QUANTITATIVE FINANCE RESEARCH CENTRE. Lie Symmetry Methods for Local Volatility Models QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE F

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1 QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE F INANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 377 Septemer 06 Lie Symmetry Methods for Local Volatility Models Mark Craddock and Martino Grasselli ISSN

2 Lie Symmetry Methods for Local Volatility Models Mark Craddock Martino Grasselli Septemer 6, 06 Astract We investigate PDEs of the form u t = σ t, xu xx gxu which are associated with the calculation of expectations for a large class of local volatility models. We find nontrivial symmetry groups that can e used to otain standard integral transforms of fundamental solutions of the PDE. We detail explicit computations in the separale volatility case when σt, x = htα + βx + γx, g = 0, corresponding to the so called Quadratic Normal Volatility Model. We also consider choices of g for which we can otain exact fundamental solutions that are also positive and continuous proaility densities. Key words: Lie symmetries, fundamental Solution, PDEs, Local Volatility Models, Normal Quadratic Volatility Model. Introduction A symmetry of a differential equation is a transformation that maps solutions to solutions. Continuous symmetries which have group properties are called Lie symmetries, since the means for computing them were developed in the late nineteenth century y Sophus Lie. A modern account of the theory can e found in Olver s ook, Olver 993 and the ooks of Bluman, such as Bluman and Kumei 989. The applications of Lie symmetry groups are extensive. These range from epidemiology to finance. Examples of these applications in stochastic analysis and finance may e found in references such as Craddock and Lennox 007, Craddock and Lennox 009, Craddock and Platen 004, Lennox 0, Craddock 009, Craddock and Lennox 0, Baldeaux and Platen 03, Goard 0, Goard 000, Goard and Mazur 03, Andersen 0, Lipton 00, Zuhlsdorff 00, Carr et al. 006, Carr et al. 03, Cordoni and Di Persio 04, Grasselli 06 and Itkin 03. This list is y no means exhaustive. School of Mathematical and Physical Sciences, University of Technology Sydney. Australia. Mark.Craddock@uts.edu.au. Department of Mathematics, University of Padova Italy and Léonard de Vinci Pôle Universitaire, Research Center, Finance Group, 9 96 Paris La Défense France. grassell@math.unipd.it. Acknowledgments: we are grateful to Giulio Miglietta, Johanes Ruf, Josef Teichmann and the participants of the Quantitative Methods in Finance conference Sydney, Decemer 05 and the London Mathematical Finance Seminar London, March 06 for useful discussions and suggestions. The usual disclaimer applies.

3 The aim of this paper is to approach the classical local volatility models from the perspective of Lie symmetry analysis of the associated ackward PDEs. The concept of a symmetry group of a PDE was introduced y Sophus Lie in the 880s, see the collection Lie 9. Lie s methods have recently een applied to prolems in stochastic analysis y, among others, Craddock and Platen 004, Craddock and Lennox 007 and Craddock and Lennox 009, who focused mainly on generalized square root models of the form dx t = fx t dt + X γ t dw t, for some constant γ. These models have ovious financial interpretations. In this paper, we focus on local martingale models of the form together with the associated paraolic ackward equation dx t = σt, X t dw t, t u = σ t, x xx u. Here and throughout x = x etc. For suitale choices of σ, this PDE has many symmetries. If we have at hand a Lie Group that leaves the PDE invariant, then we can apply it to any solution to get another solution. The key result for finding the Lie Groups admitted y the PDE is Lie s Theorem. For a PDE of order n, we write down the infinitesimal generator v of the symmetry and otain its so called n-th prolongation pr n v see elow. Lie s Theorem says that v generates a symmetry of the PDE P x, D α u = 0, x Ω R m, D α u = α u, α x xαm n + + α m n, if and only if pr n v[p x, D α u] = 0 whenever P x, D α u = 0. See Olver 993, Chapter. This condition leads to an equation in a set of independent functions of the derivatives of u. As the equation must e true for aritrary values of these independent functions, their coefficients must vanish, leading to a linear system of equations known as the Determining Equations. Once these equations are solved, one can find the corresponding Lie group admitted y the PDE. For linear paraolic PDEs on the line, Craddock proved that it is always possile to find a symmetry that maps a nonzero solution to a generalised Fourier or Laplace transform of a fundamental solution, see Craddock and Dooley 00. The question of under what conditions these fundamental solutions yield transition densities for the underlying stochastic process was addressed in Craddock and Lennox 009 and Craddock 009. In this paper we give sufficient conditions directly on σ for the existence of a Lie group admitted y the PDE. We compute symmetries which lead directly to a Fourier integral transform of the fundamental solution of the ackwards PDE. In the special case where σt, x = htα + βx + γx i.e. with σ a separale function with a polynomial of degree two dependence on the state variale X t we exponentiate the group to explicitly find the symmetries of the equation y considering separately the cases of two distinct real roots, a single real root and two distinct complex roots. This local volatility

4 case corresponds to the so called Quadratic Normal Volatility model that has een investigated, among others, y Zuhlsdorff 00, Andersen 0 and Carr et al. 03. For this model, we provide the explicit expression for the positive fundamental solution that are also proaility densities, thus giving an analytical counterpart to the proailistic justification of the tractaility of this model presented in Carr et al. 03. For pricing, we have the following generic situation. Generally one can show, for example y aritrage arguments, that the price us, t for a derivative security on an underlying S Ω R n, is given y a solution of the terminal value prolem for the paraolic PDE u t + Lu = 0, us, T = fs, S Ω R n, t [0, T ]. 3 One transforms this to the initial value prolem u t = Lu = 0, us, 0 = fs, S Ω R n, t [0, T ], 4 y letting t T t. Clearly we require a unique, continuous price. For hedging we also require the existence of a certain numer of derivatives. This amounts to asking under which circumstances the Cauchy prolem 4 has a unique, smooth solution. There is extensive literature on the question of existence and uniqueness of solutions. We cannot survey it all here, ut Theorem 6 of Friedman 964 tells us that if u t = Lu is uniformly paraolic, with smooth coefficients having ounded first and second derivatives in Ω [0, T ] then 4 has a unique solution satisfying T 0 R ux, t exp k x dxdt < for some constant k > 0. n This solution will also e smooth if f is not too pathological. Generally speaking, we get non-uniqueness for smooth solutions only when the initial data grows extremely rapidly. This is precisely the situation in the famous example of non-uniqueness for the heat equation that was constructed y Tychonoff Tychonoff 935. Financial considerations typically rule out such pathological situations. Thus for pricing, we require fundamental solutions that return smooth continuous solutions of the pricing PDE, given reasonale payoffs. In this paper we exhiit such fundamental solutions. It is also sometimes possile to otain fundamental solutions that are not continuous and which return solutions that are discontinuous at some point, typically the origin. We rule out such solutions on the grounds that the price should depend continuously on the underlying. Lie Symmetries for PDEs In this Section, we will give a short introduction to Lie Groups of transformations with a particular focus on their applications to the solution of PDE s. In this introduction we follow closely the presentation given in Bluman and Kumei 989. Another standard reference for this topic is Olver 000. Then we will focus on the relations etween symmetry analysis of PDE s and transform of solutions of PDE s as studied in Craddock and Platen 004, Craddock and Lennox 007 and Craddock and Lennox

