Dyson series for the PDEs arising in Mathematical Finance I

Transcription

1 for the PDEs arising in Mathematical Finance I 1 1 Penn State University Mathematical Finance and Probability Seminar, Rutgers, April 12, This work was supported in part by NSF, Lie Manifolds, & PDEs

2 Collaborators Anna Mazzucato (PSU), Bernd Ammann (Regensburg), Wen Cheng (PSU), Radu Constantinescu (JP Morgan), Nick Costanzino (PSU), Alexandru Ionescu (Princeton), John Liechty (Statistics, PSU), Lie Manifolds, & PDEs

3 Main Goal (joint abstract) To obtain good numerical methods for parabolic (and related) equations Main application: Finance (generalizations of the Black-Scholes equation). Issues: time dependent equation (bad); parabolic (good, Green functions,, mostly Anna s talk); degenerate (bad,, my talk);, Lie Manifolds, & PDEs

4 Outline Motivation Background: Black-Scholes eqns. SABR Model More general models, Lie Manifolds, & PDEs

5 Introduction Many of the equations arising in applications (Finance, Structural Mechanics, Fluid dynamics,... ) have degeneracies or singularities. The classical theory of Partial Differential Equations (PDEs) on R n or on smooth bounded domains is not enough to deal with degeneracies and singularities. Exact calculations are almost never possible. The degeneracies and singularities create additional difficulties when applying standard Numerical Methods., Lie Manifolds, & PDEs

6 Abstract I will discuss some theoretical and practical issues that arise in the study of PDEs with degeneracies or singularities. We combine the analysis of the classical PDEs with geometry. Analysis on :-). Ultimate goal: applications to Numerical Methods. Our method is to use Green functions (or Heat Kernels). Questions related to Finance (including and Stochastic Calculus): in Anna s talk., Lie Manifolds, & PDEs

7 Background: Black-Scholes eqns. SABR Model Green Functions Let L be an elliptic differential operator (, Black-Scholes,.. ). Consider an equation of the form t u Lu = 0. parabolic. It is customary to denote u(t) = e tl u(0). The Green function Gt L (x, y) is then the distribution kernel of e tl. More precisely u(t, x) = Gt L (x, y)u(0, y)dy. Main technique: Approximate Gt L (x, y)., Lie Manifolds, & PDEs

8 Background: Black-Scholes eqns. SABR Model 1D test A good approximation of the Green function leads to good numerical solutions (10 times better, 10 times faster) vega.math.psu.edu(nistor)% g++ main1dintegral.cpp vega.math.psu.edu(nistor)%./a.out 1 Operation count= 580, error= , errormax= , , M= 1 2 Operation count= 3268, error= , errormax= , , M= 2 3 Operation count= 18372, error= , errormax= , , M= 5 4 Operation count= 77,764, error= e-06, errormax= e , e-05, M= 10 5 Operation count= , error= e-15, errormax= e , e-10, M= 21 6 Operation count= , error= e-15, errormax= e-15 4, , M= 42 7 Operation count= , error= e-14, errormax= e-15 4, , M= 85 8 Operation count= , error= e-14, errormax= e-15 4, , M= 170 vega.math.psu.edu(nistor)% g++ main1ddirect.cpp vega.math.psu.edu(nistor)%./a.out 1 Operation count= 20, error= , errormax= , Operation count= 240, error= , errormax= , Operation count= 2240, error= , errormax= , Operation count= 19200, error= , errormax= , Operation count= , error= , errormax= , Operation count= 1,290,240, error= e-05, errormax= e , Operation count= , error= e-05, errormax= e-06 8, Operation count= , error= e-06, errormax= e-06 8, 0.25., Lie Manifolds, & PDEs

9 Background: Black-Scholes eqns. SABR Model Green functions (conclusion) Good approximation of the Green function leads to fast numerical methods. Green functions (heat kernels) are difficult to compute due to local and global issues. Local issues: parabolic rescaling and (mostly Anna s talk). Global issues are related to degeneracies and singularities (this talk, but I will forget about Green functions)., Lie Manifolds, & PDEs

10 Background: Black-Scholes eqns. SABR Model (Simplest) The Black-Scholes equation (parabolic) t u ( σ 2 2 x 2 x 2 u + rx x u ru) = 0. Laplacian in polar coordinates (ρ, θ) in 2D u = ρ 2( ρ 2 ρu 2 + ρ ρ u + θ 2 u). Schrödinger operator (3D) ( + Z ρ )u = ρ 2( ρ 2 2 ρu + 2ρ ρ u + S 2u + Z ρu )., Lie Manifolds, & PDEs

