The Uses of Differential Geometry in Finance

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1 The Uses of Differential Geometry in Finance Andrew Lesniewski Bloomberg, November The Uses of Differential Geometry in Finance p. 1

2 Overview Joint with P. Hagan and D. Woodward Motivation: Varadhan s theorem Differential geometry SABR model Geometry of no arbitrage The Uses of Differential Geometry in Finance p. 2

3 Varadhan s theorem This work has been motivated by the classical result of Varadhan. It relates the short time asymptotic of the Green s function of the backward Kolmogorov equation to the differential geometry of the state space. From the probabilistic point of view, the Green s function represents the transition probability of the diffusion, and it thus carries all the information about the process. Consequently, the geometry of the diffusion provides a natural book keeping device for calculations. The Uses of Differential Geometry in Finance p. 3

4 Varadhan s theorem Let W 1 t,...,w d t be an N-dimensional Borwnian motion with E[dW a t dw b t ] = ρ ab dt, where ρ is a constant correlation matrix. We consider a driftless, time homogeneous diffusion: dx j t = a σ j a(x t )dw a t, defined in an open subset of R N ( state space ). Define the following positive definite matrix: g ij (x) = ab ρ ab σ i a (x) σ j b (x). The Uses of Differential Geometry in Finance p. 4

5 Varadhan s theorem Let G T,X (t, x) denote the transition probability (or Green s function). It satisfies the following terminal value problem for the corresponding backward Kolmogorov equation: G T,X t (t, x) i,j g ij (x) G T,X (t, x) = 0, x i xj G T,X (T, x) = δ (x X). Substitution t T t transforms this into the initial value problem for the heat equation: G X t (t, x) = 1 2 g ij (x) i,j G X (0, x) = δ (x X). G X (t, x), x i xj The Uses of Differential Geometry in Finance p. 5

6 Varadhan s theorem Let M denote the state space of the diffusion. Varadhan s theorem states that lim t log G d (x, X)2 X (t, x) = t 0 2. Here d (x, X) is the geodesic distance on M with respect to a Riemannian metric given by the coefficients g ij (x) of the Kolmogorov equation. This gives us the leading order behavior of the Green s function: ( ) d (x, X)2 G X (t, x) exp. 2t The Uses of Differential Geometry in Finance p. 6

7 Varadhan s theorem To extract usable asymptotic information about the transition probability, more accurate analysis is necessary, but the choice of the Riemannian structure on M dictated by Varadhan s theorem turns out to be key. Indeed, that Riemannian geometry becomes an important tool in carrying out the calculations. Technically speaking, we are led to studying the asymptotic properties of the perturbed Laplace - Beltrami operator on a Riemannian manifold. The relevant techniques go by the names of the geometric optics or the WKB method. The Uses of Differential Geometry in Finance p. 7

8 Manifolds A smooth manifold is a set M along with an open covering {U α } and maps h α : U α R N such that all functions h β h 1 α : R N R N are infinitely differentiable. The maps h α are called local coordinate systems. Thus a manifold locally looks like the flat space. In the following, all our manifolds will admit one global system of coordinates. The Uses of Differential Geometry in Finance p. 8

9 Tangent bundle A tangent vector to a manifold M at a point x is a first order differential operator V (x) = V i (x) x i. i The vector space T x M of all tangent vectors at x is called the tangent space at x, the union T M = x T xm is called the tangent bundle. The Uses of Differential Geometry in Finance p. 9

10 Riemannian manifolds A Riemannian metric on a manifold is a symmetric, positive definite form ( inner product ) on the tangent bundle. The corresponding line element is ds 2 = ij g ij (x) dx i dx j. A manifold equipped with a Riemannian metric is called a Riemannian manifold. The Laplace-Beltrami operator g on a Riemannian manifold M is g f = where f C (M). 1 detg ij x i ( detg g ij f ), x j The Uses of Differential Geometry in Finance p. 10

11 Example: Poincare geometry The Poincare plane is the upper half plane H 2 = {(x, y) : y > 0} equipped with the line element ds 2 = dx2 + dy 2 y 2 This line element comes from the metric tensor given by h = 1 y For convenience, we introduce complex coordinates on H 2, z = x + iy; the defining condition then reads Imz > 0.. The Uses of Differential Geometry in Finance p. 11

12 Example: Poincare geometry By d (z, Z) we denote the geodesic distance between two points z, Z H 2, z = x + iy, Z = X + iy, i.e. the length of the shortest path connecting z and Z. There is an explicit expression for d (z, Z): coshd(z, Z) = 1 + z Z 2 2yY, where z Z denotes the Euclidean distance between z and Z. The Laplace-Beltrami operator on H 2 is given by: ( ) h = y 2 2 x y 2. The Uses of Differential Geometry in Finance p. 12

