ F /XPΨ = 0 ∀XP ∈ P} P 3. The WSp(2,R) Group

Since the elements of the algebra P are integrable vector 3.1 Heisenberg-Weyl Group. fields, a first order polarization defines a foliation of G˜. This The Heisenberg-Weyl group W is a central extension of the means that it is possible to select functional subspaces on on two dimensional Euclidean translation group. Its algebra is G˜ by requiring them to be constant along integral leaves of generated by three elements, P, Q and 1, where 1 is the the foliation. identity operator. The only non trivial commutator is 2.7.2 Higher Order Polarization [P,Q] = γ 1 γ ∈ C (10) When a first order polarization is not able to provide the required functional constraints, a higher order polarization It can be proven that the choices for γ are equivalent to select can be used. γ real or γ pure imaginary. As opposed to first-order polarizations described in (2.7.1), Let p,q ∈ R , θ ∈ C and define higher-order polarizations [8] contain higher-order differential W 1 P Q operators belonging to the left enveloping algebra. Higher (θ,x,q) ≡ exp(θ + x + p ) ˜ order polarizations do not define a foliation of G. The operators W specify8 a group composition law under 9 A higher-order polarization PH is a maximal subalgebra of multiplication . the left-invariant enveloping algebra that has no intersection 00 00 00 00 0 0 0 0 with the generators of the structural group U and commutes W (p ,x ,θ ) = W (p ,x ,θ )W(p,x,θ) with the first order polarization P. Using the the Baker-Campbell-Hausdorff formula10 , we ob- Commuting with the first order polarization ensures compati- tain bility with the action of the dynamical operators (RIVFs). As we shall see in the next sections, higher order polarizations p00 = p + p0 can be constructed by using a Casimir operator. x00 = x + x0 1 2.8 Operators θ 00 = θ + θ 0 + γ (px0 − xp0) (12) 2 . Commutativity of the right and left generators makes the The group law (11) provides a coordinate representation11 of RIVFs good candidates for quantum operators, that is, opera- the abstract operators P, Q and 1 acting on the functional tors that can reduce the representation from phase space, with 8 coordinates and momenta, to an irreducible group representa- The action of W on sl(2,R is tion with only coordinates or momenta. W(aQ + bP + η I)W−1 = aQ + bP + (η + w(a p − bx))1 Note that, since P is spanned by LIVFs, if XR is a right therefore, W acts on w as a three-dimensional space with Cartesian coordi- generator and Φ is polarized nates (p,x,θ). 9Exponentiation and composition of operators are to be considered in the P R R P P context of formal operator series. X (X Φ) = X (X Φ) = 0 ∀X ∈ P ∀Φ ∈ FP 10The Baker-Campbell-Hausdorff formula reads 1 ln(eX eY ) = X +Y + [X,Y]+ so the action of the RIVFs on the space of polarized functions 2 is well defined. Hence the quantum operators are the restric- 1 ([X,[X,Y]] − [Y,[X,Y]]) + ... tion of the RIVFs to the polarized U- invariant functional 12 space. when [X,Y] is a number

X Y X+Y+ 1 [X,Y] X Y Y X [X,Y] e e = e 2 → e e = e e e 2.9 Noether’s theorem One can prove that One can easily verify that the conection Θ is right invariant, 1 Θ(XR) = 0 for all RIVFs. The inner product of the verti- eX eY e−X = eXYe−X = Y + [X,Y] + [X,[X,Y]] + ... (11) 2 cal form Θ and the RIVFs gives the classical Hamiltonian constants of motion. 11These are right invariant generators, corresponding to the action of W0. 3 The WSp(2,R) Group 7/ 24 space Let an element g ∈ WSp(2,R) be parametrized by g(M,u,ζ) + 2 where M ∈ SL(2,R), ζ ∈ R and u ≡ (p,x) ∈ R . The Ψ(p,q,θ) = exp(θ)ψ(p,q) WSp(2,R) composition law is that provides the required Heisenberg-Weyl commutation re- g00 M00,u00,ζ 00 = g0 M0,u0,ζ 0 g(M,u,ζ) lations ∂ 1 ∂ with P 7→ Xp = − γ x ∂ p 2 ∂θ M00 = M0M ∂ 1 ∂ Q 7→ Xx = + γ p 00 0 ∂x 2 ∂θ u = u M + u ∂ 1 1 7→ X = ζ 00 = ζ 0 ζ exp − γ u0MΩuT (16) θ ∂θ 2 We get where Ω is the two-dimensional symplectic matrix in (15) and is given by the [P,Q] Heisenberg-Weyl commutator in (10). [Xp,Xx] = γ Xθ γ 3.3.1 Weyl Commutation Relations 3.2 The SL(2,R) group In the mathematics literature, such as [32] and [12], the usual SL(2,R), the special (unimodular) linear group in two real convention is that the momentum and position operator com- dimensions, has a natural representation by 2×2 real matrices mutator is [P,Q] = −i, which corresponds to an extension13 with determinant one of the 2-dimensional translations by U(1). Then, for diffusive equations such as the heat equation, one makes the theory a b M = ∈ SL(2, ), ad − bc = 1 (13) Euclidean at the last step by setting the time variable t to it. c d R In this work we set γ = 1, corresponding to a central exten- + The SL(2,R) group is an automorphism of the W group, pre- sion of the Euclidean translations by R . The reason is that serving the algebra commutators. Define new operators P0 in finance we want P to represent a change (a delta) in the and Q0 as linear combinations of P and Q, using the matrix quantity Q, wich usually represents a a price or a rate. This (13) choice has the added advantage that there is no need to consider a passage to a fictitious Euclidean time, since the time P0 P Q =a + b variable in the group represents the actual calculation time. Q0 P Q =c + d When we quote results from to the mathematics literature (for then, by direct computation, instance in section 3.4) we will assume, unless otherwise indicated, that the results are obtained with the usual commutator [P0,Q0] = [P,Q] (14) [P,Q] = −i.

SL(2,R) belongs to a class of Lie groups called the symplec- 3.3.2 Orbits and Subgroups tic groups Sp(2n,R) which leave invariant a skew-symmetric The WSp(2,R) generates the dynamics for quadratic Hamil- form and play an important role in the geometry of phase tonians ([13], [12], [2], [32] ) by the action of higher order space and Hamiltonian systems. One can verify the isomor- operators on W. The adjoint action of the group consist of six phism SL(2,R) ≈ Sp(2,R) for the two-dimensional symplec- distinct orbits [32]. Representative of these orbits are given tic matrix Ω by the following operators, that include quadratic elements 0 1 from the enveloping algebra of W M>ΩM = ΩΩ ≡ (15) − 1 0 P2 , P2 + Q, P2 + Q2 , P2 − Q2 , P, 1

3.3 The WSp(2,R) group We have the following isomorphisms: g(M,0,1) ≈ SL(2,R) WSp( , ) and g(1,u,ζ) ≈ W. The unit element is g(1,0,1) and the 2 R is a real, non-compact, connected, simple Lie −1 −1 group that is the semi direct product of the two-dimensional inverse element is g M ,−uM ,1/ζ real symplectic group Sp(2,R)(≈ SL(2,R)) by the Heisenberg- The different quantization groups considered in this paper will Weyl group W. be distinguished subroups of WSp(2,R). The WSp(2,R) group is a subgroup of the Schrodinger group In physics, the WSp(2,R) includes as subgroups the symme- in one dimension, and is called12 Group of Inhomogeneous try group of the free particle, the gravitational free-fall, as well Linear Transformations in reference [32]. as the symmetry group of the ordinary harmonic oscillator

12 13 Our composition law differs from [32] in that the extension group in (16) In quantum mechanics the cocycle in (16) is multiplied by a factor 1/} + is R , not R. for dimensionality reasons, and [P,Q] = −i}. 3 The WSp(2,R) Group 8/ 24 and the repulsive harmonic oscillator (with imaginary fre- In spite of formal similarities, existence of a Fourier transform quency). In ﬁnance, we will obtain the Black-Scholes theory, exists does not automatically imply the existence of a Laplace the Ho-Lee model and the Euclidean attractive and repulsive transform [32]. oscillators. The inverse transform is 1 Z c+∞ 3.4 Linear Canonical Transformations L −1( f )(x) = dpexp f (p) 14 2πi c−i∞ WSp(2,R) acts on functional spaces as integral transforms called Linear Canonical Transformations, whose kernel is a The integration contour is a vertical line on the complex plane, SL(2,R) matrix [32] such that all singularities of the integrand lie on the left of it.