5 . Invariance of a Differential Equation Let us consider now the second order PDE in two independent variales x, t of the form F x, t, u, u x, u t, u xx, u xt, u tt = 0. 5 We work exclusively on a second order PDE in two independent variales since this is the setting which interests us here. Of course the results we are aout to discuss are valid for ODE s as well and for differential equations of any order in general. We now give a definition that will e crucial in the following. The PDE 5 is said to admit the Lie Group of transformations X ϵ, T ϵ, U ϵ ϵ, for ϵ > 0, if the family of solutions of 5 is an invariant family of surfaces for X ϵ, T ϵ, U ϵ ϵ. In other words, this implies that if we have at hand a Lie Group G that leaves the PDE invariant, then we can apply it to any solution to get another solution. If the solution we start from is itself an invariant surface, then y applying the Lie Group to it, we will end up with the solution itself. Here the crucial concept is that the Lie Group must act on the space where the solutions of the PDE lives, in our case the x, t, u-space. We introduce a vector field v = ξx, t, u x + τx, t, u t + ϕx, t, u u, 6 which is the infinitesimal generator of G. We can extend the action of G in a natural way to act also on the derivatives of u up to second order, y essentially requiring that the chain rule holds. We call this the second prolongation of G and denote it y G. The generator of G is the second prolongation of v. This is given y see e.g. Bluman and Kumei 989 and Olver 993: pr v = v + ϕ x + ϕ t + ϕ xx + ϕ xt + ϕ tt. u x u t u xx u xt u tt Expressions for ϕ t etc are given y the prolongation formula given elow. The main result due to Lie see e.g. Bluman and Kumei 989 is that the PDE 5 admits the Lie Group of transformations X ϵ, T ϵ, U ϵ if and only if F x, t, u, u x, u t, u xx, u xt, u tt = 0 G F x, t, u, u x, u t, u xx, u xt, u tt = 0, that is the surface {F x, t, u, u x, u t, u xx, u xt, u tt = 0} is invariant for the second prolongation G of the group action G. We refer e.g. to Bluman and Kumei 989 for the construction of the second prolongation for a group. At the level of vector fields, this leads to Lie s Theorem which states that v generates a one parameter group of symmetries if and only if pr v[f x, t, u, u x, u t, u xx, u xt, u tt ] = 0 whenever F x, t, u, u x, u t, u xx, u xt, u tt = 0. The coefficients ϕ x, ϕ t etc are given y the prolongation formula, first pulished y Olver 978. ϕ J j x, u n = D J ϕj 4 p ξ i u j p i + ξ i u j J,i, 7 i= i=

6 where u j i = uj x i, and uj J,i = uj J x i, and D J is the total differentiation operator. In practice, we use the variale names rather than the multi-indices in the exponents. So we write ϕ xx rather than ϕ,. The coefficient functions ϕ x and ϕ t are ϕ x = D x ϕ ξu x τu t + ξu xx + τu xt = ϕ x + ϕ u u x ξ x u x ξ u u x ξu xx τ x u t τ u u x u t τu xt + ξu xx + τu xt = ϕ x + ϕ u ξ x u x ξ u u x τ x u t τ u u x u t. Similarly, ϕ t = D t ϕ ξu x τu t + ξu xt + τu tt, which leads to and ϕ t = ϕ t ξ t u x + ϕ u τ t u t ξ u u x u t τ u u t, ϕ xx = D xx ϕ ξu x τu t + ξu xxx + τu xxt = ϕ xx + ϕ xu ξ xx u x τ xx u t + ϕ uu ξ xu u x τ xu u x u t ξ uu u 3 x τ uu u xu t + ϕ u ξ x u xx τ x u xt 3ξ u u x u xx τ u u t u xx τ u u x u xt. The coefficients ϕ xt and ϕ tt can e otained in the same manner, ut we do not need them for our calculations.. Symmetries and Fundamental Solutions In a series of articles of increasing generality, Craddock and his coauthors studied the symmetries of some Kolmogorov ackward equations associated to a real diffusion. Namely, Craddock and Platen 004 studied the equation t u = x xx u + fx x u, x R + whereas Craddock and Lennox 007 studied the equation t u = σx γ xx u + fx x u µx r u, x R and Craddock and Lennox 009 studied the equation t u = σx γ xx u + fx x u gxu, x R. 8 In Craddock and Dooley 00, it was shown that if the PDE u t = Ax, tu xx + Bx, tu x + Cx, tu, 9 has a four dimensional Lie algera of symmetries, then there always exists a symmetry mapping a nontrivial solution u to a generalised Laplace transform of a product of a fundamental solution and u. Dividing out y u yields the desired fundamental solution. If the symmetry 5

