Group Quantization of Quadratic Hamiltonians in Finance
DDate: February 2 2021Version 1.0
Group Quantization of Quadratic Hamiltonians in Finance
Santiago Garc´ıa Abstract
The Group Quantization formalism is a scheme for constructing a functional space that is an irreducible infinitedimensional representation of the Lie algebra belonging to a dynamical symmetry group. We apply this formalismto 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 attractiveoscillators.The symmetry group used in this work has the structure of a principal bundle with base (dynamical) group asemi-direct extension of the Heisenberg-Weyl group by SL ( , R ) , and structure group (fiber) R + .By using a R + central extension, we obtain the appropriate commutator between the momentum and coordinateoperators [ ˆ p , ˆ x ] = from the beginning, rather than the quantum-mechanical [ ˆ p , ˆ x ] = − i (cid:125) . The integral transforma-tion between momentum and coordinate representations is the bilateral Laplace transform, an integral transformassociated to the symmetry group. Keywords
Black-Scholes – Lie Groups – Quadratic Hamiltonians – Central Extensions – Oscillator – Linear CanonicalTransformations – Mellin Transform – Laplace Transform [email protected] Contents
W Sp ( , R ) Group 64 Black-Scholes 95 Linear Potential. Ho-Lee Model 126 Harmonic Oscillator 157 Repulsive Oscillator 188 Conclusions 19Appendices 20A Definitions 20B Instrument Prices in Momentum Space 21C Lagrangian Formulation 21References 23
Introduction
The Group Quantization formalism ([1], [2], [3], [6]) is ascheme for constructing a functional space that is an irre-ducible infinite dimensional representation of the Lie algebrabelonging to a dynamical symmetry group.This formalism utilizes Cartan geometries in a frameworksimilar to the Hamiltonian framework, where, in finance, the coordinates represent prices, rates ... and the conjugate mo-menta operators are the corresponding deltas.We apply the Group Quantization formalism to a modified
W Sp ( , R ) group. W Sp ( , R ) is the semi direct product of thetwo-dimensional real symplectic group Sp ( , R ) ≈ SL ( , R ) by the Heisenberg-Weyl group. Our modification consists inthat the embedded 2-dimensional translation subgroup in theHeisenberg-Weyl group has been extended by R + , rather than U ( ) , which is the usual extension group in the physics andmathematical literature.Our interest in the W Sp ( , R ) group (called Group of Inho-mogeneous Linear Transformations in [32]) is that this groupis the symmetry group of the second-order parabolic differen-tial equations. Using the
W Sp ( , R ) symmetry, one can findcoordinate systems and operators that map the equations ofmotion and the corresponding solutions ([13], [32]).The use in finance of mathematical methods previously de- The infinite-dimensional representation theory of SL ( , R ) was used byBargmann ([40], [41]) for the description of the free non relativistic quantummechanical particle. a r X i v : . [ q -f i n . M F ] F e b ontents
2/ 24 veloped in in physics is often based in the formal similaritiesbetween the Black-Scholes equation and the quantum me-chanical Schrodinger equation. These formal similarities havebeen explored both from the point of view of Lie algebra in-variance ([12], [53],[36], [17] ) and global symmetries of theBlack-Scholes Hamiltonian operator( [20], [23],[22],[21]).However, the mathematical properties of a wave equation solu-tion such as the Schrodinger equation are totally different thanthe properties exhibited by a parabolic differential equation.Moreover, hermiticity and unitarity do not play a prominentrole in finance: unlike the case of quantum mechanics, infinance there is no probabilistic interpretation of solutions ofthe pricing equation, and time evolution is irreversible andnon-unitary.The Group Quantization formalism makes the most of theprincipal fiber bundle structure linked to the central extensionof a Lie group. Although this formalism shares some featureswith the
Geometric Quantization scheme ([43], [44], [45]),contrary to Geometric Quantization, the Group Quantizationformalism does not require the previous existence of a Poissonalgebra. In both formalisms, the word quantization signifies irreducible representation .As an example of the applicability of this formalism to finance,we will obtain the Black-Scholes theory, the Ho-Lee modeland the Euclidean attractive and repulsive oscillators.
Section 1 shows that a Galilean transformation on the spaceof Black-Scholes solutions constitutes a numeraire change .Strict Galilean invariance plays the role of phase invariance inquantum mechanics.We describe the main features of Group Quantization in sec-tion 2. Particularly important are the definition of the connec-tion form, the polarization algebra and the concept of higherpolarization. Group Quantization considers the action of thegroup on itself, as opposed to the group acting on an exter-nal manifold. This guarantees the existence of two sets ofcommuting generators, the right invariant fields and the leftinvariant fields. Left invariant fields provide naturally a set of polarization constraints that result in pricing equations, whilethe right invariant fields provide operators compatible withthese constraints.Section 3 gives a brief survey of the
W Sp ( , R ) group, the SL ( , R ) group and the Linear Canonical transformations.By using a R + central extension to create the embeddedHeisenberg-Weyl group, Group Quantization provides theappropriate commutator between the momentum and coor-dinate operators [ ˆ p , ˆ x ] =
1, compatible with ˆ p representing adelta, rather than the quantum-mechanical [ ˆ p , ˆ x ] = − i (cid:125) , with-out resorting to analogies with any quantum theory or rotatingto a fictitious Euclidean time .Section 4 applies the Group Quantization formalism to aparabolic SL ( , R ) subgroup in order to construct the Black-Scholes theory. We obtain polarization constraints, operators and pricing Kernel both in the momentum space and in co-ordinate space. This Heisenberg-Weyl commutator makesthe bilateral Laplace transform the integral transform map-ping momentum and coordinate spaces. We give examples ofpricing in momentum space in appendix B.Sections 5 to 7 discuss the application of Group Quantizationto theories with deformed Black-Scholes equations, i.e., Black-Scholes with a (at most quadratic) potential.The Group Quantization of the linear potential, that in financerepresents the Hee-Lo interest rate theory, is discussed insection 5. We obtain the polarized functions and pricingequations both in momentum space and in coordinate space.We compare some of our results with the similarity methodsdeveloped in [13] and [32].The harmonic and repulsive oscillators are studied in sec-tions 6 and 7. The Heisenberg-Weyl commutator [ ˆ p , ˆ q ] = SL ( , R ) subgroup, a change of scale operator, not arotation. We discuss in detail the polarization constraints inphase space and in a real analog of the Fock space. In spiteof our non-standard [ ˆ p , ˆ q ] =
1, we recover the usual ladderoperators when we construct higher order polarized functions.The Mehler kernel is obtained first by building the Kernelfrom the (Hermite) polarized functions, and later by applyingthe Linear Canonical Transformation technique described in[32].From the standpoint of the Group Quantization formalism, therepulsive oscillator can be obtained from the harmonic andoscillator by making the frequency parameter pure imaginary.Both oscillators, which as an interest rate theory representquadratic interest rates, have also been recently proposed asmodels for stock returns ( [61], [33]).Appendix A contains some general definitions in order tomake this paper more self-contained and to establish notation.Appendix C shows examples of the Lagrangian formalism andthe relationship of the connection form in the Group Quanti-zation formalism and the Poincar´e-Cartan form in classicalmechanics.We have favored clarity over mathematical rigor, and impor-tant topics such as group cohomology are just glossed over.For simplicity, we use coordinates as much as possible in-stead of a more compact notation. We refer the reader to thepublications in the bibliography for a detailed and rigoroustreatment of the mathematical concepts relevant to this article.
The Group Quantization Formalism
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1. Galilean Transformations as NumeraireChange
As a motivation for this work, we show that the action ofthe Galilei group on the space of Black-Scholes solutionsconstitutes a numeraire change
The Black-Scholes equation for a stock that pays no dividendsis ∂ V ∂ t = − σ S ∂ V ∂ S − rS ∂ V ∂ S + rV (1)where σ is the stock volatility, r is the risk free rate, S isthe stock price, t is time, and V = V ( S , t ) is the price of afinancial instrument. For simplicity, let’s assume that r and σ are constant. Under the change of variables S → S (cid:48) = Se v (cid:48) t (cid:48) t → t (cid:48) (2)where v (cid:48) is constant, the Black-Scholes equation becomes ∂ V (cid:48) ∂ t (cid:48) = − σ S (cid:48) ∂ V (cid:48) ∂ S (cid:48) − ( r + v (cid:48) ) S (cid:48) ∂ V (cid:48) ∂ S (cid:48) + rV (cid:48) (3)where V (cid:48) = V ( S (cid:48) , t (cid:48) ) .The transformations (5) can be expressed in log-stock coordi-nates as x (cid:48) = x + v (cid:48) , t v (cid:48) = v t (cid:48) = t (4)were x ≡ ln ( S ) . We recognize the familiar Galilean transfor-mations, with the position x representing the logarithm of thestock price, and the velocity v the stock’s growth rate. Giventhat the action of the Galileo group is linear in the log-stockcooordinates, it is convenient to express the Black-Scholesequation (1) in terms of the logarithm of the stock price ∂ V ∂ t = − σ ∂ V ∂ x − µ ∂ V ∂ x + rV with µ = r − σ . Equation (3) becomes ∂ V (cid:48) ∂ t (cid:48) = − σ ∂ V (cid:48) ∂ x (cid:48) − ( µ + v (cid:48) ) ∂ V (cid:48) ∂ x (cid:48) + rV (cid:48) (5)where V (cid:48) = V ( x (cid:48) , t (cid:48) ) The Black-Scholes equation is only covariant under the actionof the Galilei group. Strict Galilean invariance can be restoredby multiplying the solution V (cid:48) by an exponential factor¯ V ( x (cid:48) , t (cid:48) ) = e ε ( x (cid:48) , t (cid:48) ) V (cid:48) ( x (cid:48) , t (cid:48) ) ε ( x (cid:48) , t (cid:48) ) = v (cid:48) σ (cid:18) v (cid:48) t (cid:48) − (cid:0) x (cid:48) − µ t (cid:48) (cid:1)(cid:19) (6)One has ∂ ¯ V ∂ t (cid:48) = − σ ∂ ¯ V ∂ x (cid:48) − µ ∂ ¯ V ∂ x (cid:48) + r ¯ V Note that Φ ( x , t ) ≡ exp ( rt (cid:48) + ε ( x (cid:48) , t (cid:48) )) is a solution of equation (5) Numeraire invariance is analogous to
Phase Invariance inquantum mechanics, However, the relevant Black-Scholesgauge group is not the quantum mechanical U ( ) , but ratherthe multiplicative positive real line R + .Exponential gauge factors ( cocycles ) analog to the one inequation (6) play a pivotal role in the Group Quantization formalism.
