The Uses Of Differential Geometry In Finance

The Uses of DifferentialGeometry in FinanceAndrew LesniewskiBloomberg, November 21 2005The Uses of Differential Geometry in Finance – p. 1

OverviewJoint with P. Hagan and D. WoodwardMotivation: Varadhan’s theoremDifferential geometrySABR modelGeometry of no arbitrageThe Uses of Differential Geometry in Finance – p. 2

Varadhan’s theoremThis work has been motivated by the classical result of Varadhan.It relates the short time asymptotic of the Green’s function of thebackward Kolmogorov equation to the differential geometry ofthe state space.From the probabilistic point of view, the Green’s functionrepresents the transition probability of the diffusion, and it thuscarries all the information about the process.Consequently, the geometry of the diffusion provides a naturalbook keeping device for calculations.The Uses of Differential Geometry in Finance – p. 3

Varadhan’s theoremLet Wt1 , . . . , Wtd be an N -dimensional Borwnian motion withE[dWta dWtb ] ρab dt,where ρ is a constant correlation matrix. We consider a driftless, timehomogeneous diffusion:dXtj Xσaj (Xt )dWta ,adefined in an open subset of RN (“state space”). Define the followingpositive definite matrix:ijg (x) Xρab σai (x) σbj (x) .abThe Uses of Differential Geometry in Finance – p. 4

Varadhan’s theoremLet GT,X (t, x) denote the transition probability (or Green’s function). Itsatisfies the following terminal value problem for the correspondingbackward Kolmogorov equation: GT,X1 X ij GT,X(t, x) g (x)(t, x) 0,ij t2 i,j x xGT,X (T, x) δ (x X) .Substitution t T t transforms this into the initial value problem forthe heat equation: GX1 X ij GX(t, x) g (x)(t, x) ,ij t2 i,j x xGX (0, x) δ (x X) .The Uses of Differential Geometry in Finance – p. 5

Varadhan’s theoremLet M denote the state space of the diffusion. Varadhan’s theoremstates that2d (x, X).lim t log GX (t, x) t 02Here d (x, X) is the geodesic distance on M with respect to aRiemannian metric given by the coefficients g ij (x) of the Kolmogorovequation. This gives us the leading order behavior of the Green’sfunction:!2d (x, X)GX (t, x) exp .2tThe Uses of Differential Geometry in Finance – p. 6

Varadhan’s theoremTo extract usable asymptotic information about the transitionprobability, more accurate analysis is necessary, but the choice of theRiemannian structure on M dictated by Varadhan’s theorem turns outto be key. Indeed, that Riemannian geometry becomes an importanttool in carrying out the calculations. Technically speaking, we are led tostudying the asymptotic properties of the perturbed Laplace - Beltramioperator on a Riemannian manifold. The relevant techniques go by thenames of the geometric optics or the WKB method.The Uses of Differential Geometry in Finance – p. 7

ManifoldsA smooth manifold is a set M along with an open covering {Uα } andNNmaps hα : Uα RN such that all functions hβ h 1α : R R areinfinitely differentiable. The maps hα are called local coordinatesystems. Thus a manifold locally looks like the flat space. In thefollowing, all our manifolds will admit one global system of coordinates.The Uses of Differential Geometry in Finance – p. 8

Tangent bundleA tangent vector to a manifold M at a point x is a first order differentialoperatorX .V (x) Vi (x)i xiThe vector space Tx M of all tangent vectors at x is called the tangentSspace at x, the union T M x Tx M is called the tangent bundle.The Uses of Differential Geometry in Finance – p. 9

Riemannian manifoldsA Riemannian metric on a manifold is a symmetric, positive definiteform (“inner product”) on the tangent bundle. The corresponding lineelement isX2gij (x) dxi dxj .ds ijA manifold equipped with a Riemannian metric is called a Riemannianmanifold.The Laplace-Beltrami operator g on a Riemannian manifold M is Xp1 ij f g f det g g, xjdet g ij xiwhere f C (M).The Uses of Differential Geometry in Finance – p. 10

Example: Poincare geometryThe Poincare plane is the upper half plane H2 {(x, y) : y 0}equipped with the line element22dx dyds2 y2This line element comes from the metric tensor given by 1 1 0 h 2.y0 1For convenience, we introduce complex coordinates on H2 , z x iy;the defining condition then reads Imz 0.The Uses of Differential Geometry in Finance – p. 11

Example: Poincare geometryBy d (z, Z) we denote the geodesic distance between two pointsz, Z H2 , z x iy, Z X iY , i.e. the length of the shortest pathconnecting z and Z. There is an explicit expression for d (z, Z): z Z 2,cosh d (z, Z) 1 2yYwhere z Z denotes the Euclidean distance between z and Z.The Laplace-Beltrami operator on H2 is given by: h y2 22 2 x2 y .The Uses of Differential Geometry in Finance – p. 12