Z 3.4.3 Bargmann Transform g(M,u)( f (x)) = W(M,x,x0) f (x0)dx0 (17) R The Bargmann transform has the complex kernel with kernel (using the notation of equation (13)) 1 1 −i B = √ 2 −i 1 e−iπ/4 W(M,x,x0) = √ exp(i(ax02 −2xx0 +d x2)/(2b)) (18) 2πb 3.4.4 Mellin Transform Linear Canonical Transformations (LICs), have the important The Mellin transform M property that composition of the transforms is equivalent to Z ∞ −z−1 multiplication of their SL(2,R) kernels ([2], [32]). M ( f )(z) ≡ F(z) = y f (y)dy (20) 0 LICs kernels can be analytically continued to SL(2,C), subject to some restrictions [2]. In this work, we will use, when is obtained from the Laplace transform L by the change of indicated, equation (17) with the mapping b → −ib coordinates p → −ln(y). The resolution of identity 1 W(M,x,x0) = √ exp(−(ax02 −2xx0 +d x2)/(2b)) (19) 2πb 1 Z c+i∞ x−sysds = yδ(x − y) 2πi c−i∞ which has been modified from (18) because our choice for the Heisenberg-Weyl commutator [P,Q] = 1. leads to the Mellin inversion formula The Fourier transform, the Laplace transform, the Bargmann Z c+i∞ − 1 z transform and the Mellin transform are examples of LICs. M 1( f )(y) ≡ f (y) = y F(z)dz 2πi c−i∞ 3.4.1 Fourier Transform The Mellin transform has been more widely used in finance The Fourier transform than the double sided Laplace transform ([55], [57], [59], Z ∞ [60]). The sign of the transform variable z in (20) differs from F ( f )(z) = f (p)eipzdp −∞ the usual definition in the mathematics literature in order to calculate transforms of positive powers of the stock price. has the SL(2,R) kernel

0 1 F = −1 0

3.4.2 Double Sided Laplace Transform The double sided Laplace transform

Z ∞ L ( f )(z) = f (p)epzdp −∞ can be obtained from F by the formal map p 7→ ip. Its kernel is a SL(2,C) matrix

0 i L = i 0

14These transforms also deﬁne pseudo-differential (hyperdifferential) operators associated to the corresponding SL(2,R) kernel [32]. 4 Black-Scholes 9/ 24

4. Black-Scholes For later convenience when deﬁning the Laplace transform, we use as Galilean cocycle17 4.1 Quantization Group 1 We apply the Group Quantization formalism to the WSp(2, ) ε (g,g0) = p0x + σ 2 pp0t + σ 2 p02t (23) R G 2 subgroup generated by the parabolic SL(2,R) matrix 18 2 εN represents a numeraire choice 1 σ t 2 MBS = σ ∈ R t ∈ R (21) 0 1 0 0 εN(g,g ) ≡ δc( fN) = µ p t (24)

MBS has only one eigenvalue λ = +1 and it cannot be di- and has the generating function (see section 2.4) agonalized. M is a shear transformation on 2: it leaves BS R µ the upper plane invariant, while it displaces the lower plane fN(g) = 2 x by σ 2t. This will be significant when finding a first order σ polarization in section 4.5. 4.2 Stock Volatility Composition Law Bargmann ([40], [41]) used a central extension of the Galileo The WSp(2,R) subgroup generated by the matrix (21) consti- group with U(1) for the description of the free non relativistic ˜ tutes the Black-Scholes quantization group B. Its composition quantum mechanical particle. The cocycle used by Bargmann law is obtained from the WSp(2,R) composition law (16) and was the matrix MBS, with some cocycle modifications that we will 15 explain below 0 i 0 0 0 εB(g,g ) = − m(xv − vx + vv t) (25) 2} t00 = t +t0 which is labeled by the particle mass m. Bargmann proved 00 0 p = p + p that two cocycles of the form (25) with different values of m x00 = x0 + x + σ 2 p0t are not equivalent, they cannot be transformed into each other 0 by a coboundary. ζ 00 = ζ 0ζeεBS(g,g ) (22) The cocycle (25) equals the WSp(2,R) cocycle + where t, p,x ∈ R and ζ ∈ R When deriving the group law, 1 0 T 1 0 0 2 0 we have used that − u MBSΩu = (xp − px + σ pp t) 2 2 00 0 00 0 MBS = MBS MBS ⇒ t = t +t with the substitutions

2 } v (t, p,x) σ → i p → The composition law for gives the familiar Galilean m σ 2 transformations: the Galilei group G is the base group of B˜, + 2 G ' B˜/R , with σ p corresponding to the Galilean boost.. The cohomological importance of mass in in quantum me- Our interpretation of the group coordinates is that x represents chanics has been extensively studied in the literature. The x stock volatility plays a similar role in finance. Formally, the the (dimensionless) logarithm of the stock price, S ≡ S0 e , p is the conjugate momentum of x, t is the calendar time, Black-Scholes theory is the quantum mechanics of a free → and σ is the stock volatility. Notice that interest rates are particle with an imaginary mass. The classical limit } 0 not explicitly present in the group law. Interest rates in the corresponds to the zero volatility limit. 17 Black-Scholes theory are not dynamical quantities and they εG equals the WSp(2,R) cocycle in (16) plus the coboundary generated will be incorporated in section (4.4.2) as a coordinate change by (px)/2 (see section 2.4). in the fiber ζ. The WSp(2,R) cocycle is 16 1 1 The extension cocycle εBS is the sum of the true cocycle εG ε (g,g0) ≡ − u0M ΩuT = (xp0 − px0 + σ 2 pp0t) W 2 BS 2 and the coboundary εN. The coboundary generated by (px)/2 is 0 0 0 εBS(g,g ) = εG(g,g ) + εN(g,g ) 1 1 δ ( px) = (x0 p + xp0 + σ 2 p02t + σ 2 pp0t) c 2 2 15With no modifications, the Group Quantization formalism leads to the heat equation. 18This coboundary will provide a constant first derivative coefficient in 16All true Galilean cocycles with the same σ differ in a coboundary from the Black-Scholes equation. It is possible to add a new group coordinate, the Galilean cocycle (23), i.e., they can be undone by a change of coordinates. much like a quantum gauge potential (see [7]), which allows the introduction The exception occurs only for the one dimensional Galilean group. In one of numeraires that depend on stock price and time. This approach will be dimension there is another true cocycle which will be used in section5. explored in a future work. 4 Black-Scholes 10/ 24

4.3 Lie Algebra and it dual form dΞ acquires a horizontal component Left Invariant Vector Fields 1 1 d Ξ = dζ 7→ dζ˜ + +r dt ≡ d Ξ˜ + r dt (29) ∂ ∂ ζ ζ˜ XL = + xΞ XL = p ∂ p x ∂x The vertical connection form Θ is modified in a straightfor- ∂ ∂ XL = + σ 2 p + E(p)Ξ ward way by equation (29) t ∂t ∂x ∂ Θ 7→ Θ˜ = Θ + r dt XL ≡ Ξ = ζ ζ ∂ζ Therefore, the time component f L of the LIVFs (determined with t by Θ˜ (XL) = 0) picks a new vertical part 1 2 2 E(p) ≡ σ p + µ p ∂ ∂ 2 f L 7→ f L − rΞ˜ t ∂t t ∂t Commutators which induces the following change in the left time generator The non-zero commutators are XL 7→ XL − rΞ˜ L L 2 L L L t t [Xt ,Xp ] = −σ Xx − µΞ [Xp ,Xx ] = −Ξ (26) Note that one obtains the same Lie algebra commutators (26) in the new coordinates. The RIVFs are not modified by this 4.4 Connection coordinate change. The expression for the the vertical form Θ and the curvature In the rest of this section we use the expression of the vectors form dΘ are (see section 2.5) fields and connection in these new coordinates and we omit ˜ ˜ Θ = −xdp − E(p)dt + d Ξ the notation ζ , Θ, etc. for brevity. 2 dΘ = dp ∧ dx − (σ p + µ)dp ∧ dt (27) 4.5 First Order Polarization with 4.5.1 Black-Scholes Equation in Momentum Space 1 In section4 we noted that MBS is a shear mapping where only d Ξ ≡ dζ the p space (upper plane) is invariant, and that M cannot be ζ BS diagonalized. These features indicate that the only possible first order polarization is a polarization in p-space, as we will 4.4.1 Characteristic Module verify below. CΘ is generated by a unique field XC The first order polarization algebra P is spanned 20 by the L L x-translations and the CΘ generator XC XC = Xt + µ Xx = L ∂ ∂ P = hX ,XCi + (σ 2 p + µ) + E(p)Ξ (28) x ∂t ∂x The polarized functions Ψ ∈ FP are found by imposing the The restriction of Θ to the CΘ flows gives the Black-Scholes Lagrangian, see appendix C.1. polarization constraints on the functional space FP ˜ 4.4.2 Interest Rates FP = {Ψ : G → R/Ψ(x, p,t,ζ) = ζΨ(x, p,t)} We introduce interest rates by performing a change of coordi- One has nates19 in the fiber ζ. The mapping L ˜ rt Xx Ψ = 0 → Ψ = ζ ψ(p,t) ζ 7→ ζ = ζ e r ∈ R XCΨ = 0 → does not change the group law (22), since t00 = t +t0, however ∂ψ 1 2 2 it alters the separation between vertical and horizontal spaces. + σ p ψ + µ pψ − r ψ = 0 (30a) ∂t 2 Under this coordinate change, the structural group generator Ξ becomes Equation (30a) is the Black-Scholes equation in momentum space. ∂ ∂ Ξ = ζ 7→ Ξ˜ = ζ˜ 20 L ∂ζ ∂ζ˜ Xp cannot belong to P because [XL,XL] = −σ 2XL − µ Ξ 19This is equivalent to a non-horizontal polarization [6], with constraints t p x of the form X Ψ = aΨ,a ∈ R rather than X Ψ = 0. 4 Black-Scholes 11/ 24