2. The Group Quantization Formalism
We describe the main features of the Group Quantizationformalism ([1], [2], [3], [6]). A Quantization Group ˜ G is a connected Lie group ˜ G whichis a central extension of a Lie group G (called the dynamicalgroup ) by another Lie group U called the structural group.The central extension defines a principal bundle ( ˜ G , G , π , U ) .The structure group acts on the fiber by left multiplication. G is called base manifold of the bundle and U is called the fiber . The projection π is a continuous and surjective map π : ˜ G → G Figure 1.
One can visualize ˜ G by imagining that througheach point in the base manifold G there is a line (fiber) ofpoints with different values of the fiber coordinate ζ ∈ U .A Quantization Group ˜ G with structural group U is a Cartangeometry. The Cartan geometry is the geometry of spaces thatare locally (infinitesimally) like quotient spaces G = ˜ G / U .Although G is not a subgroup of ˜ G , ˜ G / U (cid:39) G as topo-logical spaces any ˜ g ∈ ˜ G can be decomposed in two parts,˜ g = ( g , u ) , g ∈ G , u ∈ U . This means that, locally, ˜ G lookslike the Cartesian product of G and U .In this paper, we will only consider Quantization Groups withstructural group U = R + . A Lie group is a differential manifold G endowed with a group composi-tion law F : G × G → G such as F and its inverse are smooth, differentiableapplications. The Group Quantization Formalism
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The Group Quantization method considers the action of thegroup on itself, as opposed to considering the group actingon an external manifold. This guarantees the existence of twosets of commuting group operators, the right invariant fieldsand the left invariant fields.The left and right translations, L g and R g are defined as L g : G → G / L g ( g (cid:48) ) = gg (cid:48) R g : G → G / R g ( g (cid:48) ) = g (cid:48) gL g and R g are diffeomorphisms of G. Note that L g and R g commute.Let g , g (cid:48) , g (cid:48)(cid:48) ∈ G . In an abuse of notation, we write g (cid:48)(cid:48) = g (cid:48) 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 + . Thecomposition law for ˜ G is˜ g (cid:48)(cid:48) ≡ ( g (cid:48)(cid:48) , ζ (cid:48)(cid:48) ) = ( g (cid:48) g , ζ (cid:48) ζ exp ( ε ( g (cid:48) , g )) (7)where G ∈ G , ζ ∈ R + and ε ( g (cid:48) , g ) ∈ R is the extension cocy-cle.A vector field X is called a left invariant vector field (LIVF) if L T ˜ g X ˜ g (cid:48) = X ˜ g ˜ g (cid:48) and a right invariant vector field (RIVF) if R T ˜ g X ˜ g (cid:48) = X ˜ g (cid:48) ˜ g In local coordinates, the LIVF are given by X Li = ∑ k X Li , k ∂∂ g k + λ i Ξ with Ξ = ζ ∂∂ ζ λ i ≡ ∂ ε ( g (cid:48) , g ) ∂ g i (cid:12)(cid:12)(cid:12)(cid:12) g = g (cid:48) , g (cid:48) = e X Li , k ≡ ∂ ( g (cid:48) g ) k ∂ g i (cid:12)(cid:12)(cid:12)(cid:12) g = g (cid:48) , g (cid:48) = e and the RIVFs X Ri = ∑ k X Ri , k ∂∂ g k + γ i Ξ where Ξ = ζ ∂∂ ζ γ i ≡ ∂ ε ( g (cid:48) , g ) ∂ g i (cid:12)(cid:12)(cid:12)(cid:12) g (cid:48) = g , g = e X Li , k ≡ ∂ ( g (cid:48) g ) k ∂ g (cid:48) i (cid:12)(cid:12)(cid:12)(cid:12) g (cid:48) = g , g = e The i-th component of a LIVF X Li is calculated as the deriva-tive respect to unprimed coordinates evaluated at the identity:first set g (cid:48) = g in the derivative, then set g = e . The i-th com-ponent of a LIVF X Li is calculated as the derivative respect to primed coordinates evaluated at the identity: first set g = g (cid:48) inthe derivative, then set g (cid:48) = e .Note that, by construction, RIVFs and RIVFs commute. The function ε in equation (7) is called a cocycle. Cocyclesare restricted by the group law properties.Associativity of the group law implies that ε must satisfy thefollowing functional equation ε ( g (cid:48)(cid:48) , g (cid:48) ) + ε ( g (cid:48)(cid:48) g (cid:48) , g ) = ε ( g (cid:48)(cid:48) , g (cid:48) g ) + ε ( g (cid:48) , g ) and existence of an inverse element ε ( g , e ) = ε ( e , g ) = ε ( e , e ) = e is the identity element of G . A coordinate change in the fiber U generates a mathematicallytrivial cocycle. If the fiber elements ζ undergo the coordinatechange ζ → ζ e f ( g ) in the group law (7), the extension cocycle ε gets an extra additive factor δ c ( f ) ε ( g , g (cid:48) ) → ε ( g , g (cid:48) ) + δ c ( f )( g , g (cid:48) ) with δ c ( f )( g , g (cid:48) ) = f ( gg (cid:48) ) − f ( g ) − f ( g (cid:48) ) Trivial cocycles are called coboundaries . The coboundary δ c ( f )( g , g (cid:48) ) can can be undone by a change of coordinates.Although group extensions whose cocycles differ in a cobound-ary mathematically define the same group representation, thedynamical effects generated by coboundaries are not neces-sarily trivial.One useful analogy is a change of variables in a differen-tial equation: differential equations have canonical forms towhich they can be reduced, however it is often convenient towork in non-canonical coordinates. For instance, even if theBlack-Scholes equation can be reduced to the simpler heatequation, in practice its solution is calculated using variableswhich allow to implement naturally the instrument’s pricingboundary conditions. A connection on ˜ G is a smooth choice of horizontal subspace H and vertical subspace V such that ˜ G can be decomposed asa direct sum, ˜ G = H ⊕ V .The Group Quantization formalism uses a Cartan connection,where the geodesics coincide with the flows of an integrableLIVF algebra. Concretely, the connection form in the Group Not all coboundaries are generated by a coordinate change. The notion of an affine connection on a differential manifold (
Poincar´e-Cartan form ) was introduced by the French mathematician ´Ellie Cartan intwo papers published in 1923 and 1924 [42].
The Group Quantization Formalism
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Figure 2.
Transformation along the fiber U are vertical ,quantities defined on the base manifold G (cid:39) ˜ G / U are horizontal .Quantization formalism is the vertical part Θ of the Maurer-Cartan form In this work, the vertical space is the structural group U = R + with a unique generator Ξ = ζ ∂∂ ζ (see equation (7)). Θ provides a natural definition of horizontality by requiringthat the horizontal fields belong to the kernel of Θ . i X L Θ ≡ Θ ( X L ) = i Ξ Θ ≡ Θ ( Ξ ) = X L stands for all LIVFs (Left Invariant Vector Fields).In coordinates, equations (8) read (see section 2.2) Θ = ∑ i θ i dg i + d Ξ d Ξ ≡ ζ d ζ where the θ i are solutions of a linear algebraic system ∑ k θ k X Li , k + λ i = In the Group Quantization formalism, the elements of thecharacteristic module C Θ are interpreted as evolution opera-tors. C Θ is an integrable system analogous to Hamiltonian The
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 identitymap. Γ can be written in terms of the left invariant fields at the identity X Li , i = , ... dim ( ˜ G ) and their dual forms θ Γ = ∑ i X Li ⊗ θ i θ i ( X Li ) = δ i , j The
Maurer-Cartan equationsd θ i = − ∑ jk c i jk θ j ∧ θ k . express the differential of the form in terms of the c i jk , the structure constantsof the Lie algebra ˜ G . fields in classical mechanics, and the integral flows of the C Θ elements are the geodesics of the Cartan connection. C Θ isdefined as the set of LIVFs that leave the connection form(vertical form) Θ strictly invariant X ∈ C Θ −→ d ( i X Θ ) = i X ( d Θ ) = ω = d Θ can be written in local coordinatesas ω = ∑ i , j ∂ θ i ∂ g j dg j ∧ dg i = ∑ i , j ( ∂ θ i ∂ g j − ∂ θ j ∂ g i ) dg j ⊗ dg i then, the second of equations (9) reads, using the results insection 2.5 ∑ j ( ∂ θ i ∂ g j − ∂ θ j ∂ g i ) X j = ω ( X , Y ) = Θ ([ X , Y ]) = The Group Quantization formalism constructs an irreduciblerepresentation of ˜ G starting from the ring F of functions withdomain ˜ G and range R .The reduction is achieved by restricting the arguments ofthe functional space F using a set of horizontal operatorsand building a polarized function space F P that provides arepresentation of the group.The action of vertical fields (generators of the structural group U ) X V on F P is X V Ψ = f ( X V ) Ψ ∀ Ψ ∈ F P where f ( X V ) is a function associated to the generator X V .We say that F P is a U - invariant functional space. In thisdocument, U = R + and f ( X V ) = The Group Quantization formalism provides a natural choicefor polarization by requiring invariance respect to a horizontal algebra P ( polarization algebra ) that imposes constraintson F . A vector field X is a symmetry of a 1-form Γ if X leaves Γ semi-invariant,that is, the Lie derivative of Γ with respect to X is a total differential L X Γ = d f X −→ d ( i X Γ ) + i X ( d Γ ) = d f X where f X is some function associated with the vector field X. Then, C Θ = ker Θ ∩ ker d Θ . The W Sp ( , R ) Group
6/ 24 A first-order polarization (or just polarization ) P is definedas a maximal horizontal commutative left invariant algebracontaining the characteristic module C Θ , so that the con-straints are compatible with the evolution operators.The polarized functions F P will be characterized by condi-tions of the form F P = { Ψ ∈ F / X P Ψ = ∀ X P ∈ P } Since the elements of the algebra P are integrable vectorfields, a first order polarization defines a foliation of ˜ G . Thismeans that it is possible to select functional subspaces on on˜ G by requiring them to be constant along integral leaves ofthe foliation. When a first order polarization is not able to provide therequired functional constraints, a higher order polarization can be used.As opposed to first-order polarizations described in (2.7.1),higher-order polarizations [8] contain higher-order differentialoperators belonging to the left enveloping algebra. Higherorder polarizations do not define a foliation of ˜ G .A higher-order polarization P H is a maximal subalgebra ofthe left-invariant enveloping algebra that has no intersectionwith the generators of the structural group U and commuteswith the first order polarization P .Commuting with the first order polarization ensures compati-bility with the action of the dynamical operators (RIVFs). Aswe shall see in the next sections, higher order polarizationscan be constructed by using a Casimir operator. .Commutativity of the right and left generators makes theRIVFs good candidates for quantum operators , that is, opera-tors that can reduce the representation from phase space, withcoordinates and momenta, to an irreducible group representa-tion with only coordinates or momenta.Note that, since P is spanned by LIVFs, if X R is a rightgenerator and Φ is polarized X P ( X R Φ ) = X R ( X P Φ ) = ∀ X P ∈ P ∀ Φ ∈ F P so the action of the RIVFs on the space of polarized functionsis well defined. Hence the quantum operators are the restric-tion of the RIVFs to the polarized U - invariant functionalspace. One can easily verify that the conection Θ is right invariant, Θ ( X R ) = Θ and the RIVFs gives the classical Hamiltonianconstants of motion. The classical Lagrangian L ( x , ˙ x ) is obtained by the projectiononto the base manifold G of the connection form Θ alongthe C Θ flows (trajectories). See appendix C for a discussionof the Poincar´e-Cartan form of classical mechanics and theLagrangian function.