7 algera is six dimensional, then we otain a generalised Fourier type transform. Lie proved that if a PDE of the form 9 has a six dimensional group of symmetries, then it may e transformed to the heat equation y an invertile change of variales. If the symmetry group is four dimensional, then it can e reduced to the form u t = u xx A u, A 0, 0 x which has a known fundamental solution. Thus if one is interested only in otaining a fundamental solution for a PDE, we can attempt to reduce it to 0 or the heat equation as appropriate. This procedure has een followed y e.g. Cordoni and Di Persio 04, Carr et al. 006, Goard 000 and Itkin 03. However it was shown in Craddock 009, that this will not necessarily produce a fundamental solution which is also a proaility density. There are other issues. A simple oundary value prolem on, say [0, ] may e mapped to a more difficult prolem for the heat equation or 0. To illustrate this, let σx = + x a and consider the prolem u t = σ xu xx, x R with ux, 0 = fx, where f lies in some appropriate function space, say L R. The change of variales y = tan x a, ux, t = U tan x a, t converts this to the PDE U t = U yy tan yu y, y π, π Uy, 0 = f a + tan y. If we suppose u is zero at ±, then we also require U π, t = U π, t = 0. Using the methodology of Craddock 009, it is easy to show that the PDE has a fundamental solution pt, y, z = y z t/ cos z πt e t cos y, defined on all of R except odd multiples of π/. So this does not help us. The further change of variales Uy, t = e t/ secyvy, t produces the following prolem for the one dimensional heat equation: v v t = v yy, y π, π, vy, 0 = cos yfa + tan y, π π, t = v, t = 0. This prolem can e solved y Fourier series methods, ut it is more complicated than the original prolem. In fact as we shall see, it is easier to solve the original prolem directly. A further example is the initial and oundary value prolem u t = u xx + au x, ux, 0 = fx, u x 0, t = 0. 6

8 Solution of this prolem yields the transition density for a reflected Brownian motion with drift. This is a deceptively difficult prolem as it cannot e solved y either the Fourier sine or cosine transform. Methods for its analytical solution can e found in Fokas 008. They are however well outside the scope of this paper. It is possile to solve it using Lie symmetry methods, similar to those presented here, ut we will not do so here. Here we oserve that setting ux, t = e ax/ a t/4 vx, t reduces prolem to v t = v xx, vx, 0 = e ax/ fx, v x 0, t = a ea t/4 u0, t. So we have reduced the PDE to the heat equation, ut the resulting oundary value prolem cannot e solved as we do not know the value of u0, t. The moral is that eing ale to reduce a PDE to the heat equation or 0 is not a universal panacea. We can arrive at a prolem harder than the original, or indeed one that cannot e solved at all. Therefore it is essential that we have techniques which yield solutions without the need to make any changes of variales. Note that the equation 8 corresponds to the ackward Kolmogorov equations associated to the diffusion which can e defined as a solution of the SDE dx t = fx t dt + σx γ/ dw t, killed at the rate gx t. Thus if we define [ vx, t = E x e ] t 0 gxsds fx t, then v is a solution of 8 with v., 0 = f. If we take g = 0 in the argument aove and fx = e λx then we have [ ] vx, t = E x e λxt, which is the Laplace transform of the marginal law of X t. This Laplace transform can in theory e inverted, holding x, t fixed, to get the fundamental solution of the PDE which can e interpreted as the transition proaility of the diffusion process. The articles of Craddock and coauthors cited aove sought to find solutions of the PDE s with initial datum fx = e λx y exploiting the symmetries of the PDE itself. Specifically, in the conservative case g = 0, they were ale to find all the symmetries admitted y the PDE and then they noted that there is a symmetry that maps the solution constantly equal to to a solution which is an exponential in x at t = 0. In the following section we will pursue a similar goal on an equation arising from a different linear diffusion. 3 The Prolem Following Bluman and Kumei 989 and Olver 993, we are looking for infinitesimal symmetries of the PDE when σt, x = σx: u t = σ xu xx gxu, x D R, σ > 0, 3 7

9 where σ and g are functions defined in the state space domain D R which we take to e an interval of the form D = x l ; x r. As noted aove, this is the Kolmogorov ackward equation associated to a diffusion X satisfying dx t = σx t dw t, 4 killed at rate gx t, where W is standard Brownian motion defined in a filtered proaility space Ω, F, F t t [0,T ], P. Local Volatility Models typically include the presence of a time dependence in the volatility, that is σ = σx, t. The special separale case σx, t = fthx, 5 can e managed y the time-change methodology for the Brownian motion, see also Andersen 0. The general not separale case corresponding to can e managed using the same methodology, ut it leads to extremely difficult equations. We prefer to leave this out since little insight can e gained in those cases without a considerale amount of analysis. Remark 3.. If we specify a oundary condition like ux, 0 = e λx then, using a Feynman- Kac argument, 3 can e associated to the transform { ux, t = E [exp x λx t t 0 }] gx s ds. The equation 3 can e transformed to one where the second derivative term has a constant coefficient y introducing a change of variales. Let y = x = x. We suppose dz σz that is invertile, so that x = y. Then set ux, t = Ux, t. Then so that ecomes The more general equation can e transformed to U t = U yy + u x = σx U y, u xx = σ x U yy σ x σ x U y, U t = U yy σ yu y g yu. 6 u t = σ xu xx + kxu x gxu, 7 k y σ y σ y U y g yu. 8 Craddock and Lennox 009 showed that a PDE of the form u t = σx γ u xx + fxu x gxu, 8

10 has non-trivial symmetries if and only hx = x γ fx satisfies an equation of the form σxh σh + h + σx γ gx = σax γ + B 9 σxh σh + h + σx γ gx = Ax4 γ γ + Bx γ γ + C, 0 σxh σh + h + σx γ gx = Ax4 γ γ + Bx3 3 γ 3 3 γ + Cx γ γ κ, with κ = γ 8 γ 4σ. See Craddock and Lennox 009 for the specifics. We remark that on the line, a non-trivial symmetry group will have dimensions of either four or six. Every time homogeneous, linear paraolic PDE on the line has the trivial symmetries ux, t cux, t + ϵ, which corresponds to a two dimensional symmetry group. A general result descriing conditions under which an aritrary linear paraolic PDE on the real line has non trivial symmetries follows easily. Proposition 3.. Let F y = y k y σ y σ y. Then the PDE 7 has nontrivial symmetries if and only if yf F + F + y g y = A y 4 + B y + C y 3/ + D, where y = x = x σz dz and the constants A,..., D are as in the right side of 9-, with γ = 0. This result is at the moment largely of theoretical interest. A more detailed account for higher dimensional equations is in Vu s forthcoming thesis, Vu 06. For a given σ, we can otain and determine drift functions k for a given g such that the PDE has symmetries and the results of Craddock 009 can e employed to find fundamental solutions, if the symmetry group is four dimensional. However, if we treat this as an equation for σ it is quite difficult to deal with. So here we will derive an explicit equation for σ in the case of zero drift and treat in detail a model which arises from this. 4 The Determining Equation By applying the prolongation to the PDE 3 we get [ pr v u t ] σ u xx + gu = ϕ t σxσ xu xx ξ σ xϕ xx + g xuξ + gxϕ, which is ϕ t = σxσ xu xx ξ + σ xϕ xx g xuξ gxϕ. 9