4.5.2 Polarized Functions we have

A separable solution of the Black-Scholes equation (30a) is R R 2 R R R [Xt ,Xp ] = σ Xx + µ Ξ [Xx ,Xp ] = Ξ ψ˜ (p,t) = e−Er(p)t Φ(p) 21 with Φ(p) an arbitrary function of p and All other brackets are zero . 1 Irreducibility is achieved by considering the action of the right E (p) ≡ σ 2 p2 + µ p − r r 2 invariant fields on the constants of motion Φ(p) in (31). After straightforward algebra, one finds We can write a general polarized function as an inverse Laplace transform (a Linear Canonical Integral transform) ∂ XR : Φ(p) → Φ(p) XR : Φ(p) → pΦ(p) (34) Ψ(ζ,x,t) = p ∂ p x 1 Z c+∞ ζ K(p,t)exp(px)Φ(p)dp (31) and 2πi c−i∞ 1 with −XR : Φ(p) → ( σ 2 p2 + µ p − r)Φ(p) t 2 K(p,t) = exp(−Er(p)t) (32) From (34) the price operator, xˆ, the momentum operator pˆ and The integration measure dp is the dual form of XL, the vector p the time evolution operator (Hamiltonian) Hˆ are given by field absent from the polarization algebra. We will justify the use of the inverse Laplace transform in section 4.8. ∂ 1 xˆ = pˆ = p Hˆ ≡ σ 2 pˆ2 + µ pˆ − r (35) In the Black-Scholes theory, polarized functions represent ∂ p 2 prices of financial derivatives, whereas the Φ(p) correspond to terminal (payoff) conditions. AppendixB shows how to price financial instruments in momentum space. 4.8 Operators in Coordinate Space 4.6 High Order Polarization The expressions (35) for the operators in momentum space suggest the following definition for the coordinate operator, xˆ 4.6.1 Black-Scholes Equation in Coordinate Space and the (non hermitian) momentum ,p ˆ operator in x-space The Black Scholes equation can be obtained directly in the coordinate representation by using a high-order polarization ∂ xˆ = x pˆ = [pˆ,xˆ] = 1 (36) (section 2.7.2). ∂x The second order operator Equation (36) is a direct consequence of our extension of + L L 1 2 L L the Heisenberg-Weyl subgroup by R (equation (11)). The 2- CP = Xt + µXx + σ Xx Xx 2 dimensional translation subgroup of the WSp(2,R) group acts is a Casimir operator commuting with all Black-Scholes LIVFs. on real exponentials, thus justifying the definitions (36) and L L Since Xx and Xp do not commute, CP defines two higher or- the use for the inverse bilateral Laplace transform in (31). This der polarizations. The x-space polarization is generated by is an opposition to quantum mechanics, where the Heisenberg- L hCP,Xp i Weyl subgroup acts on the unit circle instead, making the Fourier transform the mapping between the coordinate and L −px Xp Ψ = 0 → Ψ = ζ e ψ(x,t) momentum spaces. XCΨ = 0 → Using equations (36), the Black-Scholes Hamiltonian22 in ∂ψ 1 ∂ 2ψ ∂ψ coordinate space is + σ 2 + µ − rψ = 0 (33a) ∂t 2 ∂x2 ∂x 1 ∂ 2 ∂ Equation (33a) is the Black-Scholes equation in coordinate Hˆ ≡ σ 2 + µ − r (37) BS x2 x (price) space. 2 ∂ ∂

4.7 Operators in Momentum Space 21The inner product of the vertical form Θ and the RIVFs gives the (Galilean) Hamiltonian constants of motion 4.7.1 Right Invariant Vector Fields R Θ(Xx ) = 0 → p ≡ p0

R ∂ R ∂ R 1 2 2 X = X ≡ Ξ = ζ Θ(Xt ) = 0 → σ p + µ p − r ≡ E0 t ∂t ζ ∂ζ 2 R 2 ∂ Θ(Xp ) = 0 → x − (σ p + µ)t ≡ x0 XR = + pΞ x ∂x 22The non hermiticity of the Black-Scholes Hamiltonian has been exten- R ∂ 2 ∂ 2 Xp = + σ t + (σ p + µ)t Ξ sively studied in the literature ([47], [48], [54]). Non hermiticity is relatively ∂ p ∂x mild, with eigenvalues either real or appearing in complex conjugate pairs. 5 Linear Potential. Ho-Lee Model 12/ 24

4.8.1 Numeraire Coboundary Value KBS can also be written in terms of the pseudo-differential 25 The value of the numeraire parameter µ can be determined by operators associated with W(MBS) [32] requiring the stock price S ≡ ex to be a zero eigenvalue23 of 1 2 ∂2 24 τ Hˆ −rτ σ τ µτ ∂ the Hamiltonian operator . e BS f (x) = e e 2 ∂x2 e ∂x f (x) = Using the Black-Scholes Hamiltonian (37), we find Z +∞ 0 0 0 dx KBS(x,x ,τ) f (x ) (40) 1 −∞ Hˆ ex = 0 ⇒ µ = r − σ 2 BS 2 5. Linear Potential. Ho-Lee Model 4.9 Pricing Kernel 5.1 Quantization Group 0 The kernel KBS(x,x ,τ) for the Black-Scholes equation (33a) The SL(2,R) parabolic shear mapping (21) that has been used in the Black-Scholes theory will be used in this section for Z 0 0 Ψ(τ,x) = KBS(x,x ,τ)Ψ(0,x )dx constructing the linear potential model. The composition law for the new quantization group I˜ is is obtained by an in inverse Laplace transform from the momentum representation in (32) t00 = t +t0 p00 = p + p0 Z c+∞ 0 0 1 −Er(p)t p(x−x ) KBS(x,x ,t) = dpe e (38) x00 = x + x0 + σ 2 p0t 2πi c−i∞ 0 ζ 00 = ζ 0ζeεIR(g,g ) (41) The integral (38) does not exist for t > 0, in accordance to the fact that pricing problems in finance are time irreversible final where t, p,x ∈ and ∈ + with the cocycle value problems. R ζ R εIR Let t < 0 and define τ ≡ −t .Then the integral (38) exists and 0 0 0 0 εIR(g,g ) = εG(g,g ) + εN(g,g ) + εI(g,g ) we recover the well known Gaussian pricing kernel (g,g0) is the Galilean cocycle in equation (23) and (g,g0) 1 0 2 εG εN 0 −rτ 1 − 2 (x −x−µτ) KBS(x,x ,τ) = e √ e 2σ τ (39) is the coboundary term given in (24). We have added a new 2πσ 2τ 0 true cocycle (not a coboundary) εβ (g,g )

4.9.1 Derivation using LCTs 0 0 1 2 0 εβ (g,g ) = β t (x + σ p t) β ∈ R (42) The Black-Scholes pricing kernel can also be obtained with 2 the methods developed in references [13] and [32], using the . properties of the WSp(2,R) group and its representation as We interpret the group coordinates26 as x representing a short Linear Canonical Transformations. rate, p its conjugate momentum, and t the calendar time. The From equation (19), the SL(2,R) matrix (21) generates the parameter σ is the short rate volatility. LCT We will verify in the next sections that this group describes the Ho–Lee interest rate model. In quantum physics, the 0 1 0 2 2 W(MBS,x,x ) = √ exp(−(x − x) /(2σ t)) U(1) extension of this group describes the free fall (linear 2πσ 2 t potential). The Group Quantization formalism applied to the which is the heat equation kernel (or Weierstrass transform linear potential in physics can be found in [2]. W[]). 25For smooth functions 0 KBS(x,x ,τ) is obtained by multiplying the heat kernel W(MBS) a ∂ by the discount factor exp(−rτ) and setting τ = T −t, x → e ∂x f (x) = f (x + a) x + µτ. We note that x → x + µτ is the Galilean transforma- Combining the result above with the Gaussian integral tion (4) representing a numeraire change. 2 1 Z ∞ 2 ea = √ eay e−y /4dy 23This is actually a martingale condition [24]. 4π −∞ 24 In momentum space, the zero eigenvalue condition reads we get 1 1 2 2 2 2 ∂2 Z ∞ ( σ p + µ p − r)Ψ0(p) = 0 → Ψ0(p) = δ( σ p + µ p − r) 1 −y2 2 2 e ∂x2 f (x) = √ f (x − y)e dy 4π −∞ It is straightforward to prove, using the properties of the Dirac delta and the inverse Laplace transform, that if we identify Ψ0(x) with the stock price S, 26 1 Identifying group parameters is usually done after analyzing the polar- Ψ (x) ' exp(x), this requires µ = r − σ 2 ization, however in this case the identiﬁcation is simple enough. 0 2 5 Linear Potential. Ho-Lee Model 13/ 24

5.2 Lie Algrebra Equation (44a) is the Ho-Lee Equation in Momentum Space. 5.2.1 Left Invariant Vector Fields The general solution27 of (44a) is