3. The
W Sp ( , R ) Group
The Heisenberg-Weyl group W is a central extension of thetwo dimensional Euclidean translation group. Its algebra isgenerated by three elements, P , Q and , where is theidentity operator. The only non trivial commutator is [ P , Q ] = γ γ ∈ C (10)It can be proven that the choices for γ are equivalent to select γ real or γ pure imaginary.Let p , q ∈ R , θ ∈ C and define W ( θ , x , q ) ≡ exp ( θ + x P + p Q ) The operators W specify a group composition law undermultiplication . W (cid:48)(cid:48) ( p (cid:48)(cid:48) , x (cid:48)(cid:48) , θ (cid:48)(cid:48) ) = W (cid:48) ( p (cid:48) , x (cid:48) , θ (cid:48) ) W ( p , x , θ ) Using the the
Baker-Campbell-Hausdorff formula , we ob-tain p (cid:48)(cid:48) = p + p (cid:48) x (cid:48)(cid:48) = x + x (cid:48) θ (cid:48)(cid:48) = θ + θ (cid:48) + γ ( px (cid:48) − xp (cid:48) ) (12)The group law (11) provides a coordinate representation ofthe abstract operators P , Q and acting on the functional The action of W on sl ( , R is W ( a Q + b P + η I ) W − = a Q + b P + ( η + w ( a p − bx )) therefore, W acts on w as a three-dimensional space with Cartesian coordi-nates ( p , x , θ ) . Exponentiation and composition of operators are to be considered in thecontext of formal operator series. The
Baker-Campbell-Hausdorff formula readsln ( e X e Y ) = X + Y + [ X , Y ]+ ([ X , [ X , Y ]] − [ Y , [ X , Y ]]) + ... when [ X , Y ] is a number e X e Y = e X + Y + [ X , Y ] → e X e Y = e Y e X e [ X , Y ] One can prove that e X e Y e − X = e X Ye − X = Y + [ X , Y ] + [ X , [ X , Y ]] + ... (11) These are right invariant generators, corresponding to the action of W (cid:48) . The W Sp ( , R ) Group
7/ 24 space Ψ ( p , q , θ ) = exp ( θ ) ψ ( p , q ) that provides the required Heisenberg-Weyl commutation re-lations P (cid:55)→ X p = ∂∂ p − γ x ∂∂ θ Q (cid:55)→ X x = ∂∂ x + γ p ∂∂ θ (cid:55)→ X θ = ∂∂ θ We get [ X p , X x ] = γ X θ SL ( , R ) group SL ( , R ) , the special (unimodular) linear group in two realdimensions, has a natural representation by 2 × M = (cid:18) a bc d (cid:19) ∈ SL ( , R ) , ad − bc = SL ( , R ) group is an automorphism of the W group, pre-serving the algebra commutators. Define new operators P (cid:48) and Q (cid:48) as linear combinations of P and Q , using the matrix(13) P (cid:48) = a P + b QQ (cid:48) = c P + d Q then, by direct computation, [ P (cid:48) , Q (cid:48) ] = [ P , Q ] (14) SL ( , R ) belongs to a class of Lie groups called the symplec-tic groups Sp ( n , R ) which leave invariant a skew-symmetricform and play an important role in the geometry of phasespace and Hamiltonian systems. One can verify the isomor-phism SL ( , R ) ≈ Sp ( , R ) for the two-dimensional symplec-tic matrix Ω M (cid:62) Ω M = Ω Ω ≡ (cid:20) − (cid:21) (15) W Sp ( , R ) group W Sp ( , R ) is a real, non-compact, connected, simple Liegroup that is the semi direct product of the two-dimensionalreal symplectic group Sp ( , R ) ( ≈ SL ( , R ) ) by the Heisenberg-Weyl group W .The W Sp ( , R ) group is a subgroup of the Schrodinger groupin one dimension, and is called Group of InhomogeneousLinear Transformations in reference [32]. Our composition law differs from [32] in that the extension group in (16)is R + , not R . Let an element g ∈ W Sp ( , R ) be parametrized by g ( M , u , ζ ) where M ∈ SL ( , R ) , ζ ∈ R + and u ≡ ( p , x ) ∈ R . The W Sp ( , R ) composition law is g (cid:48)(cid:48) (cid:0) M (cid:48)(cid:48) , u (cid:48)(cid:48) , ζ (cid:48)(cid:48) (cid:1) = g (cid:48) (cid:0) M (cid:48) , u (cid:48) , ζ (cid:48) (cid:1) g ( M , u , ζ ) with M (cid:48)(cid:48) = M (cid:48) Mu (cid:48)(cid:48) = u (cid:48) M + u ζ (cid:48)(cid:48) = ζ (cid:48) ζ exp (cid:18) − γ u (cid:48) M Ω u T (cid:19) (16)where Ω is the two-dimensional symplectic matrix in (15) and γ is given by the [ P , Q ] Heisenberg-Weyl commutator in (10).
In the mathematics literature, such as [32] and [12], the usualconvention is that the momentum and position operator com-mutator is [ P , Q ] = − i , which corresponds to an extension of the 2-dimensional translations by U ( ) . Then, for diffusiveequations such as the heat equation, one makes the theory Euclidean at the last step by setting the time variable t to it .In this work we set γ =
1, corresponding to a central exten-sion of the Euclidean translations by R + . The reason is thatin finance we want P to represent a change (a delta) in thequantity Q , wich usually represents a a price or a rate. Thischoice has the added advantage that there is no need to con-sider a passage to a fictitious Euclidean time, since the timevariable in the group represents the actual calculation time.When we quote results from to the mathematics literature (forinstance in section 3.4) we will assume, unless otherwise indi-cated, that the results are obtained with the usual commutator [ P , Q ] = − i . The
W Sp ( , R ) generates the dynamics for quadratic Hamil-tonians ([13], [12], [2], [32] ) by the action of higher orderoperators on W . The adjoint action of the group consist of sixdistinct orbits [32]. Representative of these orbits are givenby the following operators, that include quadratic elementsfrom the enveloping algebra of W P , P + Q , P + Q , P − Q , P , We have the following isomorphisms: g ( M , , ) ≈ SL ( , R ) and g ( , u , ζ ) ≈ W . The unit element is g ( , , ) and theinverse element is g (cid:0) M − , − u M − , / ζ (cid:1) The different quantization groups considered in this paper willbe distinguished subroups of
W Sp ( , R ) .In physics, the W Sp ( , R ) includes as subgroups the symme-try group of the free particle, the gravitational free-fall, as wellas the symmetry group of the ordinary harmonic oscillator In quantum mechanics the cocycle in (16) is multiplied by a factor 1 / (cid:125) for dimensionality reasons, and [ P , Q ] = − i (cid:125) . The W Sp ( , R ) Group
8/ 24 and the repulsive harmonic oscillator (with imaginary fre-quency). In finance, we will obtain the Black-Scholes theory,the Ho-Lee model and the Euclidean attractive and repulsiveoscillators.
W Sp ( , R ) acts on functional spaces as integral transforms called Linear Canonical Transformations , whose kernel is a SL ( , R ) matrix [32] g ( M , u ) ( f ( x )) = (cid:90) R W ( M , x , x (cid:48) ) f ( x (cid:48) ) dx (cid:48) (17)with kernel (using the notation of equation (13)) W ( M , x , x (cid:48) ) = e − i π / √ π b exp ( i ( ax (cid:48) − xx (cid:48) + d x ) / ( b )) (18)Linear Canonical Transformations (LICs), have the importantproperty that composition of the transforms is equivalent tomultiplication of their SL ( , R ) kernels ([2], [32]).LICs kernels can be analytically continued to SL ( , C ) , sub-ject to some restrictions [2]. In this work, we will use, whenindicated, equation (17) with the mapping b → − ibW ( M , x , x (cid:48) ) = √ π b exp ( − ( ax (cid:48) − xx (cid:48) + d x ) / ( b )) (19)which has been modified from (18) because our choice for theHeisenberg-Weyl commutator [ P , Q ] = The Fourier transform F ( f )( z ) = (cid:90) ∞ − ∞ f ( p ) e ipz d p has the SL ( , R ) kernel F = (cid:18) − (cid:19) The double sided Laplace transform L ( f )( z ) = (cid:90) ∞ − ∞ f ( p ) e pz d p can be obtained from F by the formal map p (cid:55)→ ip . Its kernelis a SL ( , C ) matrix L = (cid:20) ii (cid:21) These transforms also define pseudo-differential (hyperdifferential) oper-ators associated to the corresponding SL ( , R ) kernel [32]. In spite of formal similarities, existence of a Fourier transformexists does not automatically imply the existence of a Laplacetransform [32].The inverse transform is L − ( f )( x ) = π i (cid:90) c + ∞ c − i ∞ d p e xp f ( p ) The integration contour is a vertical line on the complex plane,such that all singularities of the integrand lie on the left of it.