11 Since the PDE is linear, standard arguments see e.g. Bluman and Kumei 989 imply that ξ and τ will e independent of u, that is ξ = ξx, t, τ = τx, t, and ϕ will e linear in u, that is ϕx, t, u = αu + β for some αx, t, βx, t. information we otain β t + α t u + α τ t u t ξ t u x = σxσ xu xx ξ g xuξ gxαu + β + σ x β xx + α xx u + α x ξ xx u x τ xx u t + α ξ x u xx τ x u xt. Let us now identify the coefficients of the last equation: the constant term gives β t = σ xβ xx gxβ, which asically says that β satisfies the initial PDE. The coefficient of u yields while the coefficient of u x gives The coefficient of u xx gives Using this α t + τ t gx = σ x α xx + τ xx gx g xξ; ξ t = σ x α x ξ xx. 3 α τ t σ x = σxσ xξ + σ x and finally the coefficient of u xt gives which implies τ = τt, so that ecomes and from 4 we otain τ xx σ x + α ξ x 4 τ x = 0, 5 α t + τ tgx = σ xα xx g xξ 6 x ξx, t = σx τ t dy + ρt, 7 σy where ρ is an aritrary deterministic function. We now differentiate the last expression and we plug the results into equation 3 that ecomes α x = σ x ξ tx, t + ξ xxx, t = τ x t σx σy dy ρ t σx + x 4 τ tσ x σy dy + ρtσ x + 4 τ t σ x σx, We do not specify lower oundary of the integral since the constant term can e included into the aritrary time-dependent function ρ. 0

12 that we write in a more compact way y dropping the dependence on the arguments: α x = τ σ σ ρ σ + 4 τ σ σ + ρσ + 4 τ σ σ. 8 Integrating with respect to x gives α = 4 τ ρ σ σ + 4 τ σ σ + ρσ + η, 9 where η = ηt is an aritrary function of time. Also, differentiating 8 gives α xx = τ σ σ σ + σ + ρ σ σ + ρσ + σ σ 4 τ + σ σ σ σ, 30 and from 9 we get α t = 4 τ σ ρ σ + 4 τ σ σ + ρ σ + η. 3 Then after some manipulations 6 ecomes 4 τ ρ σ σ + η = σ 4 τ + ρ 4 σ σg [ σ +τ 4 σ σg σ + 8 Now notice that σσ σ σ g = 8 4 σ g σσ σ ] g. since σ σ = σ /, then we can write the determining equation granting the existence of non trivial symmetries: 4 τ ρ σ σ + η = σ 4 τ + ρσ 4 σ g [ σ σ +τ 4 σ g σ σ + 4 σ g ]. 3 Remark 4.. The case σx = x has een investigated y Craddock and Platen 004, who considered the following PDE with D = R +. Taking g = 0 in 3 gives u t = xu xx + fxu x, xτ x ρ + τ 4 + η = 3 6x 3 ρ, which agrees with Craddock and Platen 004 p. 88 when using their notation we take f = 0; ρ = ρ; σ t = η t + τ t/4.

13 Remark 4.. The case σx = σx γ/, γ, has een investigated y Craddock and Lennox 007, who considered the following PDE u t = σx γ u xx + fxu x µx r u, with D = R +, and susequently y Craddock and Lennox 009 in the more general case u t = σx γ u xx + fxu x gxu. In the latter case 3 gives x γ x σ γ τ γ ρ σ + τ γ 4 + η = g xξ τ tgx + 4 ρ σ σγ γ γ x 3 γ 3, which agrees with formula.9 in Craddock and Lennox 009 where we take f = 0; ρ = ρ σ. From 3 we see that in order to match the terms depending on x we have to compare the functional form of σ 4 σ g with the ones of σ and σ, or equivalently we have to assume some functional forms for g. We then consider separately some cases. 5 The Infinitesimal Generators In this section we show that if σ and g satisfy a given integro-differential equation, then the PDE 3 admits a symmetry group whose finite-dimensional part has dimension 6. In the following theorem we state the requirement and find the associated determining equation 3. It is also possile to have Lie symmetry algeras which are two dimensional and four dimensional. As we are interested in the quadratic local volatility model, for revity, we will focus on the six dimensional case. Theorem 5.. The PDE 3 admits a six dimensional Lie symmetry group if and only if σ and g satisfy 4 σσ g = A σ + B 33 σ σ for some constants A, B R. Proof. Under 33 the determining equation 3 ecomes 4 τ ρ σ σ + η = 4 τ + Aρ + Bρ + 3 Aτ σ + Bτ. σ 34 Note that 34 involves only the expression x /σydy which is a non constant function of x. This allows us to identify the time-dependent coefficients and arrive to the system for τ, ρ and η: 4 τ = Bτ ; ρ = Bρ + 3 Aτ ; η = 4 τ + Aρ.

14 Conversely, if σ, g do not satisfy 33, it follows immediately that either τ = 0, which in turns implies that there is no symmetry group transforming solutions which are constant in time to solutions which are not, see e.g. Baldeaux and Platen 03 p. 9, or ρ = 0 which will produce a four dimensional Lie symmetry algera. The interested reader may investigate this case. In the following we compute the infinitesimal generator of the Lie group admitted y the PDE 3 in the case where B is null. The other cases can e treated similarly and we gather them in Appendix A. Theorem 5.. If σ and g satisfy 33 with B = 0 then the PDE u t = σ xu xx gxu, x D admits a Lie symmetry group whose finite dimensional part has dimension 6. The corresponding Lie algera is generated y the following infinitesimal symmetries: t x v = σx σy dy A 4 t3 x + t t + x 3 4 σy dy + 4 At + t x 4 σ x σy dy σ x 8 At3 t 4 6 A t 4 u u ; σx x v = σy dy 3 8 σxat x + t t 3 x + 4 At σy dy + σ x x 4 σy dy 3 6 At σ x 8 A t 3 u u ; v 3 = t ; x v 4 = σxt x + σy dy + t σ x + At u u ; v 5 = σx x + σ x + At u u ; v 6 = u u. All other symmetries of 3 have infinitesimal generators of the form v β = β u, where β is an aritrary solution of the PDE. These correspond to the superposition of symmetries. i.e. u u + β is a solution when u and β are solutions. Remark 5.3. In the case g = A = 0 we recover σ = 0, corresponding to the Quadratic Normal Volatility Model. We will otain fundamental solutions in this case elow. The dimension of the Lie algera is determined y the numer of constants of integration. The function ρ appears as a second derivative, so it yields two constants of integration. If we set ρ = 0, we therefore reduce the dimension of the Lie algera from six to four. 3