Heat equation on the Poincare planeThe heat kernel on the Poincare plane satisfies: GZ (t, z) h GZ (t, z) , tGZ (0, z) Y 2 δ (z Z) .McKean’s formula gives an explicit representation for GZ (s, z): t/4GZ (t, z) e Z2(4πt)3/2 d(z,Z)2ue u/4tpdu.cosh u cosh d (z, Z)The Uses of Differential Geometry in Finance – p. 13

Asymptotic of McKean’s formulaKey for us is the following asymptotic expansion as t 0: 2 1dGZ (t, z) exp 4πs4tr d1 d coth d 121 1t Ot.2sinh d4dThe Uses of Differential Geometry in Finance – p. 14

Instantaneous covariance structureThe instantaneous covariance structure of a diffusiondXtii A (Xt , t)dt Xσai (Xt , t)dWtaaQuadratic variation of X dX i , dX j t g ij (Xt , t)dt,withE[dWta dWtb ] ρab dt,defines a time dependent Riemannian metric on M.:ijg (Xt , t) Xρab σai (Xt , t)σbi (Xt , t).abThe Uses of Differential Geometry in Finance – p. 15

SABR modelThe dynamics of the forward rate Ft isdFt Σt C(Ft )dWt ,dΣt αΣt dZt .Here Σt is the stochastic volatility parameter,C(F ) F β ,and Wt and Zt are Brownian motions withE [dWt dZt ] ρdt.We supplement the dynamics with the initial condition F0 f, Σ0 σ.The Uses of Differential Geometry in Finance – p. 16

Special case: normal SABR modelThe normal SABR model is a special case in which β 0, and ρ 0:dFt Σt dWt ,dΣt αΣt dZt .andE [dWt dZt ] 0.Backward Kolmogorov’s equation reads: 22 GT,F,Σ1 GT,F,Σ2 GT,F,Σ σ2 α 0,22 t2 f σGT,F,Σ (t, f, σ) δ (f F, σ Σ) , at t T.The Uses of Differential Geometry in Finance – p. 17

Special case: normal SABR modelAfter the transformation t T t: 2 2 GF,Σ1 GT,F,Σ2 GF,Σ σ2 α,22 t2 f σGF,Σ (0, f, σ) δ (f F, σ Σ) .This resembles the heat equation on the Poincare plane!The Uses of Differential Geometry in Finance – p. 18

Normal SABR model and Poincare geometryBrownian motion on the Poincare plane is described by:dXt Yt dWt ,dYt Yt dZt ,withE [dWt dZt ] 0.This is the normal SABR model if we make the following identifications:X t Fα 2 t ,Yt 1Σα2 t .αThe Uses of Differential Geometry in Finance – p. 19

Geometry of the full SABR modelThe state space associated with the general SABR model has asomewhat more complicated geometry. Let S2 denote the upper halfplane {(x, y) : y 0} , equipped with the following metric g:1g p21 ρ2 y 2 C (x) 1 ρC (x) ρC (x)C (x)2 .This metric is a generalization of the Poincare metric: the case of ρ 0and C (x) 1 reduces to the Poincare metric.The Uses of Differential Geometry in Finance – p. 20

Geometry of the full SABR modelThe metric g is the pullback of the Poincare metric under the followingdiffeomorphism. We choose p 0, and define a map φp : S2 H2 byφp (z) 1p1 ρ2 Zxp !du ρy , y .C (u)A consequence of this fact is that we have an explicit formula for thegeodesic distance δ (z, Z) on S2 :cosh δ (z, Z) cosh d (φp (z) , φp (Z)) R 2R x dux du2 2ρ(y Y) (y Y)X C(u)X C(u). 1 2 (1 ρ2 ) yYThe Uses of Differential Geometry in Finance – p. 21

Geometry of the full SABR modelWe use invariant notation. Let z 1 x, z 2 y, and let i / z i ,i 1, 2, denote the corresponding partial derivatives. We denote thecomponents of g 1 by g ij , and use g 1 and g to raise and lower theP ijPjiindices: zi j gij z , j g j . Explicitly, 2 1 y 2 C (x) 1 ρC (x) 2 , 2 y 2 (ρC (x) 1 2 ) .Consequently, the initial value problem can be written in the form: 1 X iKZ (s, z) ε i KZ (s, z) , s2iKZ (0, z) δ (z Z) .The Uses of Differential Geometry in Finance – p. 22