The Bargmann transform has the complex kernel B = √ (cid:18) − i − i (cid:19) The Mellin transform MM ( f )( z ) ≡ F ( z ) = (cid:90) ∞ y − z − f ( y ) dy (20)is obtained from the Laplace transform L by the change ofcoordinates p → − ln ( y ) .The resolution of identity12 π i (cid:90) c + i ∞ c − i ∞ x − s y s ds = y δ ( x − y ) leads to the Mellin inversion formula M − ( f )( y ) ≡ f ( y ) = π i (cid:90) c + i ∞ c − i ∞ y z F ( z ) dz The Mellin transform has been more widely used in financethan the double sided Laplace transform ([55], [57], [59],[60]). The sign of the transform variable z in (20) differs fromthe usual definition in the mathematics literature in order tocalculate transforms of positive powers of the stock price. Black-Scholes
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4. Black-Scholes
We apply the Group Quantization formalism to the
W Sp ( , R ) subgroup generated by the parabolic SL ( , R ) matrix M BS = (cid:20) σ t (cid:21) σ ∈ R t ∈ R (21) M BS has only one eigenvalue λ = + M BS is a shear transformation on R : it leavesthe upper plane invariant, while it displaces the lower planeby σ t . This will be significant when finding a first orderpolarization in section 4.5. Composition Law
The
W Sp ( , R ) subgroup generated by the matrix (21) consti-tutes the Black-Scholes quantization group ˜ B . Its compositionlaw is obtained from the W Sp ( , R ) composition law (16) andthe matrix M BS , with some cocycle modifications that we willexplain below t (cid:48)(cid:48) = t + t (cid:48) p (cid:48)(cid:48) = p (cid:48) + px (cid:48)(cid:48) = x (cid:48) + x + σ p (cid:48) t ζ (cid:48)(cid:48) = ζ (cid:48) ζ e ε BS ( g , g (cid:48) ) (22)where t , p , x ∈ R and ζ ∈ R + When deriving the group law,we have used that M (cid:48)(cid:48) BS = M (cid:48) BS M BS ⇒ t (cid:48)(cid:48) = t + t (cid:48) The composition law for ( t , p , x ) gives the familiar Galileantransformations: the Galilei group G is the base group of ˜ B , G (cid:39) ˜ B / R + , with σ p corresponding to the Galilean boost..Our interpretation of the group coordinates is that x representsthe (dimensionless) logarithm of the stock price, S ≡ S e x , p is the conjugate momentum of x , t is the calendar time,and σ is the stock volatility. Notice that interest rates arenot explicitly present in the group law. Interest rates in theBlack-Scholes theory are not dynamical quantities and theywill be incorporated in section (4.4.2) as a coordinate changein the fiber ζ .The extension cocycle ε BS is the sum of the true cocycle ε G and the coboundary ε N . ε BS ( g , g (cid:48) ) = ε G ( g , g (cid:48) ) + ε N ( g , g (cid:48) ) With no modifications, the Group Quantization formalism leads to theheat equation. All true Galilean cocycles with the same σ differ in a coboundary fromthe Galilean cocycle (23), i.e., they can be undone by a change of coordinates.The exception occurs only for the one dimensional Galilean group. In onedimension there is another true cocycle which will be used in section 5. For later convenience when defining the Laplace transform,we use as Galilean cocycle ε G ( g , g (cid:48) ) = p (cid:48) x + σ pp (cid:48) t + σ p (cid:48) t (23) ε N represents a numeraire choice ε N ( g , g (cid:48) ) ≡ δ c ( f N ) = µ p (cid:48) t (24)and has the generating function (see section 2.4) f N ( g ) = µσ x Bargmann ([40], [41]) used a central extension of the Galileogroup with U ( ) for the description of the free non relativisticquantum mechanical particle. The cocycle used by Bargmannwas ε B ( g , g (cid:48) ) = − i (cid:125) m ( xv (cid:48) − vx (cid:48) + vv (cid:48) t ) (25)which is labeled by the particle mass m . Bargmann provedthat two cocycles of the form (25) with different values of m are not equivalent, they cannot be transformed into each otherby a coboundary.The cocycle (25) equals the W Sp ( , R ) cocycle − u (cid:48) M BS Ω u T = ( xp (cid:48) − px (cid:48) + σ pp (cid:48) t ) with the substitutions σ → i (cid:125) m p → v σ The cohomological importance of mass in in quantum me-chanics has been extensively studied in the literature. Thestock volatility plays a similar role in finance. Formally, theBlack-Scholes theory is the quantum mechanics of a freeparticle with an imaginary mass. The classical limit (cid:125) → ε G equals the WSp ( , R ) cocycle in (16) plus the coboundary generatedby ( px ) / WSp ( , R ) cocycle is ε W ( g , g (cid:48) ) ≡ − u (cid:48) M BS Ω u T = ( xp (cid:48) − px (cid:48) + σ pp (cid:48) t ) The coboundary generated by ( px ) / δ c ( px ) = ( x (cid:48) p + xp (cid:48) + σ p (cid:48) t + σ pp (cid:48) t ) This coboundary will provide a constant first derivative coefficient inthe Black-Scholes equation. It is possible to add a new group coordinate,much like a quantum gauge potential (see [7]), which allows the introductionof numeraires that depend on stock price and time. This approach will beexplored in a future work.
Black-Scholes
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Left Invariant Vector Fields X Lp = ∂∂ p + x Ξ X Lx = ∂∂ xX Lt = ∂∂ t + σ p ∂∂ x + E ( p ) Ξ X L ζ ≡ Ξ = ζ ∂∂ ζ with E ( p ) ≡ σ p + µ p Commutators
The non-zero commutators are [ X Lt , X Lp ] = − σ X Lx − µ Ξ [ X Lp , X Lx ] = − Ξ (26) The expression for the the vertical form Θ and the curvatureform d Θ are (see section 2.5) Θ = − xd p − E ( p ) dt + d Ξ d Θ = d p ∧ dx − ( σ p + µ ) d p ∧ dt (27)with d Ξ ≡ ζ d ζ C Θ is generated by a unique field X C X C = X Lt + µ X Lx = ∂∂ t + ( σ p + µ ) ∂∂ x + E ( p ) Ξ (28)The restriction of Θ to the C Θ flows gives the Black-ScholesLagrangian, see appendix C.1. We introduce interest rates by performing a change of coordi-nates in the fiber ζ . The mapping ζ (cid:55)→ ˜ ζ = ζ e rt r ∈ R does not change the group law (22), since t (cid:48)(cid:48) = t + t (cid:48) , howeverit alters the separation between vertical and horizontal spaces.Under this coordinate change, the structural group generator Ξ becomes Ξ = ζ ∂∂ ζ (cid:55)→ ˜ Ξ = ˜ ζ ∂∂ ˜ ζ This is equivalent to a non-horizontal polarization [6], with constraintsof the form X Ψ = a Ψ , a ∈ R rather than X Ψ = and it dual form d Ξ acquires a horizontal component d Ξ = ζ d ζ (cid:55)→ ζ d ˜ ζ + + r dt ≡ d ˜ Ξ + r dt (29)The vertical connection form Θ is modified in a straightfor-ward way by equation (29) Θ (cid:55)→ ˜ Θ = Θ + r dt Therefore, the time component f Lt of the LIVFs (determinedby ˜ Θ ( X L ) =
0) picks a new vertical part f Lt ∂∂ t (cid:55)→ f Lt ∂∂ t − r ˜ Ξ which induces the following change in the left time generator X Lt (cid:55)→ X Lt − r ˜ Ξ Note that one obtains the same Lie algebra commutators (26)in the new coordinates. The RIVFs are not modified by thiscoordinate change.In the rest of this section we use the expression of the vectorsfields and connection in these new coordinates and we omitthe notation ˜ ζ , ˜ Θ , etc. for brevity. In section 4 we noted that M BS is a shear mapping where onlythe p space (upper plane) is invariant, and that M BS cannot bediagonalized. These features indicate that the only possiblefirst order polarization is a polarization in p -space, as we willverify below.The first order polarization algebra P is spanned by the x -translations and the C Θ generator X C P = (cid:104) X Lx , X C (cid:105) The polarized functions Ψ ∈ F P are found by imposing thepolarization constraints on the functional space F P F P = { Ψ : ˜ G → R / Ψ ( x , p , t , ζ ) = ζ Ψ ( x , p , t ) } One has X Lx Ψ = → Ψ = ζ ψ ( p , t ) X C Ψ = → ∂ ψ∂ t + σ p ψ + µ p ψ − r ψ = X Lp cannot belong to P because [ X Lt , X Lp ] = − σ X Lx − µ Ξ Black-Scholes
11/ 24
A separable solution of the Black-Scholes equation (30a) is˜ ψ ( p , t ) = e − E r ( p ) t Φ ( p ) with Φ ( p ) an arbitrary function of p and E r ( p ) ≡ σ p + µ p − r We can write a general polarized function as an inverse Laplacetransform (a Linear Canonical Integral transform) Ψ ( ζ , x , t ) = ζ π i (cid:90) c + ∞ c − i ∞ K ( p , t ) exp ( px ) Φ ( p ) d p (31)with K ( p , t ) = exp ( − E r ( p ) t ) (32)The integration measure d p is the dual form of X Lp , the vectorfield absent from the polarization algebra. We will justify theuse of the inverse Laplace transform in section 4.8.In the Black-Scholes theory, polarized functions representprices of financial derivatives, whereas the Φ ( p ) correspondto terminal (payoff) conditions. Appendix B shows how toprice financial instruments in momentum space. The Black Scholes equation can be obtained directly in thecoordinate representation by using a high-order polarization(section 2.7.2).The second order operator C P = X Lt + µ X Lx + σ X Lx X Lx is a Casimir operator commuting with all Black-Scholes LIVFs.Since X Lx and X Lp do not commute, C P defines two higher or-der polarizations. The x -space polarization is generated by (cid:104) C P , X Lp (cid:105) X Lp Ψ = → Ψ = ζ e − px ψ ( x , t ) X C Ψ = → ∂ ψ∂ t + σ ∂ ψ∂ x + µ ∂ ψ∂ x − r ψ = X Rt = ∂∂ t X R ζ ≡ Ξ = ζ ∂∂ ζ X Rx = ∂∂ x + p Ξ X Rp = ∂∂ p + σ t ∂∂ x + ( σ p + µ ) t Ξ we have [ X Rt , X pR ] = σ X Rx + µ Ξ [ X Rx , X pR ] = Ξ All other brackets are zero .Irreducibility is achieved by considering the action of the rightinvariant fields on the constants of motion Φ ( p ) in (31). Afterstraightforward algebra, one finds X Rp : Φ ( p ) → ∂∂ p Φ ( p ) X Rx : Φ ( p ) → p Φ ( p ) (34)and − X Rt : Φ ( p ) → ( σ p + µ p − r ) Φ ( p ) From (34) the price operator, ˆ x , the momentum operator ˆ p andthe time evolution operator (Hamiltonian) ˆ H are given byˆ x = ∂∂ p ˆ p = p ˆ H ≡ σ ˆ p + µ ˆ p − r (35) The expressions (35) for the operators in momentum spacesuggest the following definition for the coordinate operator, ˆ x and the (non hermitian) momentum , ˆ p operator in x -spaceˆ x = x ˆ p = ∂∂ x [ ˆ p , ˆ x ] = R + (equation (11)). The 2-dimensional translation subgroup of the W Sp ( , R ) group actson real exponentials, thus justifying the definitions (36) andthe use for the inverse bilateral Laplace transform in (31). Thisis an opposition to quantum mechanics, where the Heisenberg-Weyl subgroup acts on the unit circle instead, making theFourier transform the mapping between the coordinate andmomentum spaces.Using equations (36), the Black-Scholes Hamiltonian incoordinate space isˆ H BS ≡ σ ∂ ∂ x + µ ∂∂ x − r (37) The inner product of the vertical form Θ and the RIVFs gives the(Galilean) Hamiltonian constants of motion Θ ( X Rx ) = → p ≡ p Θ ( X Rt ) = → σ p + µ p − r ≡ E Θ ( X Rp ) = → x − ( σ p + µ ) t ≡ x The non hermiticity of the Black-Scholes Hamiltonian has been exten-sively studied in the literature ([47], [48], [54]). Non hermiticity is relativelymild, with eigenvalues either real or appearing in complex conjugate pairs.