15 Proof. If B = 0, the system for τ, ρ and η admits the following solution: τt = C t + C t + C 3 ; ρt = C 4 At3 3 8 C At + C 4 t + C 5 ; ηt = C 6 A t 4 C 8 A t 3 + C 4 At + C 5 A C t + C 6, 4 where C,..., C 6 are aritrary constants. From 7 we have C t + C x ξx, t = σx σy dy A 4 t3 C 3 8 At C + C 4 t + C 5, 35 and from 9 we get αx, t = C 4 x σy dy x σy dy C t + C σ x C 6 A t 4 C 8 A t 3 + C 4 At C At 3 x 4 C At + C 4 σy dy C C 5 A C 4 4 At3 3 8 C At + C 4 t + C 5 σ x t + C 6. Recall that the latter equation for α determines ϕ as ϕ = αu + β. We are looking for infinitesimal symmetries whose vector field has the following form: v = ξ x + τ t + ϕ u, then we arrive at C t + C x v = σx σy dy A 4 t3 C 3 8 At C + C 4 t + C 5 x + C t + C t + C 3 t { [ + u C x 4 σy dy 3 4 C At 3 x 4 C At + C 4 σy dy + x 4 C t + C σ x σy dy + C 4 At3 3 8 C At + C 4 t + C 5 σ x C 6 A t 4 C 8 A t 3 + C 4 At + C 5 A C ] } t + C 6 + β u. 4 Now taking the coefficients of the aritrary constants yields the result. 6 The Quadratic Normal Volatility Model The preceding material provides the tools that are needed to otain analytical results for a wide class of models. Our aim now it to demonstrate how such an analysis can proceed, y 4

16 studying a well known case. In this section we therefore consider the case B = 0, corresponding to /4σσ g = A/σ, in the specification where g = A = 0, which leads to σx = α + βx + γx 36 with γ 0 the case γ = 0 corresponds to the so called shifted lognormal model. This corresponds to the Quadratic Normal Volatility model, where the underlying process X t satisfies the following SDE: dx t = α + βx t + γx t dw t, that is a local volatility model deeply investigated in Lipton 00, Zuhlsdorff 00 and recently re-discovered y Andersen 0 and Carr et al. 03. The Quadratic Normal Volatility model admits closed form formulas for the price of vanilla options and it is in general highly tractale. The following susections constitute an analytical counterpart of the proailistic arguments of Carr et al. 03 in support of the analytical tractaility of the model. It admits indeed non trivial symmetries that are crucial in the determination of the fundamental solutions to the PDE corresponding to the proaility density of the underlying. 6. Distinct Real Roots We will consider here the case where the polynomial admits two distinct real roots l < m. In this case we can write w.l.o.g. σx = x mx l, m l, 37 m l corresponding to dx t = X t mx t l dw t, X 0 R. m l Remark 6.. The presence of a deterministic function ft such that x mx l σx, t = ft m l can e managed y the time-change methodology for the Brownian motion, see 5. This diffusion has the property that it will e asored when it hits either m or l, y the Markov property. In this case we have x x m dy = ln σy x l and σ x = x m l, m l 5

17 and we can exploit Theorem 5. to get the infinitesimal symmetries parametrized y the six aritrary coefficients C,..., C 6 : v = ξ x + τ t + ϕ u = C [ t + x mx l m l ln x m x l x + t t x m 4 ln + t x m l ln x m t x l 4 m l x l 4 ] t u u [ x mx l +C ln x m m l x l x + t t + x m l ln x m 4 m l x l u u +C 3 t +C 4 [ t x mx l x + ln x m + t m l x l [ x mx l +C 5 x + m l +C 6 u u, ] x m l u u m l ] x m l u u m l together with the usual infinite dimensional symmetries generated yβu u. In order to otain fundamental solutions we need useful symmetries. For the fundamental solution on the line, we shall require a symmetry yielding a Fourier transform. To otain such a symmetry we exponentiate v 4. Theorem 6.. Consider the PDE with u t = σ xu xx, x D, 38 σx = x mx l m l and D = R. Then the PDE 38 has a symmetry of the form { U ϵ x, t = u f ϵ x, t, t exp ln x m x l where f ϵ : D R + R is defined y m l x m x l e ϵt f ϵ x, t =. x m x l e ϵt t ϵ + t } x m ϵ That is, for ϵ sufficiently small, U ϵ is a solution of 38 whenever u is. Proof. We take the symmetry v 4 which is given y v 4 = t x mx l x + m l ln x m + t x l 6 x m l u u, m l ] x l e ϵt x m, 39 x l

18 and we exponentiate the symmetry, that is we look for the solution to the following system: d x dϵ d t dϵ dũ dϵ = t x m x l, x0 = x; m l = 0, t0 = t; = ln x m x l + t The solution of this system is given y x m l ũ, ũ0 = u. m l x m = x m x l x l eϵt 40 t = t 4 { ũ = u exp ln x m x l t ϵ t } x m ϵ x l x m. 4 x l eϵt Replacing the new parameters gives the result. 6.. Otaining Fundamental solutions Fundamental solutions are not unique and with the methods we present here it is possile to exhiit multiple fundamental solutions for certain PDEs. This is discussed in depth in Craddock and Lennox 009. In this work we restrict ourselves to exhiiting fundamental solutions which are also positive and continuous proaility densities. We oserve that the Lie algeras for our PDEs are all six dimensional, so we should look for a Fourier rather than a Laplace transform. This follows from a result in Craddock and Dooley 00. In the case of a four dimensional Lie algera we look for a Laplace transform of a fundamental solution. To otain a Fourier transform we consider 39 and make the replacement ϵ iϵ and take u =. This gives At t = 0 we have U ϵ x, t = exp { ln x m x l t U ϵ x, 0 = exp iϵ ln iϵ } x m ϵ t x m x l. x l e iϵt x m. 43 x l Let us illustrate how we may find a fundamental solution with this solution. We suppose that x l, m. In this case ln x m m x x l = ln. x l m l We seek a fundamental solution p such that m y exp i ln ϵ pt, x, ydy = exp y l { 7 ln m x t iϵ t } x m x l ϵ x l e iϵt x m. x l