From the composition law (41) and (42), we obtain 1 µ p2+ 1 σ 2 p3 ψ(p,t) = Φ(p + βt)e 2β 6β (45) ∂ XL ≡ Ξ = ζ ζ ∂ζ where Φ(p) is an arbitrary function of the momentum p. ∂ ∂ The bilateral Laplace transform relates the p-space and the XL = XL = + xΞ x ∂x p ∂ p x-space. From equation (45), the general expression of a polarized function in x-space can be written as L ∂ 2 ∂ Xt = + σ p + (E(p) + βx)Ξ (43) ∂t ∂x 1 Z c+∞ Ψ(ζ,x,t) = ζ e−βtx dpG(p,t)exp Φ(p) where 2πi c−i∞ 1 E(p) ≡ σ 2 p2 + µ p where 2 1 µ(p−βt)2+ 1 σ 2(p−βt)3 with the non-zero Lie Brackets G(p,,t) ≡ e 2β 6β L L L L 2 L [Xt ,Xx ] = −β Ξ [Xt ,Xp ] = −σ Xx − µΞ The integration contour is a vertical line on the complex plane, L L such that all singularities of the integrand lie on the left of it. [Xp ,Xx ] = −Ξ Note that, as opposed to the Black-Scholes case (section 4.3), 5.5 High Order Polarization. the time generator and the x-generator do not commute. 5.5.1 Ho-Lee Equation in Coordinate Space 5.3 Connection The pricing (evolution) equation can be found directly in x- space by using a high-order polarization (see section 2.7.2). The vertical form Θ and the curvature form dΘ are given by Let XC be the CΘ generator. The second order operator Θ = −xdp − E(p)dt − β xdt + dΞ 2 1 2 L L dΘ = dp ∧ dx − (σ p + µ)dp ∧ dt − β dx ∧ dt XP = XC + σ X X 2 x x with is a Casimir operator commuting with all LIVFs (equation 1 43). Since XL and XL do not commute, XL defines two higher d ≡ d x p P Ξ ζ L L ζ order polarizations, the p-space polarization Pp = {XP ,Xx }, L L and the x-space polarization Px = {XP ,Xp }. 5.3.1 Characteristic Module The x-space polarized functions are obtained by imposing CΘ is spanned by a unique field XC the Px polarization constraints on functions (sections) of the L L L form Ψ(z, p,x,t) = ζΨ(p,x,t) XC = Xt + µ Xq − β Xp = ∂ ∂ ∂ 1 XLΨ = 0 → Ψ = ζ V(x,t) − β + (pσ 2 + µ) + ( σ 2 p2 + µ p)Ξ p ∂t ∂ p ∂x 2 XPΨ = 0 → ∂V 1 ∂ 2V ∂V + σ 2 + µ + β xV = 0 (46a) 5.4 First Order Polarization ∂t 2 ∂x2 ∂x 5.4.1 Ho-Lee Equation in Momentum Space Equation (46a) is the Ho-Lee equation in coordinate space. As in the Black-Scholes case, there is only one first order If we write p-space polarization P spanned by the x-translations and the generator X . 1 Z c+∞ CΘ C V(x,t) = dpΦ(p,t)epx 2πi c−i∞ The polarized functions Ψ ∈ FP are found by imposing the polarization constraints on the functional space FP and substitute in (46a), we recover the first order polarization ˜ equations (44a). FP = {Ψ : I → R/Ψ(x, p,t,ζ) = ζΨ(p,q,t)} 27We will show in section 5.5 that the polarized functions can be expressed One has in the x-space using Airy functions. This can be anticipated by the identity L [16] Xx Ψ = 0 → Ψ = ζ ψ(p,t) Z c+∞ ! 1 xp+t p3 1 −x XCΨ = 0 → dpe = 1 Ai 1 c > 0 2πi c−i∞ (3t) 3 (3t) 3 ∂ψ ∂ψ 1 − β + σ 2 p2ψ + µ pψ = 0 (44a) The solutions that are not proportional to Ai grow very rapidly at infinity and ∂t ∂ p 2 they need to be constructed with different contours in the complex plane. 5 Linear Potential. Ho-Lee Model 14/ 24

Using the Feyman-Kac28 theorem, the the solution of (46a) boundary conditions constrain the values for the expansion can be written as coefﬁcients ai,bi and λi.

R T Note that the group contraction β → 0 is smooth and the group −β t X(s)ds V(x,t) = E(e φ(XT )|X(t) = x) law (41) reduces to the Black-Scholes group law, however β = 0 is a singular point in the integral for the momentum where the expectation is taken with respect to a normal process polarized functions, (45) and the polarization equation (44a). with volatility σ and drift µ (W is a Brownian montion) 5.7 Pricing Kernel dX = µdt + σ dW 5.7.1 Similarity Methods One can use the methods developed in [13] and [32] in order 5.6 Airy Expansion to ﬁnd coordinate changes mapping Black-Scholes solutions After performing the following change of function and inde- into solutions of the equation (46a) pendent variable For instance, it can be shown that if VBS(x,t) a solution of the

µ Black-Scholes equation (33a) with r = 0, then − x λ t V(x,t) ≡ e σ2 e U(y) 1 1 2 2 2 3 µ V(x,t) = Ωβ (x,t) VBS(x − βσ t ,t) y ≡ ( − λ − β x) 2 σ 2 2σ 2 with the polarization equation (46a) becomes the Airy equation βxt+ 1 β 2σ 2t3+ 1 µβt2 Ωβ (x,t) = e 6 2 (48) ∂ 2U − yU(y) = 0 (47) is a solution of (46a). Hence, the Ho-Lee pricing Kernel can ∂y2 be obtained from the Black-Scholes pricing Kernel (39) (with Therefore, the polarized functions can then be written as a r = 0) in an analogous manner superposition of Airy functions29 0 1 2 2 0 KI(x,x ,τ) = Ωβ (x,τ)KBS(x − βσ τ ,x ,τ) (49) µ ∞ 2 − 2 x λi t V(x,t) = e σ ∑ e (ai Ai(yi) + bi Bi(yi)) i=0 where we have switched to the variable τ = T −t. Notice that Ω (x,τ) is a solution of (46a). In fact, Ω (x,τ) is where a ,b ,λ ∈ and β 1 i i i R the expression for the Ho-Lee bond maturing at T

1 2 µ 2 yi ≡ ( ) 3 ( − λi − β x) −x(T−t)− σ (T−t)3+ µ (T−t)2 σ 2 2σ 2 Ω1(x,τ) = e 6 2 t ∈ [0,T]

This form is convenient for the analysis of complex boundary conditions in bond pricing problems, as shown in [38]. The 5.7.2 Pseudo-Differential Operators The pseudo-differential operator methods in section 4.9 can 28 Consider the differential operator L also be used to compute the evolution operator U. 1 ∂ 2 ∂ L ≡ ω2(x,t) + µ(x,t) From equation (46a). 2 ∂x2 ∂x 2 where µ(x,t),ω(x,t) and r(x,t) are functions of (x,t) , with x deﬁned in a 1 2 ∂ ∂ U(τ) = exp(τHI) HI ≡ σ + µ + β x real domain D ⊂ R and t a positive real number, t ∈ [0,T] Then, subject to 2 ∂x2 ∂x technical conditions, the unique solution of the PDE ∂ where τ = −t. We split HI into two operators ( + L + r(x,t)) f (x,t) = 0 x ∈ D,0 ≥ t ≤ T ∂t 1 ∂ 2 ∂ with terminal value f (x,T) = g(x), is given by the Feynman-Kac Formula A ≡ σ 2 + µ B = β x 2 ∂x2 ∂x −R T r(X,s)ds f (x,t) = E(e t g(XT )|X(t) = x) Given that the only non zero commutators are where the expectation is taken with respect to the transition density induced by the SDE ∂ [A,B] = βσ 2 + µ β [B,[A,B]] = −β 2 σ 2 dX = µ(X,t)dt + ω(X,t)dW ∂x with W a Brownian motion. it is possible to apply the left oriented extended version of the 29 There is a distinguished solution of equation (47), called Ai, that decays Baker-Campbell-Hausdorff formula [29] rapidly as y → +∞, while a second linearly independent solution Bi grows rapidly in this limit. Also, like Bessel functions, both Ai and Bi are oscillatory A+ B 1 3(2[B,[A,B]]+[A,[A,B]]) 1 2[A,B] B A with a slow decay for large values of their arguments. See reference [16] eτ τ = e 3! τ e 2 τ eτ eτ 6 Harmonic Oscillator 15/ 24

We obtain30 6.0.1 Group Composition Law

Using equation (16), MH generates the following composition H − 1 2 2 3 1 2 eτ I = e 3 β σ τ e 2 µ βτ law:

1 2 ∂2 1 τ2βσ 2 ∂ βτx σ τ µτ ∂ 00 0 × e 2 ∂x e e 2 ∂x2 e ∂x t = t +t 2 1 2 2 ∂ 1 2 ∂ ∂ 00 0 2 0 τ βσ 2 σ τ 2 µτ p = p + p cosh ωt + λ x sinh ωt = Ωβ (x,τ)e 2 ∂x e ∂x e ∂x (50) 00 0 −2 0 x = x + x cosh ωt + λ p sinh ωt 00 0 0 where Ωβ (x,τ) has been deﬁned in (48). We apply the opera- ζ = ζ ζ exp εH (g ,g) (53) tors in (50) from right to left, and use the Weierstrass theorem to recover equation (49) with the cocycle εH

0 1 0 0 1 (g,g ) = (px − xp )cosh ωt+ K (x,x0,τ) = Ω(x,τ)K (x − βσ 2τ2,x0,τ) εH I BS 2 2 1 −2 0 2 0 (λ p p − λ xx )sinh ωt 2

6. Harmonic Oscillator The matrix MH reduces to the Black-Scholes generating matrix M in the limit w → 0, and correspondingly, the quanti- As described in section 3.1, our version of the Heisenberg- BS zation group H˜ contracts to the Black-Scholes quantization Weyl group consists in the 2-dimensional translations centrally group G˜ (modulo coboundaries) in this limit. For brevity, we extended by +, such that the commutator between the mo- R have omitted the numeraire coboundary31 (24) in the compo- mentum and coordinate operators is [pˆ,xˆ] = 1. sition law (53). We will verify in this section that in our modiﬁed version of the WSp(2,R) group, the hyperbolic SL(2,R) subgroup 6.1 Polarization using Orthogonal Coordinates generates the harmonic oscillator quantization group H˜ The SL(2,R) matrix R