Linear Potential. Ho-Lee Model
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The value of the numeraire parameter µ can be determined byrequiring the stock price S ≡ e x to be a zero eigenvalue ofthe Hamiltonian operator .Using the Black-Scholes Hamiltonian (37), we findˆ H BS e x = ⇒ µ = r − σ The kernel K BS ( x , x (cid:48) , τ ) for the Black-Scholes equation (33a) Ψ ( τ , x ) = (cid:90) K BS ( x , x (cid:48) , τ ) Ψ ( , x (cid:48) ) dx is obtained by an in inverse Laplace transform from the mo-mentum representation in (32) K BS ( x , x (cid:48) , t ) = π i (cid:90) c + ∞ c − i ∞ d p e − E r ( p ) t e p ( x − x (cid:48) ) (38)The integral (38) does not exist for t >
0, in accordance to thefact that pricing problems in finance are time irreversible finalvalue problems.Let t < τ ≡ − t .Then the integral (38) exists andwe recover the well known Gaussian pricing kernel K BS ( x , x (cid:48) , τ ) = e − r τ √ πσ τ e − σ τ ( x (cid:48) − x − µτ ) (39) The Black-Scholes pricing kernel can also be obtained withthe methods developed in references [13] and [32], using theproperties of the
W Sp ( , R ) group and its representation asLinear Canonical Transformations.From equation (19), the SL ( , R ) matrix (21) generates theLCT W ( M BS , x , x (cid:48) ) = √ πσ t exp ( − ( x (cid:48) − x ) / ( σ t )) which is the heat equation kernel (or Weierstrass transform W [] ). K BS ( x , x (cid:48) , τ ) is obtained by multiplying the heat kernel W ( M BS ) by the discount factor exp ( − r τ ) and setting τ = T − t , x → x + µτ . We note that x → x + µτ is the Galilean transforma-tion (4) representing a numeraire change. This is actually a martingale condition [24]. In momentum space, the zero eigenvalue condition reads ( σ p + µ p − r ) Ψ ( p ) = → Ψ ( p ) = δ ( σ p + µ p − r ) It is straightforward to prove, using the properties of the Dirac delta and theinverse Laplace transform, that if we identify Ψ ( x ) with the stock price S , Ψ ( x ) (cid:39) exp ( x ) , this requires µ = r − σ K BS can also be written in terms of the pseudo-differentialoperators associated with W ( M BS ) [32] e τ ˆ H BS f ( x ) = e − r τ e σ τ ∂ ∂ x e µτ ∂∂ x f ( x ) = (cid:90) + ∞ − ∞ dx (cid:48) K BS ( x , x (cid:48) , τ ) f ( x (cid:48) ) (40)
5. Linear Potential. Ho-Lee Model
The SL ( , R ) parabolic shear mapping (21) that has been usedin the Black-Scholes theory will be used in this section forconstructing the linear potential model.The composition law for the new quantization group ˜ I is t (cid:48)(cid:48) = t + t (cid:48) p (cid:48)(cid:48) = p + p (cid:48) x (cid:48)(cid:48) = x + x (cid:48) + σ p (cid:48) t ζ (cid:48)(cid:48) = ζ (cid:48) ζ e ε IR ( g , g (cid:48) ) (41)where t , p , x ∈ R and ζ ∈ R + with the cocycle ε IR ε IR ( g , g (cid:48) ) = ε G ( g , g (cid:48) ) + ε N ( g , g (cid:48) ) + ε I ( g , g (cid:48) ) ε G ( g , g (cid:48) ) is the Galilean cocycle in equation (23) and ε N ( g , g (cid:48) ) is the coboundary term given in (24). We have added a newtrue cocycle (not a coboundary) ε β ( g , g (cid:48) ) ε β ( g , g (cid:48) ) = β t ( x (cid:48) + σ p (cid:48) t ) β ∈ R (42).We interpret the group coordinates as x representing a shortrate, p its conjugate momentum, and t the calendar time. Theparameter σ is the short rate volatility.We will verify in the next sections that this group describesthe Ho–Lee interest rate model. In quantum physics, the U ( ) extension of this group describes the free fall (linearpotential). The Group Quantization formalism applied to thelinear potential in physics can be found in [2]. For smooth functions e a ∂∂ x f ( x ) = f ( x + a ) Combining the result above with the Gaussian integral e a = √ π (cid:90) ∞ − ∞ e ay e − y / dy we get e ∂ ∂ x f ( x ) = √ π (cid:90) ∞ − ∞ f ( x − y ) e − y dy Identifying group parameters is usually done after analyzing the polar-ization, however in this case the identification is simple enough.
Linear Potential. Ho-Lee Model
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From the composition law (41) and (42), we obtain X L ζ ≡ Ξ = ζ ∂∂ ζ X Lx = ∂∂ x X Lp = ∂∂ p + x Ξ X Lt = ∂∂ t + σ p ∂∂ x + ( E ( p ) + β x ) Ξ (43)where E ( p ) ≡ σ p + µ p with the non-zero Lie Brackets [ X Lt , X Lx ] = − β Ξ [ X Lt , X Lp ] = − σ X Lx − µ Ξ [ X Lp , X Lx ] = − Ξ Note that, as opposed to the Black-Scholes case (section 4.3),the time generator and the x -generator do not commute. The vertical form Θ and the curvature form d Θ are given by Θ = − x d p − E ( p ) dt − β x dt + d Ξ d Θ = d p ∧ dx − ( σ p + µ ) d p ∧ dt − β dx ∧ dt with d Ξ ≡ ζ d ζ C Θ is spanned by a unique field X C X C = X Lt + µ X Lq − β X Lp = ∂∂ t − β ∂∂ p + ( p σ + µ ) ∂∂ x + ( σ p + µ p ) Ξ As in the Black-Scholes case, there is only one first order p -space polarization P spanned by the x -translations and the C Θ generator X C .The polarized functions Ψ ∈ F P are found by imposing thepolarization constraints on the functional space F P F P = { Ψ : ˜ I → R / Ψ ( x , p , t , ζ ) = ζ Ψ ( p , q , t ) } One has X Lx Ψ = → Ψ = ζ ψ ( p , t ) X C Ψ = → ∂ ψ∂ t − β ∂ ψ∂ p + σ p ψ + µ p ψ = of (44a) is ψ ( p , t ) = Φ ( p + β t ) e β µ p + β σ p (45)where Φ ( p ) is an arbitrary function of the momentum p .The bilateral Laplace transform relates the p -space and the x -space. From equation (45), the general expression of apolarized function in x -space can be written as Ψ ( ζ , x , t ) = ζ e − β tx π i (cid:90) c + ∞ c − i ∞ d p G ( p , t ) e xp Φ ( p ) where G ( p , , t ) ≡ e β µ ( p − β t ) + β σ ( p − β t ) The integration contour is a vertical line on the complex plane,such that all singularities of the integrand lie on the left of it.