19 Oserve that taking ϵ = 0 we get m l pt, x, ydy = U 0 =, so that p integrates to one and hence is a proaility density. Putting z = ln solution such that m y y l Fourier inversion gives and noting that dy = ez m l e z + dz, we look for a fundamental e izϵ pt, x, z ez m l e z + dz = U ϵx, t. pt, x, z ez m l e z + = π e iϵy U ϵ x, tdϵ, and this is a straightforward Gaussian integral. Inverting and making the appropriate sustitution ack to the y variales gives the fundamental solution t ln m y y l m lx l m x x l pt, x, y = πty l y m + exp ln m y y l + t + 4 ln m x x l for the equation 38 with l < x < m. Oserve that for t > 0, lim y l,m pt, x, y = 0, so the apparent singularities at m, l are in fact zeroes. It is also clear that this fundamental solution is positive and continuous. Oserve also that a solution given y ux, t = m l fypt, x, ydy, with pt, x, y given y 44, will satisfy the oundary conditions um, t = ul, t = 0 which corresponds to the fact that these are asoring points for the diffusion. It is worth asking what happens if we take a different initial solution? There is no general rule. For some PDEs we otain another fundamental solution that is not a proaility density. In other cases we arrive ack at the same fundamental solution that our first choice of initial solution produces. This first case is discussed extensively with examples in Craddock and Lennox 009. The reader may check that taking the stationary solution u 0 x = x in this case actually returns the same fundamental solution. We can use the same symmetry to otain other fundamental solutions. We illustrate y otaining such a solution on m,. We do so y otaining a Fourier cosine transform. Taking the real part of 43 we see that V ϵ x, t = exp ϵ t [ x l cos ϵ gx t + m x cos ϵ gx + t ], m l 8t, 44 8

20 with gx = ln m x l x, is also a solution of 38. Oserve that V ϵ x, 0 = cos We look for a solution of 38 y setting m cos ϵ ln y m y l pt, x, ydy = V ϵ x, t. ln x m x l As efore taking ϵ = 0 gives m pt, x, ydy =. Now the sustitution z = ln y m y l produces the Fourier cosine transform cosϵzp t, x, mez l m le z e z e z dz = V ϵx, t. 0 We therefore have p t, x, mez l m le z e z e z = π pt, x, y = exp πt 0 cosϵzv ϵ x, tdϵ. Evaluating the inverse cosine transform and replacing with the original variales we arrive at the fundamental solution ln y m y l t + ln x m x l t. Kt, x, y 45 in which Kt, x, y = m lx l m x l x ln m y y l t x m ln y m y l t x l y ly m. It is not hard to see that lim y m pt, x, y = 0 and if ux, t = m fypt, x, ydy then um, t = 0. We may find other fundamental solutions for this case y these methods. For example we can otain a fundamental solution on, l. However we shall leave this to the interested reader. Also, one could exponentiate symmetries generated y other vector fields and get different fundamental solutions. In Appendix B we discuss this possiility y taking v. 6. Single Real Root We now consider the case where the polynomial admits one single root a of multiplicity. In this case we can write w.l.o.g. σx = x a, 46 9

21 corresponding to dx t = X t a dw t, X 0 R. If we impose the initial condition X 0 = a, then X t = a is the unique solution of this SDE y the Yamada-Watanae Theorem, see Kleaner 005. Thus y the Markov property, if X t reaches a, then it will remain at a. Of course one may condition the diffusion to e, say reflected left or right at a, ut we will not consider these cases here. So we have x σy dy = x a and σ x = x a, and we can exploit theorem 5. again to get the infinitesimal symmetries parametrized y the six aritrary coefficients C,..., C 6. If we exponentiate v 4 we have a symmetry that can e used to produce a fundamental solution. Theorem 6.3. Consider the PDE with u t = σ xu xx, x D, 47 σx = x a and D = {x R}. Then the PDE 47 has a symmetry of the form { U ϵ x, t = u f ϵ x, t, t exp x a ϵ + t } ϵ + x aϵt, 48 where f ϵ : D R + R is defined y f ϵ x, t = x a + x aϵt. That is, for ϵ sufficiently small, U ϵ is a solution of 47 whenever u is. Proof. We take the symmetry v 4 which is given y v 4 = tx a x + + tx a u u x a and we exponentiate the symmetry, that is we look for the solution to the following system: d x dϵ d t dϵ dũ dϵ = t x a, x0 = x; = 0, t0 = t; = x a + t x a ũ, ũ0 = u. 0

22 The solution of this system is given y x a = t = t; ũ = u exp Replacing the new parameters x, t gives x a x aϵt ; { x a ϵ t } ϵ x aϵt. t = t; x a = x a + x aϵ t ; x = f ϵ x, t, therefore the function ũ can e written as follows { ũ x, t = uf x, t, t exp x a ϵ + t } ϵ + x aϵ t. We relael the parameters and arrive at 48. Using the same type of Fourier transform argument as efore, we otain the fundamental solution. We let { ux, t = exp i x a ϵ } ϵ t ix aϵt, e a solution and we look for a fundamental solution such that { ϵ exp i pt, x, ydy = exp i y a x a ϵ } ϵ t ix aϵt. 49 We make the change of variales z = y a. Writing = a + a we see that under the + change of variales a 0 Hz dz and a 0 + Hz dz so that 49 ecomes { } e iϵz p t, x, az + z dz z = exp i x a ϵ ϵ t ix aϵt. Inverting the Fourier transform and converting ack to the original variales produces the fundamental solution x a pt, x, y = exp x y πty a 3 tx a y a, for the PDE u t = x a4 u xx. 50