−2 √ cosh ωt λ sinh ωt ω 1 λ −1/λ MH = 2 λ ≡ (51) R = √ (54) λ sinh ωt cosh ωt σ 2 λ 1/λ

transforms MH into a change of scale (squeeze) operator where ω, σ,t ∈ R By contrast, in quantum mechanics ([3], [5]), where the 2- eωt 0 D = R−1 M R = dimensional translations are centrally extended by U(1), the H H 0 e−ωt commutator between the momentum and coordinate operators is pure imaginary, [pˆ,xˆ] = −i}, and the harmonic oscillator The change of coordinates quantization group is generated by an elliptic SL(2, ) sub- R 1 1 group A =√ ( p − λ x) 2 λ −2 1 1 cos ωt λ sin ωt B =√ ( p + λ x) (55) (52) λ −λ 2 sin ωt cos ωt 2 splits the phase space (p,x) into orthogonal subspaces. The The SL(2,R) matrix (51) acts as a squeeze mapping of the group law (53) is written as Euclidean plane, whereas (52) corresponds to a rotation. Both subgroups can be transformed into each other by the the map- t00 = t +t0 ping ω → iω. Notice that the mapping between subgroups is A00 = A + A0e−ωt not the Euclidean time rotation t → it. B00 = B + B0eωt 00 0 ε (g0,g) 30We have interchanged the exponential factors ζ = ζ ζ e H

1 τ2βσ2 ∂ βτx e 2 ∂x e with 1 since the commutator is a number ε (g,g0) = (B0Aeωt − BA0 e−ωt ) H 2 1 τ2βσ2 ∂ βτx 1 τ3β 2σ2 βτx 1 τ2βσ2 ∂ e 2 ∂x e = e 2 e e 2 ∂x 31The generating function for this coboundary is µ x, and the ﬁnal expres- −2 0 0 0 sion is µ(λ p sinh ωt + x cosh ωt − x ), with µ real. 6 Harmonic Oscillator 16/ 24

Left Invariant Vector Fields Left Invariant Vector Fields

L ∂ ∂ − ∂ L ∂ ∂ ∂ 2 2 X = − ω A + ω B Xt = + ω λ x + ω λ p t ∂t ∂A ∂B ∂t ∂ p ∂x 1 L ∂ 1 L ∂ 1 L ∂ X = + BΞ X = − AΞ Xx = − pΞ A ∂A 2 B ∂B 2 ∂x 2 ∂ 1 XL = + xΞ with Lie Brackets p ∂ p 2

L L L L L L [Xt ,XA ] = wXA [Xt ,XB ] = −wXB L L −2 L L L 2 L L L [Xt ,Xp ] = −ω λ Xx [Xt ,Xx ] = −ω λ Xp [XA ,XB ] = −Ξ L L [Xp ,Xx ] = −Ξ

All other brackets are zero. Connection The vertical form Θ and the curvature form dΘ are given by Connection

1 1 Θ = (AdB − BdA) − wABdt + dΞ Θ = (pdx − xdp) − E(p,x)dt + d Ξ 2 2 −2 2 d Θ = dA ∧ dB − wAdB ∧ dt − wBdA ∧ dt dΘ = dp ∧ dx − ω λ pdp ∧ dt + ω λ xdx ∧ dt (57)

L with CΘ is spanned by the time generator Xt . 1 −2 2 2 2 E(p,x) ≡ ω(λ p − λ x ) First Order Polarization 2 1 d Ξ ≡ dζ (58) We choose the ﬁrst order polarization algebra spanned by ζ

L L P = hXB ,Xt i L The time translations Xt are the only generator of the characteristic module CΘ. The polarized functions Ψ ≡ ζ Ψ(A,B,t) are found by imposing First Order Polarization

L AB There are two ﬁrst order polarizations P in phase space XB Ψ = 0 → Ψ = ζ e 2 Ψ(A,t) L L −1 L L ∂Ψ ∂Ψ hXt ,Y+i / Y+ ≡ λ Xp + λ Xx (59a) X Ψ = 0 → − ωA = 0 (56a) t ∂t ∂A L L −1 L hXt ,Y−i / Y− ≡ λ Xp − λ Xx

The readers familiar with quantum mechanics will ﬁnd equa- We choose (59a) as polarization. Let Ψ a function of the form tion (56a) formally similar to quantum harmonic oscillator Ψ(ζ, p,q,t) = ζ Ψ(p,q,t) coherent state equations in Fock space ([3], [5]), except that here the coordinate A is real, not complex. Y+Ψ = 0 → We give the solution of these polarization constraints in equa- ∂Ψ ∂Ψ 1 λ − λ −1 + (λx − λ −1 p)Ψ = 0 (60) tion (62). Equation (56a) coincides with equation (61) ob- ∂ p ∂x 2 tained in the next section 6.2. The solution of (60) is

6.2 First Polarization in Phase Space 1 AB Ψ(x, p,t,ζ) = ζψ(A,t)e 2 We use now the original phase space coordinates (p,q) in the composition law (53). where A = A(p,q) and B = B(p,q) are the orthogonal coordi- Since the generating matrix M does not leave invariant either nates (55). Then the p or the x space, a ﬁrst order polarization cannot map the L ∂ψ ∂ψ phase space (p,x) into either the momentum or the coordinate X Ψ = 0 → − ω A = 0 (61) t ∂t ∂A space. However, the results in this section will be illustrative of a phase space formulation for the harmonic oscillator and The constraint (61) is the previously found (56a) using orthog- will also be useful for ﬁnding a higher order polarization. onal coordinates. 6 Harmonic Oscillator 17/ 24

6.3 First Order Polarized Functions HI can be written in terms of raising and lowering (ladder) Using the separable solution of (61) operators by using the diagonalizing SL(2,R) matrix (54)

ω nt n ωnt + e A e n ∈ 1 † † N H = ω (a a + aa ) I 2 the polarized functions are expressed as an infinite series with ∞ 1 ω nt n AB Ψ(ζ, p,x,t) = ζ e αn A e 2 (62) ∑ 1 −1 † 1 −1 n=0 a ≡ √ (λ pˆ − λ xˆ) a ≡ √ (λ pˆ + λ xˆ) 2 2 where αn ∈ R are the expansion coefficients. First order polarization in non orthogonal (p,q) coordinates where leads naturally to a financial theory in phase space, with ∂ pˆ = xˆ = x 1 AB 1 ( −2 p2− 2 x2) ∂x F(p,q) = e 2 = e 4 λ λ Note that a,a† are the regular ladder operators that are used the analog of a Husimi quasi-probability. The expression for the construction of the harmonic oscillator Fock space for the quantum harmonic coherent states in the Fock basis in quantum mechanics. The Heisenberg-Weyl commutation and the Bargmann-Segal transform can be obtained32 from relations [pˆ,xˆ] = 1 imply the commutator [a,a†] = 1 since (62) by mapping the phase space (p,q) into by the simple C SL(2, ) is an automorphism of the Heisenberg-Weyl sub- correspondence p → ip. R group (equation(14)). 6.4 High Order Polarization 6.4.2 Polarized Functions in Coordinate Space 6.4.1 Harmonic Oscillator Equation in Coordinate Space It is well know that the solution of equation (64) can be repre- The pricing equation can be obtained directly in x-space by sented as a series in Hermite functions ϕ using a high-order polarization. n ∞ The second order operator 1 −ω(n+ )τ ψ(τ,x) = ∑ e 2 αn ϕn(λx) (66) 1 −2 L L 1 2 L L n=0 X = X + ωλ X X − ωλ X X P t 2 x x 2 p p with αn ∈ R the expansion coefficients. The Hermite functions is a Casimir operator commuting with all LIVFs. ϕn have the following expression([16]) L L Since Xx and Xp do not commute, XP defines two higher order polarizations, hX ,XLi and hX ,XLi. The coordinate space 1 − 1 x2 P x P p ϕn(x) = p √ Hn(x)e 2 (67) L nn representation is generated by hXP,Xp i 2 ! π

L −px/2 where Hn are the Hermite polynomials. ϕn form an orthonor- Xp Ψ = 0 → Ψ(ζ, p,q,t) = ζe ψ(x,t) mal basis in L2(R) and fulﬁll the eigenvalue condition XPΨ = 0 → 2 2 ∂ψ 1 −2 ∂ ψ 2 2 ∂ 2 + ω λ − λ x ψ = 0 (63a) − x ϕn(x) = −(2n + 1)ϕn(x) ∂t 2 ∂x2 ∂x2

Since in finance one is usually interested in final value prob- The relationship between the (first) polarized functions (62) lems, we make the change τ = T −t, where T is a maturity and the polarized functions (66) is given by a modified Bargmann time and t ≤ T. Equation (63a) becomes transform ∂ψ = HI ψ (64) 1 ∂τ ϕn(λ x) = p √ × n! π wit HI the harmonic oscillator Hamiltonian in coordinate Z i∞ Z ∞ √ 1 n −B2/2+ 2λ xB−λ 2 x2/2 −AB space dp dxA e e 2πi −i∞ −∞ 2 1 −2 ∂ 2 2 HI ≡ ω λ 2 − λ x (65) 2 ∂x where A = A(p,q) and B = B(p,q) are the variables defined in 32( The zero energy 1/2 is the change or coordinates in the fiber, ζ → (55). The integral over the momentum p is an inverse double ζ exp(t/2) sided Laplace transform. 7 Repulsive Oscillator 18/ 24