The pricing (evolution) equation can be found directly in x -space by using a high-order polarization (see section 2.7.2).Let X C be the C Θ generator. The second order operator X P = X C + σ X Lx X Lx is a Casimir operator commuting with all LIVFs (equation43). Since X Lx and X Lp do not commute, X LP defines two higherorder polarizations, the p -space polarization P p = { X LP , X Lx } ,and the x -space polarization P x = { X LP , X Lp } .The x -space polarized functions are obtained by imposingthe P x polarization constraints on functions (sections) of theform Ψ ( z , p , x , t ) = ζ Ψ ( p , x , t ) X Lp Ψ = → Ψ = ζ V ( x , t ) X P Ψ = → ∂ V ∂ t + σ ∂ V ∂ x + µ ∂ V ∂ x + β xV = V ( x , t ) = π i (cid:90) c + ∞ c − i ∞ d p Φ ( p , t ) e px and substitute in (46a), we recover the first order polarizationequations (44a). We will show in section 5.5 that the polarized functions can be expressedin the x -space using Airy functions. This can be anticipated by the identity[16] 12 π i (cid:90) c + ∞ c − i ∞ d pe xp + tp = ( t ) Ai (cid:32) − x ( t ) (cid:33) c > Ai grow very rapidly at infinity andthey need to be constructed with different contours in the complex plane. Linear Potential. Ho-Lee Model
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Using the Feyman-Kac theorem, the the solution of (46a)can be written as V ( x , t ) = E ( e − β (cid:82) Tt X ( s ) ds φ ( X T ) | X ( t ) = x ) where the expectation is taken with respect to a normal processwith volatility σ and drift µ ( W is a Brownian montion) dX = µ dt + σ dW After performing the following change of function and inde-pendent variable V ( x , t ) ≡ e − µσ x e λ t U ( y ) y ≡ (cid:18) σ (cid:19) ( µ σ − λ − β x ) the polarization equation (46a) becomes the Airy equation ∂ U ∂ y − yU ( y ) = V ( x , t ) = e − µσ x ∞ ∑ i = e λ i t ( a i Ai ( y i ) + b i Bi ( y i )) where a i , b i , λ i ∈ R and y i ≡ ( σ ) ( µ σ − λ i − β x ) This form is convenient for the analysis of complex boundaryconditions in bond pricing problems, as shown in [38]. The Consider the differential operator LL ≡ ω ( x , t ) ∂ ∂ x + µ ( x , t ) ∂∂ x where µ ( x , t ) , ω ( x , t ) and r ( x , t ) are functions of ( x , t ) , with x defined in areal domain D ⊂ R and t a positive real number, t ∈ [ , T ] Then, subject totechnical conditions, the unique solution of the PDE ( ∂∂ t + L + r ( x , t )) f ( x , t ) = x ∈ D , ≥ t ≤ T with terminal value f ( x , T ) = g ( x ) , is given by the Feynman-Kac Formulaf ( x , t ) = E ( e − (cid:82) Tt r ( X , s ) ds g ( X T ) | X ( t ) = x ) where the expectation is taken with respect to the transition density inducedby the SDE dX = µ ( X , t ) dt + ω ( X , t ) dW with W a Brownian motion. There is a distinguished solution of equation (47), called Ai , that decaysrapidly as y → + ∞ , while a second linearly independent solution Bi growsrapidly in this limit. Also, like Bessel functions, both Ai and Bi are oscillatorywith a slow decay for large values of their arguments. See reference [16] boundary conditions constrain the values for the expansioncoefficients a i , b i and λ i .Note that the group contraction β → β = One can use the methods developed in [13] and [32] in orderto find coordinate changes mapping Black-Scholes solutionsinto solutions of the equation (46a)For instance, it can be shown that if V BS ( x , t ) a solution of theBlack-Scholes equation (33a) with r =
0, then V ( x , t ) = Ω β ( x , t ) V BS ( x − β σ t , t ) with Ω β ( x , t ) = e β xt + β σ t + µβ t (48)is a solution of (46a). Hence, the Ho-Lee pricing Kernel canbe obtained from the Black-Scholes pricing Kernel (39) (with r =
0) in an analogous manner K I ( x , x (cid:48) , τ ) = Ω β ( x , τ ) K BS ( x − β σ τ , x (cid:48) , τ ) (49)where we have switched to the variable τ = T − t .Notice that Ω β ( x , τ ) is a solution of (46a). In fact, Ω ( x , τ ) isthe expression for the Ho-Lee bond maturing at T Ω ( x , τ ) = e − x ( T − t ) − σ ( T − t ) + µ ( T − t ) t ∈ [ , T ] The pseudo-differential operator methods in section 4.9 canalso be used to compute the evolution operator U .From equation (46a). U ( τ ) = exp ( τ H I ) H I ≡ σ ∂ ∂ x + µ ∂∂ x + β x where τ = − t . We split H I into two operators A ≡ σ ∂ ∂ x + µ ∂∂ x B = β x Given that the only non zero commutators are [ A , B ] = β σ ∂∂ x + µ β [ B , [ A , B ]] = − β σ it is possible to apply the left oriented extended version of theBaker-Campbell-Hausdorff formula [29] e τ A + τ B = e τ ( [ B , [ A , B ]]+[ A , [ A , B ]]) e τ [ A , B ] e τ B e τ A Harmonic Oscillator
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We obtain e τ H I = e − β σ τ e µ βτ × e τ βσ ∂∂ x e βτ x e σ τ ∂ ∂ x e µτ ∂∂ x = Ω β ( x , τ ) e τ βσ ∂∂ x e σ τ ∂ ∂ x e µτ ∂∂ x (50)where Ω β ( x , τ ) has been defined in (48). We apply the opera-tors in (50) from right to left, and use the Weierstrass theoremto recover equation (49) K I ( x , x (cid:48) , τ ) = Ω ( x , τ ) K BS ( x − β σ τ , x (cid:48) , τ )
6. Harmonic Oscillator
As described in section 3.1, our version of the Heisenberg-Weyl group consists in the 2-dimensional translations centrallyextended by R + , such that the commutator between the mo-mentum and coordinate operators is [ ˆ p , ˆ x ] = W Sp ( , R ) group, the hyperbolic SL ( , R ) subgroupgenerates the harmonic oscillator quantization group ˜ H M H = (cid:18) cosh ω t λ − sinh ω t λ sinh ω t cosh ω t (cid:19) λ ≡ √ ωσ (51)where ω , σ , t ∈ R By contrast, in quantum mechanics ([3], [5]), where the 2-dimensional translations are centrally extended by U ( ) , thecommutator between the momentum and coordinate operatorsis pure imaginary, [ ˆ p , ˆ x ] = − i (cid:125) , and the harmonic oscillatorquantization group is generated by an elliptic SL ( , R ) sub-group (cid:18) cos ω t λ − sin ω t − λ sin ω t cos ω t (cid:19) (52)The SL ( , R ) matrix (51) acts as a squeeze mapping of theEuclidean plane, whereas (52) corresponds to a rotation. Bothsubgroups can be transformed into each other by the the map-ping ω → i ω . Notice that the mapping between subgroups isnot the Euclidean time rotation t → it . We have interchanged the exponential factors e τ βσ ∂∂ x e βτ x since the commutator is a number e τ βσ ∂∂ x e βτ x = e τ β σ e βτ x e τ βσ ∂∂ x Using equation (16), M H generates the following compositionlaw: t (cid:48)(cid:48) = t + t (cid:48) p (cid:48)(cid:48) = p + p (cid:48) cosh ω t + λ x (cid:48) sinh ω tx (cid:48)(cid:48) = x + x (cid:48) cosh ω t + λ − p (cid:48) sinh ω t ζ (cid:48)(cid:48) = ζ (cid:48) ζ exp (cid:0) ε H ( g (cid:48) , g ) (cid:1) (53)with the cocycle ε H ε H ( g , g (cid:48) ) = ( px (cid:48) − xp (cid:48) ) cosh ω t + ( λ − p p (cid:48) − λ x x (cid:48) ) sinh ω t The matrix M H reduces to the Black-Scholes generating ma-trix M BS in the limit w →
0, and correspondingly, the quanti-zation group ˜ H contracts to the Black-Scholes quantizationgroup ˜ G (modulo coboundaries) in this limit. For brevity, wehave omitted the numeraire coboundary (24) in the compo-sition law (53). The SL ( , R ) matrix RR = √ (cid:18) λ − / λλ / λ (cid:19) (54)transforms M H into a change of scale (squeeze) operator D H = R − M H R = (cid:20) e ω t e − ω t (cid:21) The change of coordinates A = √ ( λ p − λ x ) B = √ ( λ p + λ x ) (55)splits the phase space (p,x) into orthogonal subspaces. Thegroup law (53) is written as t (cid:48)(cid:48) = t + t (cid:48) A (cid:48)(cid:48) = A + A (cid:48) e − ω t B (cid:48)(cid:48) = B + B (cid:48) e ω t ζ (cid:48)(cid:48) = ζ (cid:48) ζ e ε H ( g (cid:48) , g ) with ε H ( g , g (cid:48) ) = ( B (cid:48) A e ω t − B A (cid:48) e − ω t ) The generating function for this coboundary is µ x , and the final expres-sion is µ ( λ − p (cid:48) sinh ω t + x (cid:48) cosh ω t − x (cid:48) ) , with µ real. Harmonic Oscillator
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Left Invariant Vector Fields X Lt = ∂∂ t − ω A ∂∂ A + ω B ∂∂ BX LA = ∂∂ A + B Ξ X LB = ∂∂ B − A Ξ with Lie Brackets [ X Lt , X LA ] = wX LA [ X Lt , X LB ] = − wX LB [ X LA , X LB ] = − Ξ Connection
The vertical form Θ and the curvature form d Θ are given by Θ = ( A dB − B dA ) − wAB dt + d Ξ d Θ = dA ∧ dB − wA dB ∧ dt − wB dA ∧ dt C Θ is spanned by the time generator X Lt . First Order Polarization
We choose the first order polarization algebra spanned by P = (cid:104) X LB , X Lt (cid:105) The polarized functions Ψ ≡ ζ Ψ ( A , B , t ) are found by impos-ing X LB Ψ = → Ψ = ζ e AB Ψ ( A , t ) X Lt Ψ = → ∂ Ψ ∂ t − ω A ∂ Ψ ∂ A = A is real, not complex.We give the solution of these polarization constraints in equa-tion (62). Equation (56a) coincides with equation (61) ob-tained in the next section 6.2. We use now the original phase space coordinates ( p , q ) in thecomposition law (53).Since the generating matrix M does not leave invariant eitherthe p or the x space, a first order polarization cannot map thephase space ( p , x ) into either the momentum or the coordinatespace. However, the results in this section will be illustrativeof a phase space formulation for the harmonic oscillator andwill also be useful for finding a higher order polarization. Left Invariant Vector Fields X Lt = ∂∂ t + ω λ x ∂∂ p + ω λ − p ∂∂ xX Lx = ∂∂ x − p Ξ X Lp = ∂∂ p + x Ξ [ X Lt , X Lp ] = − ω λ − X Lx [ X Lt , X Lx ] = − ω λ X Lp [ X Lp , X Lx ] = − Ξ All other brackets are zero.