23 Taking ϵ = 0 in 49 shows that this fundamental solution is also a proaility density, since the right side is equal to one at ϵ = 0 and the left is the integral of the fundamental solution. It is also not hard to check that taking the solution { u x, t = exp i x a ϵ } ϵ t x a, leads to the same fundamental solution. Oserve that a solution defined y x a ux, t = φy exp x y πty a 3 tx a y a dy, has the property ua, t = 0. So this density should correspond to a process which is asored at x = a, which is true of this process. One can also check that lim y a pt, x, y = 0. So we have a positive fundamental solution, which is a continuous proaility density. Now let us study the process restricted to a,. We do this y otaining a Fourier cosine transform. Taking the real part of 49 we see that U ξ x, t = exp ξ t ξ ξ cos ξtx a sin x a x a is a solution of 50. Oserve that U ξ x, 0 = cos ξ x a. We therefore look for a solution such that ξ cos pt, x, ydy = exp ξ t ξ ξ cos ξtx a sin. y a x a x a 0 a Taking ξ = 0 gives a pt, x, ydy =, so such a fundamental solution will e a proaility density. Setting z = y a gives cosξzpt, x, /z + a dz z = exp ξ t ξ ξ cos ξtx a sin. x a x a This is a Fourier cosine transform. We can then recover the fundamental solution y pt, x, /z + a/z = cosξz exp ξ t ξ ξ cos ξtx a sin dξ. π x a x a This leads to 0 pt, x, y = [ x a x + y a πt y a exp tx a y a exp tx a ty a. exp x y tx a y a A fundamental solution on, a can also e otained. We leave this to the interested reader. ]

24 6.3 Distinct Complex Roots We now consider the case where the polynomial admits two distinct complex roots a ± i and the process starts at X 0 R. In this case we can write without loss of generality x a σx = + corresponding to Xt a dx t = + dw t, X 0 D, with D = R. In this case we have x x a dy = arctan σy and σ x = x a, and we can exploit Theorem 6.3. again to get the infinitesimal symmetries parametrized y the six aritrary coefficients C,..., C 6. If we exponentiate v 4 we have a symmetry that leads to the Fourier transform of a fundamental solution, just as in the two distinct roots case. Theorem 6.4. Consider the PDE u t = σ xu xx, x D, 5 with x a σx = + and D = R. Then the PDE 5 has a symmetry of the form { U ϵ x, t = u f ϵ x, t, t exp arctan where f ϵ : D R + R is defined y f ϵ x, t = a + tan x a arctan ϵ + t } cos arctan x a ϵ ϵt cos arctan x a, 5 x a ϵt. That is, for ϵ sufficiently small, U ϵ is a solution of 5 whenever u is. Proof. We take the non trivial symmetry associated to v 4 which is given y x a x a v 4 = + t x + arctan + t x a u u 53 3

25 and we exponentiate the symmetry, that is we look for the solution to the following system: d x x a = + t, x0 = x; dϵ d t dϵ dũ dϵ = 0, t0 = t; x a = arctan + t x a ũ, ũ0 = u. The solution of this system is given y x a x a = tan arctan t = t; { x a ũ = u exp arctan Replacing the new parameters x, t gives + ϵt ; t = t; x a = tanarctan x a ϵt; x = f ϵ x, t, therefore the function ũ can e written as follows { x a ũ x, t =uf x, t, t exp arctan We relael the parameters and we arrive at 5. ϵ t } cos arctan x a ϵ cos arctan x a + ϵt. ϵ + t } cos arctan x a ϵ cos arctan x a 6.3. A Fourier Series Representation of the Fundamental Solution ϵ t. 54 This case is interesting in that we actually otain a Fourier series representation of our fundamental solution, which is of course the Fourier transform on the circle. As this is a situation that has not een investigated previously in the literature, we will present the details. From the previous symmetry we are ale to exhiit the solution ux, t = x a + e ϵ t iϵ arctan x a cosh ϵt + i arctan x a, 55 y taking our initial solution in equation 5 to e u = and making the replacement ϵ iϵ. We have x a ux, 0 = exp iϵ arctan. 56 4

26 We therefore seek a fundamental solution pt, x, y such that uy, 0pt, x, ydy = x a + e ϵ t iϵ arctan x a cosh ϵt + i arctan It is clear from the fact U 0 = that this fundamental solutions satisfies and so is a proaility density. We make the change of variales z = arctan y a and this ecomes π π e iϵz pt, x, a + tan zsec zdz = x a + e ϵ t iϵ arctan x a Putting ϵ = n, z z/ leads to the expression π π π cosh ϵt + i arctan x a. x a. 57 pt, x, ydy = e niz pt, x, a + tanz/sec z/dz = x a π + e n t ni arctan x a x a cosh nt + i arctan, which is the n-th Fourier coefficient of the kernel function. Fourier inversion then gives pt, x, a + tanz/ = cos z/ e niz x a π + e n t ni arctan x a n= x a cosh nt + i arctan. From this one easily otains a Fourier series for the fundamental solution pt, x, y = cos arctan y a y a ni π e arctan x a + e n t n= x a ni e arctan x a cosh nt + i arctan = πa + y e ni arctan y x +y ax a e n t n= [ cosh nt ia x sinhnt]. 7 Other tractale models It is possile to otain other models from this which are also tractale. In fact using our methods we can generate them quite easily. We consider the equation 4 σσ g = A σ + B σ σ. 58 5

27 Suppose we set A = B = 0 then integrate. This produces the nonlinear ODE 4 σσ 8 σ g = C, 59 where C is a constant of integration. For specific choices of g this equation can e solved. For example, gx = µ/x implies that σx = 4 µ/3 x. And one can perform the analysis we considered here in that case. However we can also reverse the process. If we specify σ in equation 58 in advance and determine g from this. For this choice of g and σ we can then compute appropriate fundamental solutions. Of course one might not have any specific financial applications in mind for every functional E other useful models may e found in this way. 8 Conclusions [ exp t 0 gx sds ], ut potentially In this paper we gave conditions on σ for the tractaility of a local volatility model and in the special case of the Quadratic Normal Volatility model we were ale to explicitly exponentiate the admitted Lie group of transformations to find the symmetries of the PDE and the corresponding fundamental solutions. By doing so we provided an analytical counterpart to the proailistic justification of the tractaility of this model given in Carr et al. 03. In the future, we aim at finding symmetries of more general models e.g. in the case where B is not zero and at inverting the resulting integral transforms to provide fundamental solutions for such models. A Infinitesimal generator for B 0 In this appendix we compute the infinitesimal generator of the Lie group admitted y the PDE 3 when the constant B in 33 is not null. A. Case B > 0 Theorem A.. If σ and g satisfy 33 with B > 0 then the PDE u t = σ xu xx gxu, x D admits a Lie symmetry group whose finite dimensional part has dimension 6. The corresponding Lie algera is generated y the following infinitesimal symmetries: [ v = σx B sin x Bt σy dy A sin ] [ Bt x + cos ] Bt t B [ + B cos x Bt σy dy + A cos x Bt σy dy B x sin Btσ x σy dy A sin Btσ x B ] B + sin Bt A B cos Bt u u ; 6