6.5 Pricing Kernel A discussion of heat kernels and Mehler-type formulas based The the orthogonality of the expansion (66) makes possible to on group-invariant solutions can be found in reference [14]. 0 express the pricing kernel KI(x,x ,τ), deﬁned by Chapter 9 of [32] gives Baker-Campbell-Hausdorff relations between pseudo-differential operators for the harmonic oscil- Z 0 0 lator based on composition of SL(2, ) matrices. Ψ(τ,x) = KI(x,x ,τ)Ψ(0,x )dx R 6.6 Financial Interpretation as a sum of products of Hermite polynomials in x and x0 Using the Feynman-Kac formula, we can write the solution ∞ −ω n 1 τ 0 − ω (e ) of (63a) as K (x,x ,τ) = λ e 2 τ I ∑ 2nn! =0 − 1 γ R T X(s)2 ds ψ(x,t) = E(e 2 t ψ(XT ,T)|X(t) = x) 0 1 2 (x2+x02) × H ( x)H ( x )e 2 λ n λ n λ (68) 2 2 with γ ≡ ω /σ and where the expectation is taken with re- By applying the Mehler formula ([39]) spect to a normal process with volatility σ. This equation

∞ n describes a derivative with a payoff that is discounted quadrat- ρ 1 2 2 ically with the oscillator level x. A drift term can be easily ∑ n Hn(x)Hn(y)exp( (x + y )) = =0 2 n! 2 added with a coboundary generated by x. 1 4ρ xy − (1 + ρ2)(x2 + y2) A more interesting approach is to use the harmonic oscillator p exp 2 1 − ρ2 2(1 − ρ ) as a process describing stocks with non-normal returns with a correlation given by the Mehler Kernel (69). The reference to the equation (68), with ρ = exp(−ωτ), one obtains the [61] proposes a quantum harmonic oscillator as a model for Mehler kernel, a generalized bivariate Gaussian probability the market force which draws a stock return from short-run density ﬂuctuations to the long-run equilibrium.

0 λ 1 2 KI(x,x ,τ) = √ exp( λ xAx|) (69) 7. Repulsive Oscillator 2π sinh ωτ 2 where we have made the change τ = T −t, τ ≥ 0, and x ≡ Using the arguments in section6 for the harmonic oscillator, 0 + (x,x ). A is the SL(2,R) matrix because of the R central extension in the Heisenberg-Weyl subgroup of WSp(2, ), the quantization group for the repul- R −coth ωτ csch ωτ ˜ A = sive oscillator HR is generated by the elliptic SL(2,R) matrix csch ωτ −coth ωτ −2 cos ωt λ sin ωt MR = with coth and csch the hyperbolic cotangent and cosecant, −λ 2 sin ωt cos ωt respectively. √ where ω, σ,t ∈ ,λ ≡ ω/σ A is singular when ω → 0. However R M is obtained from the harmonic oscillator generating matrix R 1 −1 1 (51) by the mapping lim ω A = ω→0 τ 1 −1 ω 7→ i ω (70) As expected from the group law (53), the Mehler kernel maps into the heat kernel (hence the Black-Scholes theory) when For brevity, in this article we will not apply the full Group ω → 0 Quantization scheme to the repulsive oscillator. Most results for the repulsive oscillator can be readily obtained from the 1 0 2 0 1 2 (x−x ) lim KI(x,x ,τ) = √ e 2σ τ harmonic oscillator results and the correspondence (70). ω→0 2 2πσ τ The repulsive oscillator group composition law is 00 0 6.5.1 Derivation using LCTs t = t +t 00 0 2 0 The Mehler kernel (69) can also be obtained using the LCT p = p + p cos ωt + λ x sin ωt 00 0 −2 0 associated ([13], [32] ) to the harmonic oscillator generating x = x + x cos ωt + λ p sin ωt matrix (51). From (19) 00 0 0 ζ = ζ ζ exp εR(g ,g) 0 1 02 0 2 W(MH,x,x ) = √ exp(−(ax −2xx +d x )/(2b)) with the cocycle ε 2πb R 0 1 0 0 where a,b,c,d refer to the elements of (51). By direct substi- εR(g,g ) = (px − xp )cos ωt+ tution, we get 2 1 −2 0 2 0 0 0 (λ p p − λ xx )sin ωt W(MH,x,x ) = KI(x,x ,τ) 2 8 Conclusions 19/ 24

One can easily verify that the high-polarization equation for Group Quantization generates representations of a financial the repulsive oscillator is (compare with (63a)) theory in different functional spaces, allowing for alternative pricing frameworks such as the use of Laplace and Mellin 2 2 ∂ψ 1 2 ∂ ψ 2 ω transforms. In the case of Black-Scholes and Ho-Lee, the + σ + γ x ψ = 0 γ ≡ (71) ∂t 2 ∂x2 σ2 polarized functions have the meaning of prices of financial instruments, while their meaning for the oscillators is object whose separable solutions are time exponentials and Parabolic of active research. cylinder functions in x ([16], [32], [35], [31]). Note that upon the analytic continuation (70) the Hermite functions (67) turn As we have shown in this article, among other interesting into Parabolic cylinder functions. features, the Group Quantization formalism provides naturally the functional constraints (polarization algebra or higher The quantum repulsive oscillator [34] is a well known model polarization) for deriving the pricing equations. This makes in the literature. A complete survey of the repulsive oscillator Group Quantization a versatile methodology for construct- in the context of WSp(2,R) can be found in [32]. Chapter ing a financial theory from symmetry arguments alone, using 9 of [32] gives special Baker-Campbell-Hausdorff relations solely the principal bundle structure of a centrally extended between pseudo-differential operators for the repulsive oscil- Lie group. lator.

7.1 Financial Interpretation Using the Feynman-Kac theorem, the solution of (71) can be written as

1 γ R T X(s)2 ds ψ(x,t) = E(e 2 t ψ(XT ,T)|X(t) = x) where the expectation is taken with respect to a normal process with volatility σ. This equation describes a derivative with a payoff that grows quadratically with the oscillator level x. Reference [33] uses a repulsive anharmonic oscillator model to explain the distribution of ﬁnancial returns in a stock market when the market exhibits an upward trend.

8. Conclusions We have presented a methodology, the Group Quantization formalism, for constructing a financial theory from symmetry arguments. The WSp(2,R) group has been used to describe the Black-Scholes model, the Ho-Lee model and the harmonic and repulsive oscillators. + The choice of R as the structural group in the Heisenberg- Weyl group ensures the appropriate commutator between the momentum and coordinate operators [pˆ,xˆ] = 1. This choice is compatible with the financial interpretation ofp ˆ as a delta. + In addition, extending translations by R formulates the theory directly in real (calculation) time, without the need to switch to an Euclidean time in order to derive pricing quantities. The case of Black-Scholes has been studied in detail. We have constructed the polarized functions and the position, momentum and Hamiltonian operators both in coordinate and in momentum spaces. The role of variance as cohomological invariant has been identified. First polarizations for the harmonic oscillator have been de- rived in phase space and in orthogonal coordinates. A high polarization was necessary to obtain polarized functions in coordinate (price) space. A Definitions 20/ 24

Appendices For a 1-form Γ ∂Γi dΓ = dxi ∧ dx j 1. Deﬁnitions ∂x j

Vector Fields where the wedge operator ∧ is the antisymmetric combination

Let M be a N-dimensional manifold with local coordinates dxi ∧ dx j = dxi ⊗ dx j − dx j ⊗ dxi xi,i = 1,2...N. A vector field X is an application that associates a first order differential operator X(x) to a point x ∈ M. One can define in an analogous way differentials of higher X(x) can be expressed in local coordinates as a linear combi- order forms. Note that the antisymmetry of the wedge operator 2 2 nation of the base fields implies that d f = 0 and d Γ = 0. Also, i f a n-form acts on n − k k vector fields one obtains a k-form. For instance the ∂ 1-form d f acting on a field X gives a zero-form ei ≡ i = 1,2...N ∂xi ∂ f ∂ ∂ f that is d f (X) = dxi(Xj(x) ) = Xi(x) = X( f ) ∂xi ∂x j ∂xi ∂ X(x) = Xi(x) Analogously, one can check that dΓ acting on X alone gives a ∂xi 1-form with Xi(x),i = 1,2...N differentiable functions on M. The ∂Γi dΓ(X,.) = (Xidx j − Xjdxi) space of the vector fields is called the tangent space of M, T ∂x j (M). For brevity we will write X instead of X(x) The integral curves of X are the solution to the set of ordinary We use the inner product notation to denote the action of a differential equations n-for Ω on a vector field X

dx iX (Ω) = Ω(X) i = X (x) ds i where s is an integration parameter. Note that the invariance Lie Derivative condition X f = 0 implies that f is constant along the integral The Lie derivative evaluates the change of vector fields and curves of X. forms along the flow defined by another vector field. The Lie derivative of a function f with respect to a vector field Forms X is A 1-form Γ is an application that associates to every point x ∈ M an element of the dual space of T (M). LX f = X( f ) = d f (X)

Γ : x → Γ(x) / Γ(x)(X(x)) = f (x) For two vector fields X, Y , the Lie derivative of Y with respect to X, LXY , is a vector field defined by with f (x) a differentiable function. L Y = −L X ≡ [X,Y] = As with vector fields, we write Γ for Γ(x). The space of 1- X Y forms is called the cotangent space of M and is denoted by ∂Yj ∂Xj ∂ ∗ ∗ Xi −Yi T (M). A convenient representation for the basis of T (M) is ∂xi ∂xi ∂x j

ui ≡ dxi i = 1,2...N The Lie derivative of a 1-form Γ with respect to a vector field X,LX Γ is also a 1-form, and has the meaning of the rate of and its action on the basis of T (M) is change of Γ along the integral lines (flow lines) of X. One finds, in local coordinates ∂ ui(e j) =≡ dxi( ) = δi, j i, j = 1,2...N ∂x j LX Γ = (LX Γ)idxi