Connection Θ = ( pdx − xd p ) − E ( p , x ) dt + d Ξ d Θ = d p ∧ dx − ω λ − p d p ∧ dt + ω λ x dx ∧ dt (57)with E ( p , x ) ≡ ω ( λ − p − λ x ) d Ξ ≡ ζ d ζ (58)The time translations X Lt are the only generator of the charac-teristic module C Θ . First Order Polarization
There are two first order polarizations P in phase space (cid:104) X Lt , Y + (cid:105) / Y + ≡ λ X Lp + λ − X Lx (59a) (cid:104) X Lt , Y − (cid:105) / Y − ≡ λ X Lp − λ − X Lx We choose (59a) as polarization. Let Ψ a function of the form Ψ ( ζ , p , q , t ) = ζ Ψ ( p , q , t ) Y + Ψ = → λ ∂ Ψ ∂ p − λ − ∂ Ψ ∂ x + ( λ x − λ − p ) Ψ = Ψ ( x , p , t , ζ ) = ζ ψ ( A , t ) e AB where A = A ( p , q ) and B = B ( p , q ) are the orthogonal coordi-nates (55). Then X Lt Ψ = → ∂ ψ∂ t − ω A ∂ ψ∂ A = Harmonic Oscillator
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Using the separable solution of (61) e ω nt A n e ω nt n ∈ N + the polarized functions are expressed as an infinite series Ψ ( ζ , p , x , t ) = ζ ∞ ∑ n = e ω nt α n A n e AB (62)where α n ∈ R are the expansion coefficients.First order polarization in non orthogonal ( p , q ) coordinatesleads naturally to a financial theory in phase space, with F ( p , q ) = e AB = e ( λ − p − λ x ) the analog of a Husimi quasi-probability. The expressionfor the quantum harmonic coherent states in the Fock basisand the Bargmann-Segal transform can be obtained from(62) by mapping the phase space ( p , q ) into C by the simplecorrespondence p → ip . The pricing equation can be obtained directly in x -space byusing a high-order polarization.The second order operator X P = X t + ω λ − X Lx X Lx − ω λ X Lp X Lp is a Casimir operator commuting with all LIVFs.Since X Lx and X Lp do not commute, X P defines two higher orderpolarizations, (cid:104) X P , X Lx (cid:105) and (cid:104) X P , X Lp (cid:105) . The coordinate spacerepresentation is generated by (cid:104) X P , X Lp (cid:105) X Lp Ψ = → Ψ ( ζ , p , q , t ) = ζ e − px / ψ ( x , t ) X P Ψ = → ∂ ψ∂ t + ω (cid:18) λ − ∂ ψ∂ x − λ x ψ (cid:19) = τ = T − t , where T is a maturitytime and t ≤ T . Equation (63a) becomes ∂ ψ∂ τ = H I ψ (64)wit H I the harmonic oscillator Hamiltonian in coordinatespace H I ≡ ω (cid:18) λ − ∂ ∂ x − λ x (cid:19) (65) ( The zero energy 1 / ζ → ζ exp ( t / ) H I can be written in terms of raising and lowering (ladder)operators by using the diagonalizing SL ( , R ) matrix (54) H I = ω ( a † a + a a † ) with a ≡ √ ( λ − ˆ p − λ ˆ x ) a † ≡ √ ( λ − ˆ p + λ ˆ x ) where ˆ p = ∂∂ x ˆ x = x Note that a , a † are the regular ladder operators that are usedfor the construction of the harmonic oscillator Fock spacein quantum mechanics. The Heisenberg-Weyl commutationrelations [ ˆ p , ˆ x ] = [ a , a † ] = SL ( , R ) is an automorphism of the Heisenberg-Weyl sub-group (equation(14)). It is well know that the solution of equation (64) can be repre-sented as a series in Hermite functions ϕ n ψ ( τ , x ) = ∞ ∑ n = e − ω ( n + ) τ α n ϕ n ( λ x ) (66)with α n ∈ R the expansion coefficients. The Hermite functions ϕ n have the following expression([16]) ϕ n ( x ) = (cid:112) n n ! √ π H n ( x ) e − x (67)where H n are the Hermite polynomials. ϕ n form an orthonor-mal basis in L ( R ) and fulfill the eigenvalue condition (cid:18) ∂ ∂ x − x (cid:19) ϕ n ( x ) = − ( n + ) ϕ n ( x ) The relationship between the (first) polarized functions (62)and the polarized functions (66) is given by a modified Bargmanntransform ϕ n ( λ x ) = (cid:112) n ! √ π × π i (cid:90) i ∞ − i ∞ d p (cid:90) ∞ − ∞ dx A n e − B / + √ λ xB − λ x / e − AB where A = A ( p , q ) and B = B ( p , q ) are the variables defined in(55). The integral over the momentum p is an inverse doublesided Laplace transform. Repulsive Oscillator
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The the orthogonality of the expansion (66) makes possible toexpress the pricing kernel K I ( x , x (cid:48) , τ ) , defined by Ψ ( τ , x ) = (cid:90) K I ( x , x (cid:48) , τ ) Ψ ( , x (cid:48) ) dx as a sum of products of Hermite polynomials in x and x (cid:48) K I ( x , x (cid:48) , τ ) = λ e − ω τ ∞ ∑ = ( e − ω τ ) n n n ! × H n ( λ x ) H n ( λ x (cid:48) ) e λ ( x + x (cid:48) ) (68)By applying the Mehler formula ([39]) ∞ ∑ = ρ n n n ! H n ( x ) H n ( y ) exp ( ( x + y )) = (cid:112) − ρ exp (cid:18) ρ x y − ( + ρ )( x + y ) ( − ρ ) (cid:19) to the equation (68), with ρ = exp ( − ω τ ) , one obtains theMehler kernel, a generalized bivariate Gaussian probabilitydensity K I ( x , x (cid:48) , τ ) = λ √ π sinh ω τ exp ( λ xAx (cid:124) ) (69)where we have made the change τ = T − t , τ ≥
0, and x ≡ ( x , x (cid:48) ) . A is the SL ( , R ) matrix A = (cid:18) − coth ω τ csch ω τ csch ω τ − coth ω τ (cid:19) with coth and csch the hyperbolic cotangent and cosecant,respectively. A is singular when ω →
0. Howeverlim ω → ω A = τ (cid:18) − − (cid:19) As expected from the group law (53), the Mehler kernel mapsinto the heat kernel (hence the Black-Scholes theory) when ω → ω → K I ( x , x (cid:48) , τ ) = √ π σ τ e σ τ ( x − x (cid:48) ) The Mehler kernel (69) can also be obtained using the LCTassociated ([13], [32] ) to the harmonic oscillator generatingmatrix (51). From (19) W ( M H , x , x (cid:48) ) = √ π b exp ( − ( ax (cid:48) − xx (cid:48) + d x ) / ( b )) where a , b , c , d refer to the elements of (51). By direct substi-tution, we get W ( M H , x , x (cid:48) ) = K I ( x , x (cid:48) , τ ) A discussion of heat kernels and Mehler-type formulas basedon group-invariant solutions can be found in reference [14].Chapter 9 of [32] gives Baker-Campbell-Hausdorff relationsbetween pseudo-differential operators for the harmonic oscil-lator based on composition of SL ( , R ) matrices. Using the Feynman-Kac formula, we can write the solutionof (63a) as ψ ( x , t ) = E ( e − γ (cid:82) Tt X ( s ) ds ψ ( X T , T ) | X ( t ) = x ) with γ ≡ ω / σ and where the expectation is taken with re-spect to a normal process with volatility σ . This equationdescribes a derivative with a payoff that is discounted quadrat-ically with the oscillator level x . A drift term can be easilyadded with a coboundary generated by x .A more interesting approach is to use the harmonic oscillatoras a process describing stocks with non-normal returns witha correlation given by the Mehler Kernel (69). The reference[61] proposes a quantum harmonic oscillator as a model forthe market force which draws a stock return from short-runfluctuations to the long-run equilibrium.
7. Repulsive Oscillator
Using the arguments in section 6 for the harmonic oscillator,because of the R + central extension in the Heisenberg-Weylsubgroup of W Sp ( , R ) , the quantization group for the repul-sive oscillator ˜ H R is generated by the elliptic SL ( , R ) matrix M R = (cid:18) cos ω t λ − sin ω t − λ sin ω t cos ω t (cid:19) where ω , σ , t ∈ R , λ ≡ √ ω / σ M R is obtained from the harmonic oscillator generating matrix(51) by the mapping ω (cid:55)→ i ω (70)For brevity, in this article we will not apply the full GroupQuantization scheme to the repulsive oscillator. Most resultsfor the repulsive oscillator can be readily obtained from theharmonic oscillator results and the correspondence (70).The repulsive oscillator group composition law is t (cid:48)(cid:48) = t + t (cid:48) p (cid:48)(cid:48) = p + p (cid:48) cos ω t + λ x (cid:48) sin ω tx (cid:48)(cid:48) = x + x (cid:48) cos ω t + λ − p (cid:48) sin ω t ζ (cid:48)(cid:48) = ζ (cid:48) ζ exp (cid:0) ε R ( g (cid:48) , g ) (cid:1) with the cocycle ε R ε R ( g , g (cid:48) ) = ( px (cid:48) − xp (cid:48) ) cos ω t + ( λ − p p (cid:48) − λ x x (cid:48) ) sin ω t Conclusions
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One can easily verify that the high-polarization equation forthe repulsive oscillator is (compare with (63a)) ∂ ψ∂ t + σ ∂ ψ∂ x + γ x ψ = γ ≡ ω σ (71)whose separable solutions are time exponentials and Paraboliccylinder functions in x ([16], [32], [35], [31]). Note that uponthe analytic continuation (70) the Hermite functions (67) turninto Parabolic cylinder functions.The quantum repulsive oscillator [34] is a well known modelin the literature. A complete survey of the repulsive oscillatorin the context of W Sp ( , R ) can be found in [32]. Chapter9 of [32] gives special Baker-Campbell-Hausdorff relationsbetween pseudo-differential operators for the repulsive oscil-lator. Using the Feynman-Kac theorem, the solution of (71) can bewritten as ψ ( x , t ) = E ( e γ (cid:82) Tt X ( s ) ds ψ ( X T , T ) | X ( t ) = x ) where the expectation is taken with respect to a normal processwith volatility σ . This equation describes a derivative with apayoff that grows quadratically with the oscillator level x .Reference [33] uses a repulsive anharmonic oscillator modelto explain the distribution of financial returns in a stock marketwhen the market exhibits an upward trend.