13 Heat equation on the Poincare plane The heat kernel on the Poincare plane satisfies: t G Z (t, z) = h G Z (t, z), G Z (0, z) = Y 2 δ (z Z). McKean s formula gives an explicit representation for G Z (s, z): G Z (t, z) = e t/4 2 (4πt) 3/2 d(z,z) ue u2 /4t coshu coshd(z, Z) du. The Uses of Differential Geometry in Finance p. 13

14 Asymptotic of McKean s formula Key for us is the following asymptotic expansion as t 0: G Z (t, z) = 1 4πs exp d sinhd ( d2 [ ) 4t ( ) d cothd 1 d t + O ( t 2)]. The Uses of Differential Geometry in Finance p. 14

15 Instantaneous covariance structure The instantaneous covariance structure of a diffusion dx i t = A i (X t, t)dt + a σ i a(x t, t)dw a t Quadratic variation of X < dx i, dx j > t = g ij (X t, t)dt, with E[dW a t dw b t ] = ρ ab dt, defines a time dependent Riemannian metric on M.: g ij (X t, t) = ab ρ ab σ i a(x t, t)σ i b(x t, t). The Uses of Differential Geometry in Finance p. 15

16 SABR model The dynamics of the forward rate F t is df t = Σ t C(F t )dw t, dσ t = ασ t dz t. Here Σ t is the stochastic volatility parameter, C(F) = F β, and W t and Z t are Brownian motions with E [dw t dz t ] = ρdt. We supplement the dynamics with the initial condition F 0 = f, Σ 0 = σ. The Uses of Differential Geometry in Finance p. 16

17 Special case: normal SABR model The normal SABR model is a special case in which β = 0, and ρ = 0: df t = Σ t dw t, dσ t = ασ t dz t. and E [dw t dz t ] = 0. Backward Kolmogorov s equation reads: G T,F,Σ t σ2 ( 2 G T,F,Σ f 2 + α 2 2 G T,F,Σ σ 2 G T,F,Σ (t, f, σ) = δ (f F, σ Σ), at t = T. ) = 0, The Uses of Differential Geometry in Finance p. 17

18 Special case: normal SABR model After the transformation t T t: G F,Σ t = 1 2 σ2 ( 2 G T,F,Σ f 2 + α 2 2 G F,Σ σ 2 G F,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

19 Normal SABR model and Poincare geometry Brownian motion on the Poincare plane is described by: dx t = Y t dw t, dy t = Y t dz t, with E [dw t dz t ] = 0. This is the normal SABR model if we make the following identifications: X t = F α 2 t, Y t = 1 α Σ α 2 t. The Uses of Differential Geometry in Finance p. 19

20 Geometry of the full SABR model The state space associated with the general SABR model has a somewhat more complicated geometry. Let S 2 denote the upper half plane {(x, y) : y > 0}, equipped with the following metric g: g = 1 1 ρ2 y 2 C (x) 2 1 ρc (x) ρc (x) C (x) 2 This metric is a generalization of the Poincare metric: the case of ρ = 0 and C (x) = 1 reduces to the Poincare metric.. The Uses of Differential Geometry in Finance p. 20

21 Geometry of the full SABR model The metric g is the pullback of the Poincare metric under the following diffeomorphism. We choose p 0, and define a map φ p : S 2 H 2 by φ p (z) = ( 1 1 ρ 2 ( x p ) ) du C (u) ρy, y. A consequence of this fact is that we have an explicit formula for the geodesic distance δ (z, Z) on S 2 : coshδ (z, Z) = coshd(φ p (z), φ p (Z)) = 1 + ( x X du C(u) ) 2 2ρ (y Y ) x X 2 (1 ρ 2 ) yy du C(u) + (y Y )2. The Uses of Differential Geometry in Finance p. 21

22 Geometry of the full SABR model We 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 the components of g 1 by g ij, and use g 1 and g to raise and lower the indices: z i = j g ijz j, i = j gij j. Explicitly, ( ) 1 = y 2 C (x) ρc (x) 2, 2 = y 2 (ρc (x) ). Consequently, the initial value problem can be written in the form: s K Z (s, z) = 1 2 ε i i i K Z (s, z), K Z (0, z) = δ (z Z). The Uses of Differential Geometry in Finance p. 22

23 Geometry of the full SABR model Except for the normal case, the operator i i i does not coincide with the Laplace-Beltrami operator g on S 2. One verifies that i i i f = g f 1 detg = g f ij 1 1 ρ 2 y2 CC f x. ( ) detg g ij f x j x i We treat the Laplace-Beltrami part by mapping it to the Laplace-Beltrami operator in the Poincare plane. The first order operator is treated as a regular perturbation of the Laplace-Beltrami operator. The Uses of Differential Geometry in Finance p. 23