2-forms, 3-forms , etc. are linear combinations of tensor where products of 1-forms. A function f is considered a zero-form. ∂Γi ∂Xj An important operation on forms is the differential d. The (LX Γ)i = Xj + Γ j ∂x ∂x differential of a n-form is either a (n+1)-form or zero. For a j i function f, d is the ordinary differential or, in a more concise notation

∂ f LX Γ = d(Γ(X)) + dΓ(X,.) ≡ iX dΓ + d(iX Γ) d f (x) = dxi ∂xi C Lagrangian Formulation 21/ 24

Lie Algebra B.2 Call Option Price A Lie algebra G is a vector space over a field F equipped with We price a call option with strike X and maturity T using the a bilinear map [,] : (G,G) → G such that [X,Y] = −[Y,X] and Mellin transform. The payoff is, from (73) [X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0 (Jacobi identity).This Z +∞ X1−p map is called Lie bracket. The Lie algebra for a Lie group G C(p,0) = (S − X)+S−p−1dS = is defined using the tangent vectors at the identity e as vector 0 p(p − 1) space of and the Lie derivative as bracket operator. One has with ℜ(p) > 1. The expression of the call option price in the k L [Xi,Xj] = ci, jXk i, j = 1,2...dimG Mellin space is, using (74) ck ∈ ,i, j,k = , ... G X1−p 1 where the coefficients i, j F 1 2 dim are called C(p,τ) = e−rτ exp(−( σ 2 p2 + µ p)τ) structure constants. p(p − 1) 2 T Since Xg is a LIVF if and only if Xg = Lg Xe and an RIVF if T where τ ≡ T −t, t ≤ T. and only if Xg = Rg Xe, there is a biunivocal correspondence between the LIVFs (RIVFs) and the set of the tangent vectors The Black-Scholes call option formula in the price (coordi- at the identity e of G, with the following brackets nate) representation is obtained by the inverse Mellin transform L L k L R R k R [Xi ,Xj ] = ci, jXk [Xi ,Xj ] = −ci, jXk Z c+∞ −rτ 1 p R L V(S,t) = e C(p,τ)S dp [Xi ,Xj ] = 0 ∀i, j = 1,2...dimG 2πi c−i∞ where c ∈ (1,+∞) and ℜ(p) > 1. 2. Instrument Prices in Momentum Space Details of the calculation of this integral can be found in reference [55]. In this section we present examples of derivatives pricing in the less familiar momentum (Laplace) space. The price of a financial instrument Ψ(x,t) maturing at T is 3. Lagrangian Formulation found by performing an inverse Laplace transform In the Group Quantization formalism, Θ| , the projection 1 Z c+∞ C Ψ(x,t) = dpexpK(p,τ)Ψ(p,T) (72) onto the base manifold of the vertical form Θ along the inte- 2πi c−i 34 ∞ gral trajectories of the characteristic module CΘ provides where τ ≡ T − t, t ≤ T and Ψ(p,T) = Ψ(p)exp(E(p)T) is the classical action S ([1]) the payoff at maturity in the Laplace space Z c+∞ Z Z 1 xp Ψ(x,T) = dpe Ψ(p,T) (73) S = Θ|C = L (x,x˙)dt 2πi c−i∞ The classical Lagrangian L (x,x˙) is obtained by the projection B.1 Instrument Prices using the Mellin Transform onto the base manifold. Note that Θ and Θ + d f generate The inverse Mellin transform33 gives the instrument’s price equivalent Lagrangians that differ in a total derivative and do in terms of the stock value instead than the log-stock. The not change the action S pricing equations (72) and (73) read now The vertical (connection) form dΘ restricted to the quotient space G˜/U is analogous to the Poincare-Cartan´ form of classi- 1 Z c+∞ V(S,t) = dp exp(−E(p)τ)V(p,T)Sp cal mechanics ΘPC ,which can be written in local coordinates 2πi c−i∞ q,p as 1 Z c+∞ V(S,T) = dpSpV(p,T) (74) 2πi c−i∞ ΘPC = ∑ pidqi − H(p,q)dt (75) i with where H(p,q) is the Hamiltonian function, and q,p are conju- 1 1 E (p) ≡ σ 2 p2 + µ p − r µ = 1 − σ 2 gate pairs of coordinates and momenta. In classical mechanics, r 2 2 34 ˜ ˜ The final condition in Mellin space is given by If the quantization group G is finite, the quotient space P = G/CΘ and the quotient connection form ΘP = Θ/CΘ, where CΘ is the char- Z +∞ acteristic module of Θ, define a fiber bundle (P,U,π) where the curva- −p−1 C(p,0) = V(S,T)S dS ture form Ω = dΘP is a symplectic form over P/U. Taking the quotient 0 by the integral flows of CΘ allows the definition of local coordinates in 1 33 −1 P/U, (p ,q ),i = 1,2... dim(P/U) representing the canonical momenta M exists only for complex values of y so that c ≡ ℜ(y) > 0 is within i i 2 certain (possibly multiple) convergence strips. Each strip leads to different and canonical coordinates. of the vertical form Θ along the integral flows −1 results for M ( f ) ([57], [58], [59], [60]). (trajectories) of the characteristic module CΘ. C Lagrangian Formulation 22/ 24 the equations of motion are just the flows associated to the C.2 Euclidean Oscillator Lagrangian Hamiltonian field Following the same steps than in section C.1, and using the ∂ ∂H ∂ ∂H ∂ expressions (57) and (58), we find that the CΘ trajectories XH ≡ + ∑ − ∂t i ∂ pi ∂qi ∂qi ∂ pi dt dp 2 dx −2 = 1 = ω λ x = ω λ p so along Hamiltonian trajectories ds ds ds dqi ∂H dpi ∂H q˙i ≡ = p˙i ≡ = − generates the following Lagrangian for the Euclidean har- dt ∂ pi dt ∂qi monic oscillator Note that ΘPC has the structure of a Legendre transformation if one identifies p ≡ ∂ /∂q˙ , where is the Lagrangian 1 2 1 2 2 i L i L LH (x,x˙) = x˙ + ω x function. Along the Hamiltonian flows 2σ 2 2 | = (q˙,q)dt ΘPC H L As expected, the mapping ω → iω provides the Lagrangian The differential of Θ|PC along the Hamiltonian trajectories is for the Euclidean repulsive oscillator a symplectic form. 1 2 1 2 2 LR(x,x˙) = x˙ − ω x dΘPC|H = ∑dpi ∧ dqi 2σ 2 2 i The Lagrangian function is used for the application of path integral methods in computational finance ( [24], [26], [27], Acknowledgements [15]). The author is grateful to Peter Carr for very many valuable insights and support, and to Gregory Pelts for sharing his C.1 Black-Scholes Lagrangian knowledge on the geometric structure of finance. We find the Black Scholes Lagrangian by restricting the Black- Scholes connection Θ (57) to the characteristic module trajectories on the base manifold. Disclaimer L CΘ is generated by the time translations Xt whose integral flows on the base manifold are (see (28)) The views expressed herein are those of the author and do not reflect the views of my current employer, Wells Fargo dt dp dx = 1 = 0 = σ 2 p + µ Securities, or affiliate entities. ds ds ds 2 → t = s p = p0 x = x0 + (σ p0 + µ)t (76) where x0, p0 , x0, ζ0 are integration constants and s is the integration parameter. We first add the total differential d(px) to Θ so the equations adopt the Poincare-Cartan´ expression (75) Θ → d(xp) + Θ = pdx − E(p)dt then 1 Θ| = p(σ 2 p + µ)dt − σ 2 p2 + µ p − r dt = C 2 1 (x˙− µ)2dt − rdt 2σ 2 where we have used equation (76) dx x˙− µ ≡ x˙ = σ 2 p + µ → p = dt σ 2 with the result 1 L (x,x˙) = (x˙− µ)2 − r (77) 2σ 2 The solution x(τ) of the Lagrangian (77) coincides with the expected value of the coordinate (log-price) using the KBS kernel in (39)) Z ∞ 1 1 − (x0−x−µτ)2 0 hxî = √ dxx e 2σ2τ = x + µτ 2πσ 2τ −∞ References 23/ 24