Geometry of the full SABR modelExcept for the normal case, the operator i i i does not coincidewith the Laplace-Beltrami operator g on S2 . One verifies thatPXi fX p1ijdet g g i f g f xidet g ij xji g f p11 ρ2y 2 CC ′ f. xWe treat the Laplace-Beltrami part by mapping it to theLaplace-Beltrami operator in the Poincare plane. The first orderoperator is treated as a regular perturbation of the Laplace-Beltramioperator.The Uses of Differential Geometry in Finance – p. 23

Asymptotics of the Green’s functionUsing the asymptotic expansion of McKean’s formula we derive thefollowing asymptotic expansion for the Green’s function:2 rδ1δpexp 2λsinh δ2πλ 1 ρ2 Y 2 C (X) 2δ1 δ coth δ 1 3 (1 δ coth δ) δ1 q q λ . .2sinh δ88δ8δ sinh δKZ (t, z) Here λ tα2 , and′q (z, Z) yC (x)2 (1 ρ2 )3/2Y ZxX du ρ (y Y ) .C (u)The Uses of Differential Geometry in Finance – p. 24

Probability distribution in the SABR modelSTEP 1. We integrate the asymptotic joint density over the terminalvolatility variable Y to find the marginal density for the forward x:PX (t, x, y) Z KZ (t, z) dY01 r δ 2 /2λδsinh δ δpe 1 q2sinhδ2πλ 1 ρ C (X) 0 21δ coth δ 1 3 (1 δ coth δ) δdY q. λ 1 8δ2δ sinh δY2ZHere again, δ (z, Z) denotes the geodesic distance on S2 .The Uses of Differential Geometry in Finance – p. 25

Probability distribution in the SABR modelSTEP 2. We evaluate this integral asymptotically by using the steepestdescent method. Assume that φ (u) is positive and has a uniqueminimum u0 in (0, ) with φ′′ (u0 ) 0. Then, as ǫ 0,Z φ(u)/ǫf (u)edu 0(s2πǫ φ(u0 )/ǫe φ′′ (u0 )"f ′′ (u0 )φ(4) (u0 ) f (u0 )f (u0 ) ǫ ′′2φ (u0 )8φ′′ (u0 )2#)2 f ′ (u0 ) φ(3) (u0 ) 5φ(3) (u0 ) f (u0 )2 O ǫ.23′′′′2φ (u0 )24φ (u0 )The Uses of Differential Geometry in Finance – p. 26

Probability distribution in the SABR modelSTEP 3. The exponent:φ (Y ) 12δ (z, Z) ,2has a unique minimum at Y0 given byY0 ywherep1ζ yζ 2 2ρζ 1 ,ZxXdu.C (u)Y0 is the “most likely value” of Y , and thus Y0 C (X) (when expressed inthe original units) is the leading contribution to the observed impliedvolatility.The Uses of Differential Geometry in Finance – p. 27

Probability distribution in the SABR modelSTEP 4. Let D (ζ) denote the value of δ (z, Z) with Y Y0 . Explicitly,pζ 2 2ρζ 1 ζ ρ.D (ζ) log1 ρWe also introduce the notation:pI (ζ) ζ 2 2ρζ 1 cosh D (ζ) ρ sinh D (ζ) .The Uses of Differential Geometry in Finance – p. 28

Probability distribution in the SABR model2 Then, to within O λ , we find that (2DyC ′ (x) Dexp 1 p2λ2 1 ρ2 I112πλ yC (X) I 3/2"1yC ′ (x) D6ρyC ′ (x) λ 1 p pcosh (D)22282 1 ρ I1 ρ I#) ! ′223yC (x) 5 ρ D sinh (D)3 1 ρp . . 22ID2 1 ρ IPX (t, x, y) The Uses of Differential Geometry in Finance – p. 29

Probability distribution in the SABR modelIn terms of the original variables:2 (D11σC ′ (f ) Dexp 2PF (t, f, σ) 1 p3/22tv2πτ σC (F ) I2v 1 ρ2 I"1 26ρσC ′ (f )σC ′ (f ) D τv 1 p pcosh (D)22282v 1 ρ Iv 1 ρ I#) !2′23 1 ρ3σC (f ) 5 ρ D sinh (D)p . .22ID2v 1 ρ I The Uses of Differential Geometry in Finance – p. 30

Line bundles over manifoldsA line bundle over a manifold M is a manifold L together with a mapπ : L M such that:Each fiber π 1 (x) is isomorphic to R.Each x M has a neighborhood U M and a diffeomorphismπ 1 (U ) U R.The simplest example of a line bundle is the trivial line bundle,L M R. A general line bundle looks locally like a trivial line bundle.A smooth map φ : M L is called a section of L if π φ Identity.The set of all sections is denoted by Γ(L).The Uses of Differential Geometry in Finance – p. 31