24 Asymptotics of the Green s function Using the asymptotic expansion of McKean s formula we derive the following asymptotic expansion for the Green s function: 1 K Z (t, z) = 2πλ 1 ρ 2 Y 2 C (X) exp [ 1 δ ( 1 sinhδ q 8 + δ cothδ 1 8δ 2 ( δ2 2λ ) δ sinhδ 3 (1 δ cothδ) + δ2 8δ sinhδ ) ] q λ Here λ = tα 2, and yc (x) q (z, Z) = 2 (1 ρ 2 ) 3/2 Y ( x X ) du C (u) ρ (y Y ). The Uses of Differential Geometry in Finance p. 24

25 Probability distribution in the SABR model STEP 1. We integrate the asymptotic joint density over the terminal volatility variable Y to find the marginal density for the forward x: P X (t, x, y) = 0 K Z (t, z)dy 1 = 2πλ 1 ρ 2 C (X) 1 ( 8 λ 1 + δ cothδ 1 δ 2 e δ2 /2λ 0 δ sinhδ 3 (1 δ cothδ) + δ2 δ sinhδ Here again, δ (z, Z) denotes the geodesic distance on S 2. [ 1 δ q sinhδ q )] dy Y 2. The Uses of Differential Geometry in Finance p. 25

26 Probability distribution in the SABR model STEP 2. We evaluate this integral asymptotically by using the steepest descent method. Assume that φ (u) is positive and has a unique minimum u 0 in (0, ) with φ (u 0 ) > 0. Then, as ǫ 0, 0 f (u)e φ(u)/ǫ 2πǫ du = φ (u 0 ) e φ(u 0)/ǫ { [ f (u 0 ) f (u 0 ) + ǫ 2φ (u 0 ) φ(4) (u 0 ) f (u 0 ) 8φ (u 0 ) 2 ] f (u 0 )φ (3) (u 0 ) 2φ (u 0 ) 2 + 5φ(3) (u 0 ) 2 f (u 0 ) 24φ (u 0 ) 3 + O ( ǫ 2)}. The Uses of Differential Geometry in Finance p. 26

27 Probability distribution in the SABR model STEP 3. The exponent: φ (Y ) = 1 2 δ (z, Z)2, has a unique minimum at Y 0 given by where Y 0 = y ζ 2 2ρζ + 1, ζ = 1 y x X du C (u). Y 0 is the most likely value of Y, and thus Y 0 C (X) (when expressed in the original units) is the leading contribution to the observed implied volatility. The Uses of Differential Geometry in Finance p. 27

28 Probability distribution in the SABR model STEP 4. Let D (ζ) denote the value of δ (z, Z) with Y = Y 0. Explicitly, D (ζ) = log ζ2 2ρζ ζ ρ 1 ρ. We also introduce the notation: I (ζ) = ζ 2 2ρζ + 1 = coshd (ζ) ρ sinhd (ζ). The Uses of Differential Geometry in Finance p. 28

29 Probability distribution in the SABR model Then, to within O ( λ 2), we find that P X (t, x, y) = 1 } 1 exp { { D2 1 + yc (x) D 2πλ yc (X)I3/2 2λ 2 1 ρ 2 I [ 1 8 λ 1 + yc (x) D 2 1 ρ 2 I + 6ρyC (x) 1 ρ2 I cosh(d) 2 ( 3 ( 1 ρ 2 ) I + 3yC (x) ( 5 ρ 2) D 2 1 ρ 2 I 2 ) ] sinh(d) D +... }. The Uses of Differential Geometry in Finance p. 29

30 Probability distribution in the SABR model In terms of the original variables: P F (t, f, σ) = 1 } 1 exp { { D2 2πτ σc (F)I3/2 2tv σc (f) D 2v 1 ρ 2 I [ 1 8 τv2 1 + σc (f) D 2v 1 ρ 2 I + 6ρσC (f) v 1 ρ 2 I cosh(d) 2 ( 3 ( 1 ρ 2 ) I + 3σC (f) ( 5 ρ 2) D 2v 1 ρ 2 I 2 ) sinh(d) D ] +... }. The Uses of Differential Geometry in Finance p. 30

31 Line bundles over manifolds A line bundle over a manifold M is a manifold L together with a map π : L M such that: Each fiber π 1 (x) is isomorphic to R. Each x M has a neighborhood U M and a diffeomorphism π 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