References [16] Abramowitz, Milton, and Irene A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and [1] V. Aldaya and J. A. de Azcarraga, Quantization as a conse- Mathematical Tables, Dover Publications, Ninth printing, quence of the symmetry group: An approach to geometric 1970. quantization, J. Math. Phys. 23, 1982. [17] Roman Kozlov On Lie Group Classification of a Scalar [2] V. Aldaya and J. A. de Azcarraga´ and K.B. Wolf Quanti- Stochastic Differential Equation, Journal of Nonlinear zation, symmetry, and natural polarization, J. Math. Phys. Mathematical Physics, 18,1, 2010. 25 (3), March 1984, [18] A. Pazy, Semigroups of Linear Operator and Applications [3] V. Aldaya, J. A. de Azcarraga´ and S. Garc´ıa, Group mani- to Partial Differential Equations, Applied Mathematical fold approach to field quantisation, Journal of Physics A: Sciences, Springer-Verlag, New York, 1983. group Mathematical and General, 21, 23, 1988. [19] F. Soto-Eguibar and H. M. Moya-Cessa, Solution of the [4] Santiago Garc´ıa, Hidden invariance of the free classical Schrodinger Equation for a Linear Potential using the Ex- particle, https://arxiv.org/abs/hep-th/9306040 American tended Baker-Campbell-Hausdorff Formula, Appl. Math. Journal of Physics, 62, 6, 1994. Inf. Sci. 9, No. 1, 175-181, [5] V.Aldaya, J.Bisquert, J.Guerrero, J.Navarro-Salas Group- [20] Juan M Romero, Elio Mart´ınez Miranda and Ulises La- theoretical construction of the quantum relativistic har- vana, Conformal symmetry in quantum finance, J. Phys. monic oscillator, Reports on Mathematical Physics Vol- Conf. Ser. 512, 2014. ume 37, Issue 3, June 1996, Pages 387-418, 1997. [21] Juan M Romero, Elio Mart´ınez Miranda and Ulises [6] Aldaya V., Guerrero J., Marmo G. Quantization on a Lie Lavana, Schrodinger group and quantum finance Group: Higher-Order Polarizations, In: Gruber B., Ramek https://arxiv.org/abs/1304.4995 M. (eds) Symmetries in Science X. Springer, Boston, MA, [22] U. Niederer, The maximal kinematical invariance group arXiv:physics/9710002v1, 1998. of the free Schrodinger equation, Helv. Phys. Acta, 45, [7] M. Calixto, V.Aldaya; and M. Navarro, Quantum Field 1972. Theory in Curved Space from a Second Quantization of a [23] C.R. Hagen, Scale and conformai transformations in Group, Int.J.Mod.Phys. A15 , https://arxiv.org/abs/hep- galilean-covariant field theory, Phys. Rev. 5,2, 1972. th/9701180, Mar 2000. [24] [8] Belal Baaquie, Quantum Finance: Path Integrals and V. Aldaya, J. Guerrero, Lie Group Representations and Hamiltonians for Options and Interest Rates, Cambridge Quantization, Reports on Mathematical Physics, 47, 2001. University Press, 2004. [9] J. Wei and E. Norman, Lie algebraic solution of linear dif- [25] Belal Baaquie, Interest Rates and Coupon Bonds in Quan- ferential equations, J. Math. Phys., 4(4):575–581,, 1963. tum Finance, Cambridge University Press, 2009. [10] Szymon Charzynski and Marek Kus, Wei-Norman equa- [26] Belal Baaquie, Financial modeling and quantum mathe- tions for a unitary evolution, Journal of Physics A: Mathe- matics, Computers and Mathematics with Applications 65 matical and Theoretical, Volume 46, Number 26, 2013. (2013) 1665–1673, [11] C. F. Lo, Lie-Algebraic Approach for Pricing Zero- [27] Belal Baaquie,Claudio Coriano, Marakani Srikant, Quan- Coupon Bonds in Single-Factor Interest Rate Models, tum Mechanics, Path Integrals and Option Pricing: Reduc- Journal of Applied Mathematics,Volume 2013, Article ing the Complexity of Finance, arXiv:cond-mat/0208191 ID 276238, 2013. [cond-mat.soft], 2002. [12] E.G. Kalnins, W. Miller, Jr., G.S. Pogosyan Complete sets [28] R. K. Gazizov, N. H. Ibragimov, Lie Symmetry Analysis of invariants for dynamical systems that admit a separa- of Differential Equations in Finance, Nonlinear Dynam- tion of variables, Journal of Mathematical Physics 43(7), ics,17,4, 1998 pp.–3592-3609, July 2002. [29] F. Casas, A. Murua M. Nadini, Efficient computation of [13] W. Miller, Jr., The Scrodinger and Heat Equations, En- the Zassenhaus formula, Computer Physics Communica- cyclopedia of Mathematics and its Applications,Vol. 4, tions, Volume 183, Issue 11, pp 2386–2391, November Cambridge University Press, 2010. 2012 [14] F. Gungor, Equivalence and Symmetries for Lin- [30] K.B. Wolf, On Time-Dependent Quadratic Quantum ear Parabolic Equations and Applications Revisited, Hamiltonians, SIAM Journal on Applied Mathematics, Vol. arXiv:1501.01481 [math-ph], 2017. 40, No. 3, Jun., 1981, pp. 419-431 [15] Vadim Linetsky, The Path Integral Approach to Financial [31] C. Munoz, J. Rueda-Paz and K.B. Wolf, Discrete repul- Modeling and Options Pricing, Computational Economics, sive oscillator wavefunctions, J. Phys. A: Math. Theor. 42 11,1, 1997. (2009) 485210 (12pp) References 24/ 24

[32] K.B. Wolf, Integral Transforms in Science and Engineer- [50] Masao Nagasawa, Scrodinger Equations and Diffusion ing, Plenum, 1979 Theory, Birkhauser-Verlag, 1993 [33] N. Jaroonchokanan and S. Suwanna, Inverted anhamonic [51] S. Garc´ıa and F. Garc´ıa, Imaginary Mass, Black-Scholes oscillator model for distribution of financial returns, J. Variance, and Group Quantization (March 14, 2018), Phys.: Conf. Ser. 1144 012101, 2018 Available at SSRN: https://ssrn.com/abstract=3140485 [34] G. Barton, Quantum mechanics of the inverted oscillator or http://dx.doi.org/10.2139/ssrn.3140485 potential, Ann. Phys. (NY) 166 322–63, 1986 [52] C. Ozemir and F. Gungor, Symmetry classification of [35] A. Wunsche, Associated Hermite Polynomials Related to variable coefficient cubic-quintic nonlinear Schrodinger Parabolic Cylinder Functions, Advances in Pure Mathe- equations, arXiv:1201.4033v1, 2012. matics, 9, 15-42, 2019 [53] R. K. Gazizov, N. H. Ibragimov, Lie Symmetry Analysis [36] A Paliathanasis, RM Morris and PGL Leach, Lie symme- of Differential Equations in Finance, Nonlinear Dynam- tries of 1+2 nonautonomous evolution equations in Finan- ics,17,4, 1998. cial Mathematics, https://arxiv.org/pdf/1605.01071.pdf, [54] A. Blasi, G. Scolarici‡and L. Solombrino, Pseudo- 2016. Hermitian Hamiltonians, indefinite inner product spaces [37] Nicholas Wheeler, Classical/Quantum Dynamics in a and their symmetries, arXiv:quant-ph/0310106v2, 2004. Uniform Gravitational Field: A. Unobstructed Free Fall, [55] Radha Panini, Ram Prasad Srivastav, Option pricing Preprint, Reed College Physics Department August 2002 with Mellin transnforms, Mathematical and Computer [38] Goldstein, Robert S. and Keirstead, William P., Modelling,40,43-56, 2004. On the Term Structure of Interest Rates in the [56] P. Carr, R. Jarrow, R. Myneni, Alternative characteriza- Presence of Reflecting and Absorbing Boundaries , tions of American put options, Mathematical Finance 2, Available at SSRN: https://ssrn.com/abstract=19840 or 87–105. 1992 http://dx.doi.org/10.2139/ssrn.19840, June 1997 [57] J. Bertrand, P. Bertrand and J. Ovarlez, The Transforms [39] Mehler, F. G. , Ueber die Entwicklung einer Function von and Applications Handbook: Second Edition., The Mellin beliebig vielen Variabeln nach Laplaceschen Functionen Transform,Ed. Alexander D. Poularikas,Boca Raton: CRC hoherer¨ Ordnung, Journal fur¨ die Reine und Angewandte Press LLC , 2000. Mathematik (in German) (66), 1866 [58] Paul L. Butzer A Direct Approach to the Mellin Trans- [40] V. Bargmann, Ann. Math. 59, 1, 1954. form, Journal of Fourier Analysis and Applications, July [41] V. Bargmann, Commun. Pure Appl. Math. 14, 187 (1961); 1997. 20, 1, (1967). [59] R. Frontczak, Pricing Options in Jump Diffusion Mod- [42] Elie´ Cartan, Les variet´ es´ a` connexion affine et la theorie´ els Using Mellin Transforms., Journal of Mathematical de la relativite´ gen´ eralis´ ee,´ I et II, Ann. Ec. Norm. 40, Finance,3,366-373, 2013. 1923, and 41, 1924. [60] P. Buchen, An Introduction to Exotic Option Pricing, [43] J. M. Souriau, Structure des systemes dynamiques Dunod, CHAPMAN & HALL/CRC, Financial Mathematics Series, Paris, 1970. 2012. [61] [44] B. Kostant, Quantization and Unitary Representations, K. Ahn, M.Y. Choi, B. Dai, S. Sohn and B. Yang, Mod- Lecture Notes in Mathematics 170, Springer-Verlag, New eling stock return distributions with a quantum harmonic York, 1970. oscillator, EPL 120 38003, https://doi.org/10.1209/0295- 5075/120/38003, 2018 [45] D. J.Simms and N. M. J. Woodhouse, Lecture Notes in Geometric Quantization (Springer-Verlag, Berlin, 1976. [46] Pierre-Henry Labordere,` Analysis, Geometry, and Mod- eling in Finance: Advanced Methods in Option Pricing, Chapman and Hall/CRC Financial Mathematics Series, 2008. [47] T.K. Jana, P. Roy, Pseudo Hermitian formulation of Black- Scholes equation, https://arxiv.org/abs/1112.3217, 2011 [48] N. Bebiano, J. da Providencia and J.P. da Providencia, Non-Hermitian quantum mechanics of bosonic operators, https://arxiv.org/abs/1705.11153, 2017 [49] Asim O. Barut, Ryszard Raczka, Theory of Group Repre- sentations and Applications, Ars Polona, 1980