11 Background: Black-Scholes eqns. SABR Model Simplest cont. Common to these examples: the appearance of x x and x 2 2 x = (x x ) 2 x x ; x never appears alone; elliptic, but not uniformly elliptic. The principal symbol σ(x, ξ) of a second order differential operator is elliptic if there exist C x > 0 σ(x, ξ) C x ξ 2, x, ξ It is uniformly elliptic if C x can be chosen independent of x. The principal symbol of (x x ) 2 is σ(x, ξ) = x 2 ξ 2., Lie Manifolds, & PDEs

12 Background: Black-Scholes eqns. SABR Model Treatment of the simplest examples The change of variables x = e y, y R transforms x x into y (Kondratiev transform). The Black-Scholes operator L BS := σ2 2 x 2 2 x + rx x r becomes L 0 := σ2 2 2 y + (r σ2 2 ) y r a uniformly elliptic, constant coefficient operator. The equation can be solved explicity. t u L 0 u = 0, Lie Manifolds, & PDEs

13 Background: Black-Scholes eqns. SABR Model SABR PDE The Black-Scholes model (and PDE) assumes the volatility σ to be constant. We need to allow σ to be non-constant and even vary randomly: stochastic volatility. : Heston, SABR (Hagan, Kumar, Lesniewski, Woodward). Slightly simplified SABR: t u σ2 v 2 ( x 2 x 2 u + 2ρx x v u + v 2 u ) = 0. 2 Kondratiev s transform x = e y gives t u σ2 v 2 ( 2 2 y u + 2ρ y v u + v 2 u 2 y u ) = 0., Lie Manifolds, & PDEs

14 Background: Black-Scholes eqns. SABR Model SABR PDE cont. Ignoring the lower order term 2 y u and setting ρ = 0, we obtain L 0 = σ2 v 2 ( 2 2 y + v 2 ), the Laplacian on the hyperbolic space v 2 (dv 2 + dy 2 ). Explicit formulas exist for e tl 0. (Notation: u(t) = e tl 0f solves t u = L 0 u, with the initial condition u(0) = f.) HKLW: to include the lower order term., Lie Manifolds, & PDEs

15 More general models Let L = L 0 + V. Iterating Duhamel s gives time-ordered (Dyson) expansion: e L 0+V = e L τ τ τd e (1 τ 1)L 0 Ve τ 1L 0 dτ 1 e (1 τ 1)L 0 Ve (τ 1 τ 2 )L 0 Ve τ 2L 0 dτ 2 dτ τd 1 0 e (1 τ 1)L 0 Ve (τ 1 τ 2 )L 0... e (τ d 1 τ d )L 0 Ve τ d L 0 dτ 0... dτ 1 e (1 τ 1)L 0 Ve (τ 1 τ 2 )L 0... e (τ d τ d+1 )L 0 Ve τ d+1l dτ d+1... dτ 1 Difficult to compute in general. What is the error if we ignore the last term? Answers in Anna s talk., Lie Manifolds, & PDEs

16 More general models Questions for more general models Practical question: to study similar, but more general models with several (many?) underlyings (spread options). Related theoretical question: How to deal with more general equations t u Lu = 0 Lu = a ij xi xj u + lower order terms? for the principal part, for the lower order part., Lie Manifolds, & PDEs

17 More general models and the Varadhan metric Lu = a ij xi xj u + lower order terms. Assume a ij = a ji (symmetric) and let [a ij ] = [a ij ] 1. Varadhan metric: g = ij aij dx i dx j. Length of a curve γ : [a, b] R n is then l(γ) = b a a ij γ i (t)γ j (t) dt d(x, y) = inf l(γ), for l(a) = x, l(b) = y can be recovered from e tl as t 0 (Varadhan)., Lie Manifolds, & PDEs

18 Metric and degeneracy For the Black-Scholes-Merton model, the Varadhan metric is g = (dx) 2 /x 2 (ignoring the volatility factor) d(a, b) = b a (dx) 2 x 2 = ln b ln a. (1) If we introduce y = ±d(x, 1) = ln x as a new variable, we recover the Kondratiev transform (x = e y ). An analogous transformation works for all one dimensional models., Lie Manifolds, & PDEs

19 Metric and degeneracy, cont. For the CEV model L CEV u = x 2β 2 x u + rx x u ru, β 0, the resulting metric will not be complete for β < 1. Different properties. The SABR model has a also the parameter β, the metric is again incomplete for β < 1. For β = 1 and after the transformation x = e y, the Varadhan metric for the SABR model is the hyperbolic metric., Lie Manifolds, & PDEs

20 Lie algebras Our examples have more structure than just the metric. Black-Scholes type (L BS = σ2 2 x 2 2 x +...): differential operators generated by x x on [0, ). SABR type: differential operators generated by X := vx x and V := v v on [0, ) 2. Also, [X, V ] = XV VX = vx x v v v v vx x = v 2 x x v v 2 x v x vx x = X. In other words, the linear span of X and V is closed under the Lie bracket [, ], and hence it is a Lie algebra., Lie Manifolds, & PDEs