32 Connections on line bundles A connection on a line bundle L M is a way to calculate derivatives of sections of L along tangent vectors. For a, b R, and φ, ψ Γ(L), For f C (M), and φ Γ (L), (aφ + bψ) = a φ + b ψ. (fφ) = df φ + f φ. For example, on a trivial bundle L = M R, all connections are of the form = d + ω, where ω is a 1-form on M. The Uses of Differential Geometry in Finance p. 32

33 Numeraire line bundle Each nonzero x M R N determines a direction in R N. Define a line bundle L by L = {(x, λx) : x M, λ R}, and π (x, λx) = x. Action of the group R + on L: Λ (x) φ (x), where Λ (x) is the ratio of two numeraires. The Uses of Differential Geometry in Finance p. 33

34 Holonomy and arbitrage For φ Γ(L) there is a 1-form ω such that (φ) = ωφ. If φ (x) = Λ(x)φ(x), then and so (φ ) = (Λφ) = (dλ + Λω)φ, ω = Λ 1 dλ + ω = d log Λ + ω. The absence of arbitrage is related to the flatness of ω, dω = 0. The Uses of Differential Geometry in Finance p. 34

35 LIBOR market model We are given N LIBOR forwards L 1 t,...,l N t, expiring at T j, with the dynamics dl j t = j (L t, t)dt + C j (L j t, t)dw j t, with E[dL i t dl j t] = ρ ij dt, Choose the drifts so that this dynamics is arbitrage free! The instantaneous Riemannian metric is given by g ij (L) = ρ ij C i (L i )C j (L j ). The Uses of Differential Geometry in Finance p. 35

36 LIBOR market model We let P j (L) denote the price of the zero coupon bond expiring at T j, P j (L) = 1 i j δ i L i, where δ i is the day count fraction. Therefore, if Λ(L) = P k(l) P j (L), the corresponding connection forms differ by d log j+1 i k δ i L i The Uses of Differential Geometry in Finance p. 36

37 LIBOR market model Therefore, and thus ω i = δ i 1 + δ i L i, if j + 1 i k, t δ i 1 + δ i L i, if k + 1 i j, t 0, otherwise. j = i g ji ω i = C j i ρ ji C i ω i. The Uses of Differential Geometry in Finance p. 37

38 LIBOR market model As a consequence, the dynamics of the LMM has the familiar form: Under Q k, dl j t = C j (L j t, t) ρ ji δ i C i (L i t, t) j+1 i k 1 + δ i L i dt + dw j t, if j < k, t dw j t, if j = k, ρ ji δ i C i (L i t, t) k+1 i j 1 + δ i L i dt + dw j t, if j > k. t The Uses of Differential Geometry in Finance p. 38

39 Extended LIBOR market model We consider an extension of the LIBOR market model with stochastic volatility parameters denoted by σ 1 t,...,σ N t : with dl j t = j (L t, σ t, t)dt + C j (L j t, σ j t,t)dw j t, dσ j t = Γ j (L t, σ t, t)dt + D j (L j t, σ j t,t)dz j t, E[dW i t dw j t ] = ρ ij dt, E[dW i t dz j t ] = r ij dt, E[dZ i t dz j t ] = π ij dt. Choose the drifts so that this dynamics is arbitrage free! The Uses of Differential Geometry in Finance p. 39

40 Extended LIBOR market model The state space of this model has dimension 2N and the instantaneous Riemannian metric is given by: g ij (L, σ) = ρ ij C i (L i, σ i )C j (L j, σ j ), if 1 i, j N, π ij C i (L i, σ i )D j (L j, σ j ), if 1 i N, N + 1 j 2N, π ij D i (L i, σ i )C j (L j, σ j ), if N + 1 i 2N, 1 j N, r ij D i (L i, σ i )D j (L j, σ j ), if N + 1 i, j 2N. Calculation of the connection form ω is identical to that of the LIBOR model. The Uses of Differential Geometry in Finance p. 40

41 Extended LIBOR market model Under Q k, dl j t = C j (L j t, σ j t,t) dσ j t = D j (L j t, σ j t, t) ρ ji δ i C i (L i t, σt, i t) j+1 i k 1 + δ i L i dt + dw j t, if j < k, t dw j t, if j = k, ρ ji δ i C i (L i t, σt, i t) k+1 i j 1 + δ i L i dt + dw j t, if j > k, t r ji δ i C i (L i t, σt, i t) j+1 i k 1 + δ i L i dt + dz j t, if j < k, t dz j t, if j = k, r ji δ i C i (L i t, σt, i t) k+1 i j 1 + δ i L i dt + dz j t, if j > k. t The Uses of Differential Geometry in Finance p. 41