8. Conclusions
We have presented a methodology, the Group Quantizationformalism, for constructing a financial theory from symmetryarguments. The
W Sp ( , R ) group has been used to describethe Black-Scholes model, the Ho-Lee model and the harmonicand repulsive oscillators.The choice of R + as the structural group in the Heisenberg-Weyl group ensures the appropriate commutator between themomentum and coordinate operators [ ˆ p , ˆ x ] =
1. This choiceis compatible with the financial interpretation of ˆ p as a delta.In addition, extending translations by R + formulates the the-ory directly in real (calculation) time, without the need toswitch to an Euclidean time in order to derive pricing quanti-ties.The case of Black-Scholes has been studied in detail. Wehave constructed the polarized functions and the position,momentum and Hamiltonian operators both in coordinate andin momentum spaces. The role of variance as cohomologicalinvariant has been identified.First polarizations for the harmonic oscillator have been de-rived in phase space and in orthogonal coordinates. A highpolarization was necessary to obtain polarized functions incoordinate (price) space. Group Quantization generates representations of a financialtheory in different functional spaces, allowing for alternativepricing frameworks such as the use of Laplace and Mellintransforms. In the case of Black-Scholes and Ho-Lee, thepolarized functions have the meaning of prices of financialinstruments, while their meaning for the oscillators is objectof active research.As we have shown in this article, among other interestingfeatures, the Group Quantization formalism provides natu-rally the functional constraints (polarization algebra or higherpolarization) for deriving the pricing equations. This makesGroup Quantization a versatile methodology for construct-ing a financial theory from symmetry arguments alone, usingsolely the principal bundle structure of a centrally extendedLie group. Definitions
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Appendices
1. Definitions
Vector Fields
Let M be a N-dimensional manifold with local coordinates x i , i = , . . . N . A vector field X is an application that asso-ciates a first order differential operator X ( x ) to a point x ∈ M . X ( x ) can be expressed in local coordinates as a linear combi-nation of the base fields e i ≡ ∂∂ x i i = , . . . N that is X ( x ) = X i ( x ) ∂∂ x i with X i ( x ) , i = , . . . N differentiable functions on M. Thespace of the vector fields is called the tangent space of M, T(M). For brevity we will write X instead of X ( x ) The integral curves of X are the solution to the set of ordinarydifferential equations dx i ds = X i ( x ) where s is an integration parameter. Note that the invariancecondition X f = f is constant along the integralcurves of X. Forms
A 1-form Γ is an application that associates to every point x ∈ M an element of the dual space of T (M). Γ : x → Γ ( x ) / Γ ( x )( X ( x )) = f ( x ) with f ( x ) a differentiable function.As with vector fields, we write Γ for Γ ( x ) . The space of 1-forms is called the cotangent space of M and is denoted by T ∗ ( M ) . A convenient representation for the basis of T ∗ ( M ) is u i ≡ dx i i = , . . . N and its action on the basis of T (M) is u i ( e j ) = ≡ dx i ( ∂∂ x j ) = δ i , j i , j = , . . . N d f ( x ) = ∂ f ∂ x i dx i For a 1-form Γ d Γ = ∂ Γ i ∂ x j dx i ∧ dx j where the wedge operator ∧ is the antisymmetric combination dx i ∧ dx j = dx i ⊗ dx j − dx j ⊗ dx i One can define in an analogous way differentials of higherorder forms. Note that the antisymmetry of the wedge operatorimplies that d f = d Γ =
0. Also, i f a n -form acts on n − k k vector fields one obtains a k -form. For instance the1-form d f acting on a field X gives a zero-form d f ( X ) = ∂ f ∂ x i dx i ( X j ( x ) ∂∂ x j ) = X i ( x ) ∂ f ∂ x i = X ( f ) Analogously, one can check that d Γ acting on X alone gives a1-form d Γ ( X , . ) = ∂ Γ i ∂ x j ( X i dx j − X j dx i ) We use the inner product notation to denote the action of an-for Ω on a vector field X i X ( Ω ) = Ω ( X ) Lie Derivative
The Lie derivative evaluates the change of vector fields andforms along the flow defined by another vector field.The Lie derivative of a function f with respect to a vector fieldX is L X f = X ( f ) = d f ( X ) For two vector fields X, Y , the Lie derivative of Y with respectto X, L X Y , is a vector field defined by L X Y = − L Y X ≡ [ X , Y ] = (cid:18) X i ∂ Y j ∂ x i − Y i ∂ X j ∂ x i (cid:19) ∂∂ x j The Lie derivative of a 1-form Γ with respect to a vector fieldX, L X Γ is also a 1-form, and has the meaning of the rate ofchange of Γ along the integral lines (flow lines) of X. Onefinds, in local coordinates L X Γ = ( L X Γ ) i dx i where ( L X Γ ) i = X j ∂ Γ i ∂ x j + Γ j ∂ X j ∂ x i or, in a more concise notation L X Γ = d ( Γ ( X )) + d Γ ( X , . ) ≡ i X d Γ + d ( i X Γ ) Lagrangian Formulation
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Lie Algebra
A Lie algebra G is a vector space over a field F equipped witha bilinear map [ , ] : ( G , G ) → G such that [ X , Y ] = − [ Y , X ] and [ X , [ Y , Z ]] + [ Y , [ Z , X ]] + [ Z , [ X , Y ]] = Lie bracket . The Lie algebra for a Lie group Gis defined using the tangent vectors at the identity e as vectorspace of and the Lie derivative as bracket operator. One has [ X i , X j ] = c ki , j X Lk i , j = , . . . dim G where the coefficients c ki , j ∈ F , i , j , k = , . . . dim G are called structure constants .Since X g is a LIVF if and only if X g = L Tg X e and an RIVF ifand only if X g = R Tg X e , there is a biunivocal correspondencebetween the LIVFs (RIVFs) and the set of the tangent vectorsat the identity e of G , with the following brackets [ X Li , X Lj ] = c ki , j X Lk [ X Ri , X Rj ] = − c ki , j X Rk [ X Ri , X Lj ] = ∀ i , j = , . . . dim G
2. Instrument Prices in Momentum Space
In this section we present examples of derivatives pricing inthe less familiar momentum (Laplace) space.The price of a financial instrument Ψ ( x , t ) maturing at T isfound by performing an inverse Laplace transform Ψ ( x , t ) = π i (cid:90) c + ∞ c − i ∞ d p e xp K ( p , τ ) Ψ ( p , T ) (72)where τ ≡ T − t , t ≤ T and Ψ ( p , T ) = Ψ ( p ) exp ( E ( p ) T ) isthe payoff at maturity in the Laplace space Ψ ( x , T ) = π i (cid:90) c + ∞ c − i ∞ d p e xp Ψ ( p , T ) (73) B.1 Instrument Prices using the Mellin Transform
The inverse Mellin transform gives the instrument’s pricein terms of the stock value instead than the log-stock. Thepricing equations (72) and (73) read now V ( S , t ) = π i (cid:90) c + ∞ c − i ∞ d p exp ( − E ( p ) τ ) V ( p , T ) S p V ( S , T ) = π i (cid:90) c + ∞ c − i ∞ d p S p V ( p , T ) (74)with E r ( p ) ≡ σ p + µ p − r µ = − σ The final condition in Mellin space is given by C ( p , ) = (cid:90) + ∞ V ( S , T ) S − p − dS M − exists only for complex values of y so that c ≡ ℜ ( y ) > convergence strips . Each strip leads to differentresults for M − ( f ) ([57], [58], [59], [60]). B.2 Call Option Price
We price a call option with strike X and maturity T using theMellin transform. The payoff is, from (73) C ( p , ) = (cid:90) + ∞ ( S − X ) + S − p − dS = X − p p ( p − ) with ℜ ( p ) >
1. The expression of the call option price in theMellin space is, using (74) C ( p , τ ) = e − r τ X − p p ( p − ) exp ( − ( σ p + µ p ) τ ) where τ ≡ T − t , t ≤ T .The Black-Scholes call option formula in the price (coordi-nate) representation is obtained by the inverse Mellin trans-form V ( S , t ) = e − r τ π i (cid:90) c + ∞ c − i ∞ C ( p , τ ) S p d p where c ∈ ( , + ∞ ) and ℜ ( p ) >
3. Lagrangian Formulation
In the Group Quantization formalism, Θ | C , the projectiononto the base manifold of the vertical form Θ along the inte-gral trajectories of the characteristic module C Θ provides the classical action S ([1] ) S = (cid:90) Θ | C = (cid:90) L ( x , ˙ x ) dt The classical Lagrangian L ( x , ˙ x ) is obtained by the projectiononto the base manifold. Note that Θ and Θ + d f generateequivalent Lagrangians that differ in a total derivative and donot change the action S The vertical (connection) form d Θ restricted to the quotientspace ˜ G / U is analogous to the Poincar´e-Cartan form of classi-cal mechanics Θ PC ,which can be written in local coordinates q , p as Θ PC = ∑ i p i dq i − H ( p , q ) dt (75)where H ( p , q ) is the Hamiltonian function, and q , p are conju-gate pairs of coordinates and momenta. In classical mechanics, If the quantization group ˜ G is finite, the quotient space P = ˜ G / C Θ and the quotient connection form Θ P = Θ / C Θ , where C Θ is the char-acteristic module of Θ , define a fiber bundle ( P , U , π ) where the curva-ture form Ω = d Θ P is a symplectic form over P / U . Taking the quotientby the integral flows of C Θ allows the definition of local coordinates in P / U , ( p i , q i ) , i = , ... dim ( P / U ) representing the canonical momentaand canonical coordinates. of the vertical form Θ along the integral flows(trajectories) of the characteristic module C Θ . Lagrangian Formulation
22/ 24 the equations of motion are just the flows associated to theHamiltonian field X H ≡ ∂∂ t + ∑ i ∂ H ∂ p i ∂∂ q i − ∂ H ∂ q i ∂∂ p i so along Hamiltonian trajectories˙ q i ≡ dq i dt = ∂ H ∂ p i ˙ p i ≡ d p i dt = − ∂ H ∂ q i Note that Θ PC has the structure of a Legendre transformationif one identifies p i ≡ ∂ L / ∂ ˙ q i , where L is the Lagrangianfunction. Along the Hamiltonian flows Θ PC | H = L ( ˙ q , q ) dt The differential of Θ | PC along the Hamiltonian trajectories isa symplectic form. d Θ PC | H = ∑ i d p i ∧ dq i The Lagrangian function is used for the application of pathintegral methods in computational finance ( [24], [26], [27],[15]).
C.1 Black-Scholes Lagrangian
We find the Black Scholes Lagrangian by restricting the Black-Scholes connection Θ (57) to the characteristic module trajec-tories on the base manifold. C Θ is generated by the time translations X Lt whose integralflows on the base manifold are (see (28)) dtds = d pds = dxds = σ p + µ → t = s p = p x = x + ( σ p + µ ) t (76)where x , p , x , ζ are integration constants and s is theintegration parameter.We first add the total differential d ( px ) to Θ so the equationsadopt the Poincar´e-Cartan expression (75) Θ → d ( xp ) + Θ = p dx − E ( p ) dt then Θ | C = p ( σ p + µ ) dt − (cid:18) σ p + µ p − r (cid:19) dt = σ ( ˙ x − µ ) dt − rdt where we have used equation (76) dxdt ≡ ˙ x = σ p + µ → p = ˙ x − µσ with the result L ( x , ˙ x ) = σ ( ˙ x − µ ) − r (77)The solution x ( τ ) of the Lagrangian (77) coincides with theexpected value of the coordinate (log-price) using the K BS kernel in (39)) (cid:104) ˆ x (cid:105) = √ πσ τ (cid:90) ∞ − ∞ dx x e − σ τ ( x (cid:48) − x − µτ ) = x (cid:48) + µτ C.2 Euclidean Oscillator Lagrangian
Following the same steps than in section C.1, and using theexpressions (57) and (58), we find that the C Θ trajectories dtds = d pds = ω λ x dxds = ω λ − p generates the following Lagrangian for the Euclidean har-monic oscillator L H ( x , ˙ x ) = σ (cid:18) ˙ x + ω x (cid:19) As expected, the mapping ω → i ω provides the Lagrangianfor the Euclidean repulsive oscillator L R ( x , ˙ x ) = σ (cid:18) ˙ x − ω x (cid:19) Acknowledgements
The author is grateful to Peter Carr for very many valuableinsights and support, and to Gregory Pelts for sharing hisknowledge on the geometric structure of finance.
Disclaimer
The views expressed herein are those of the author and donot reflect the views of my current employer, Wells FargoSecurities, or affiliate entities. eferences
23/ 24
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