28 [ B x v = σx cos Bt [ + B sin x Bt σy dy B x + cos Btσ x v 3 = t ; + A B sin Bt A cos ] [ Bt x + sin ] Bt t B σy dy + + A sin Bt σy dy ] B cos Bt x σy dy A B cos Btσ x u u ; v 4 = σx cos Bt x [ B x + sin Bt σy dy + cos Btσ x + A sin ] Bt u u ; B v 5 = σx sin Bt x [ + B cos Bt v 6 = u u. x σy dy + sin Btσ x A cos ] Bt u u ; B In oth cases, there is an infinitesimal symmetry v β = β u, making the Lie algera infinitedimensional. Proof. If B > 0, the system for τ, ρ and η admits the following solution: τt = C cos Bt + C sin Bt + C 3 ; ρt = C 4 cos Bt + C 5 sin Bt + A C cos Bt B ηt = B C + A B C sin Bt B C A B C A +C 4 B sin A Bt C 5 B cos Bt + C 6 ; where C,..., C 6 are aritrary constants. From 7 we have [ B ξx, t = σx C sin Bt + C cos x Bt σy dy A C sin Bt; B cos Bt +C 4 cos Bt + C 5 sin Bt + A C cos Bt A C sin Bt B B ] 60 7

29 and from 9 we get αx, t = B C cos Bt + C sin x Bt σy dy + BC4 sin Bt BC 5 cos Bt + AC sin Bt + AC cos Bt x B σy dy + C sin Bt + C cos x Bt σ x σy dy + C 4 cos Bt + C 5 sin Bt + A C cos Bt A C sin Bt σ x B B B + C + A B C sin B Bt C A B C cos Bt A +C 4 B sin A Bt C 5 B cos Bt + C 6. Now taking the coefficients of the aritrary constants yields the result. A. Case B < 0 Theorem A.. If σ and g satisfy 33 with B < 0 then the PDE u t = σ xu xx gxu, x D admits a Lie symmetry group whose finite dimensional part has dimension 6. The corresponding Lie algera is generated y the following infinitesimal symmetries: [ v = σx B exp x Bt σy dy + A exp ] Bt x B [ + B exp x Bt σy dy + A exp x Bt σy dy B B + exp x Btσ x + A exp Bt B σy dy + ] u u + A B exp Btσ x [ exp ] Bt t ; 8

30 [ B x v = σx exp Bt [ + B exp x Bt σy dy B + exp x Btσ x B v 3 = t ; A exp Bt B A exp ] Bt x B σy dy A exp Bt σy dy ] u u + x σy dy A B exp Btσ x [ exp ] Bt t ; [ B exp Bt x σy dy + exp Btσ x v 4 = σx exp Bt x + v 5 = A exp ] Bt u u ; B σx exp [ Bt x + B exp x Bt σy dy + exp Btσ x + A exp ] Bt u u ; B v 6 = u u. Proof. If B < 0, the system for τ, ρ and η admits the following solution: τt = C exp Bt + C exp Bt + C 3 ; ρt = C 4 exp Bt + C 5 exp Bt + A C exp Bt A C exp Bt; B B ηt = B + A B A C 4 exp Bt + B C exp B Bt A B C 5 exp Bt; A C exp Bt B where C,..., C 6 are aritrary constants. From 7 we have [ ξx, t = σx B C exp Bt C exp x Bt σy dy +C 4 exp Bt + C 5 exp Bt + A C exp Bt A C exp ] Bt, B B 9

31 and from 9 we get αx, t = B C exp Bt + C exp x Bt σy dy BC4 + exp Bt BC 5 exp Bt +AC exp Bt AC exp x Bt σy dy B C exp Bt C exp x Bt σ x σy dy + C 4 exp Bt + C 5 exp Bt + A C exp Bt A C exp Bt σ x B B B + + A C exp B Bt B A C 4 exp Bt + A C 5 exp Bt. B B A C exp Bt B 0. B Now taking the coefficients of the aritrary constants yields the result for the case B < Another symmetry in the distinct real roots case In this appendix we analyze another symmetry for the PDE 38 starting from another vector field generating the Lie symmetry. Theorem B.. Consider the PDE 38 with D = {x R : 38 has a symmetry of the form U ϵ x, t = x m x l +4ϵt x m m l + 4ϵt where f ϵ : D R + R is defined y exp ϵ log x m x l + 4ϵt x > m > l}. Then the PDE + t u f ϵ x, t, t, + 4ϵt 6 f ϵ x, t = m l x m +4ϵt x l x m x l +4ϵt. That is, for ϵ sufficiently small, U ϵ is a solution of 38 whenever u is. 30

32 Proof. We take the vector field y x mx l v = 4t m l + ln x m x l x + 4t t t ln x m + t x m l ln x m t x l m l x l u u and we exponentiate the symmetry, that is we look for the solution to the following system: d x dϵ d t dϵ dũ dϵ x m x l = 4 t m l ln x m, x0 = x; 6 x l = 4 t, t0 = t; 63 = t ln x m + t x m l ln x m t ũ, ũ0 = u. 64 x l m l x l The calculations are straightforward, though somewhat tedious. From 63 we immediately get t t = 4ϵt, then 6 ecomes m ld x x m x l ln x m x l = 4t dϵ, x0 = x, 4ϵt which through the change of variale y = x m/ x l transforms into thus giving Then 64 ecomes dũ ũ = t dy y ln y x m x l 4ϵt x m 4tϵ ln x l = d ln 4ϵt, = x m 4ϵt. x l + t + 4tϵ x m 4ϵt x l x m x l 4ϵt ln x m x l with ũ0 = u. Integrating and relaeling the parameters completes the proof. t dϵ, 4tϵ The symmetry can e extended in a straightforward way to values of x to the left of m y taking asolute values. We can otain a fundamental solution from this symmetry. The style of argument is essentially the same as in for example Craddock and Lennox 007. If we look for a fundamental solution on m, using this symmetry then we are lead to a Laplace transform, which yields the fundamental solution 45. We also leave this to the interested reader. In fact we will otain the fundamental solution already otained for the case x > m, so we do not pursue it here. 3

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