21 : Motivation We shall consider differential operators generated by a Lie algebra of vector fields. No difference between 0 and in [0, ). Let everything act on [0, ]. More generally, we shall consider differential operators that act on spaces (manifolds) M of the form M = [0, ] k. Even more generally, it is enough for M to be locally of this form: manifold with corners. A rich theory is possible (Cordes, Melrose, Parenti, Schulze,... Kondratiev, Mazya,... ), Lie Manifolds, & PDEs

22 : Preliminaries To define we need the following: M is a compact manifold with corners (locally like [0, 1] n ). M denotes the interior of M: M = M faces. We denote by Γ(E) the space of smooth sections of E, so Γ(T M) is the space of smooth vector fields on M. V Γ(T M) and [V, V] V, Lie algebra of vector fields., Lie Manifolds, & PDEs

23 Definition of A Lie manifold is pair (M, V) consisting of a compact manifold with corners M and a subspace V Γ(T M) of vector fields that are tangent all faces of M and satisfy: V is closed under the Lie bracket [, ]; V is a projective C (M) module; the vector fields X 1,..., X n that locally generate V around an interior point p also give a local basis of T p M. Cordes, many interesting particular cases in Melrose s, Geometric scattering theory, Parenti, Schulze. Formalized by Ammann-Lauter-N. and A.-L.-N.-Vasy., Lie Manifolds, & PDEs

24 Cylindrical ends V = V b := the space of vector fields on M that are tangent to the boundary M. M = a manifold with smooth boundary M = {x = 0}. At the boundary M = {x = 0}, a local basis is given by x x, y2,..., yn. (y 2,..., y n are local coordinates on M.) There is no condition on these vector fields in the interior (valid for all ). Black-Scholes type operators fit into this framework. The Riemannian metric is that of a manifold with cylindrical ends., Lie Manifolds, & PDEs

25 Asymptotically Hyperbolic manifolds V = xγ(t M) = the space of vector fields on M that vanish on the smooth boundary M = {x = 0}. At the boundary M = {x = 0}, a local basis is given by x x, x y2,..., x yn. No condition in the interior (all ). The SABR model fits into this framework. Pseudodifferential calculus: Lauter, Mazzeo, Schulze., Lie Manifolds, & PDEs

26 Asymptotically Euclidean manifolds V = xv b = the space of vector fields on M that vanish on the smooth boundary M = {x = 0} and whose normal covariant derivative to the boundary also vanishes. At the boundary M = {x = 0}, a local basis is given by x 2 x, x y2,..., x yn. The resulting geometry for M = S n 1 is that of an asymptotically Euclidean manifold. Pseudodifferential calculus: Melrose, Parenti s, Schrohe., Lie Manifolds, & PDEs

27 Why? Let Diff(V) = the algebra of differential operators generated by the vector fields in V and C (M S ). The point of introducing is that one can prove harmonic analysis results for suitable operator in Diff(V): uniform ellipticity and regularity; mapping and embedding properties for function spaces; structure of resolvents; Fredholm conditions., Lie Manifolds, & PDEs

28 Metric The local basis X 1, X 2,..., X n V in the neighborhood of any given point of M will determine a metric g V on M up to Lipschitz equivalence. (M = interior of M) We obtain a natural L 2 (M) space. An elliptic operator P Diff m (V) will be called V-elliptic if the resulting Varadhan metric is Lipschitz equivalent to g V. The Laplacian associated to g V will be in Diff(V) and V-elliptic., Lie Manifolds, & PDEs

29 Functions spaces Sobolev spaces: K m (M) := {v, X 1 X 2... X k v L 2 (M), X 1,..., X k V, k m}. We have elliptic regularity: Theorem.[Ammann-Ionescu-N.] (i) K m (M) = H m (M, g V ). (ii) If P Diff m (V) is elliptic and v K r (M) for some r, then Pv K s m (M) v K s (M)., Lie Manifolds, & PDEs

30 Poisson s equation on polyhedra P a polyhedral domain in R n (curved). Let r be the distance to the singular points of the boundary. Sobolev spaces: K m a (P) := {v, r α a α v L 2 (P), α m}. We have well-posedness: Theorem.[Bacuta-Mazzucato-N.-Zikatanov] There exista η = η Ω > 0 such that : K m+1 a+1 (P) H1 0 Km 1 a 1 (P) is an isomorphism for all a < η and m 0. We allow interfaces and cracks., Lie Manifolds, & PDEs

31 Heat kernels: Summary Green functions give good approximations to solutions of parabolic equations. We want to obtain good formulas for the Green function (Heat kernel) of degenerate parabolic equations. Local estimates: Dyson s formula (Anna s talk). Global estimates (degeneracies): (program, in progress)., Lie Manifolds, & PDEs