Implied Volatility, Fundamental solutions, asymptotic analysis and - - PowerPoint PPT Presentation
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop, September 11, 2008 Peter Laurence Dipartimento di Matematica e Facolt a di Statistica, Uni. Roma 1 Implied Volatility,
Dipartimento di Matematica e Facolt´ a di Statistica, Uni. Roma 1
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 1/4
Rapid review Implied Volatility, local volatility, mimicking behaviour. Practitioners like closed form formulas for calibration. Review Heat Kernel approach to solving stochastic volatility models. Hagan-Lesniewski, Henry-Labordère. Refined asymptotics for a class of generalized SABR like models. Joint work with Gérard Ben Arous and Tai-Ho Wang. Influence of curvature. Interaction between symmetry and heat kernel approach. Distance function: Work in progress with Matveev and Wang.
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 2/4
The goal, from PDE point of view, is to solve parabolic equations in one or several spatial dimensions. On [0, T], where T is the maturity of European option, solve: ut + aij(x)uxixj + biuxi − ru = 0, x ∈ Ω ⊂ Rn, t ∈ [0, T] u(x, T) = ψ(x) final condition The matrix {aij} is usually degenerate, so the operator above is often not uniformly parabolic. Researchers in PDE are used to seeing the equation expressed as initial value (rather than final value) problem. This can be achieved, by making the change of variables: τ = T − t. The problem then reads as: uτ − aij(x)uxixj − biuxi + ru = 0, x ∈ Ω ⊂ Rn, t ∈ [0, T] u(x, 0) = ψ(x) initial condition
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 3/4
We are interested in finding a fundamental solution of such parabolic equations: F(x, t, ξ, T), x, ξ ∈ Ω, t ∈ [0, T] Often Ω is Rn or Rn
+.
Here F satisfies the parabolic equation in the variables x and t and has a delta function final condition: F(x, T, ξ, T) = δξ(x)
barrier options, and in this case, we seek the Green’s function, rather than the Fundamental
the Neumann heat kernel.
principal part at the boundary (Stochastic CEV models) or in the entire domain (Asian Options).
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 4/4
Quote from article by Mike Giles and Ronnie Sircar in Siam NEWS, October 2007: “ The major challenges in computational finance arise not from difficult geometries, as in many physical problems, but from the need for rapid calculation of an EXPECTATION or the solution of its associated Kolmogorov partial differential equation.” “ Efficiency is at the forefront, because models are re-estimated as new market data arrives and calibration (or “marking to market”) embeds the expectation/PDE calculation in an iterative solution to an inverse problem”’ Interpretation of last sentence: from traded market prices back out parameters in coefficients of parabolic operator and/or back out the functional form of these coefficients.
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 5/4
Closed Form or quasi-closed form solutions: A Mathematician’s Toolbox Recent years have seen surge in attempts to find closed form or quasi-closed form solutions to certain parabolic problems arising in option pricing. In today’s talk, we will concentrate on one of these,asymptotics. The other approach, which we will not discuss in detail, for lack of time,is:
Changes of Variables, Transformation Groups and Lie Symmetry Analysis. Literature (Partial List): Albanese and Kusnetsov: Reducing time homogeneous one dimensional diffusions to standard form and solving via special functions. Transformations of Markov Processes and Classification Scheme for Solvable Driftless
Carr, Laurence and Wang: Reducing time inhomogeneous diffusions to standard form. Via Lie symmetry considerations. Comptes Rendus de l’Académie des Sciences, 2006. Linetsky: Time homogeneous one dimensional diffusions. Approach via eigenfunction
Ait-Sahalia: Annals of Statistics, 2007 “Closed-Form Likelihood Expansions for Multivariate Diffusions”. Reduction method to heat equation with lower order terms.(related to Lie).
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 6/4
and Woodward, Andreasen-Andersen), with Ft forward price. dFt = F β
t ytdW1t
dyt = αytdW2t < dW1t, dW2t >= ρdt Calibrates to smile, but for only one maturity.
dFt = γ(t) C(Ft)ytdW1t dyt = ν(t) ytdW2t < dW1t, dW2t >= ρ(t)dt, time dependent parameters. Calibratable to implied volatility surface for several maturities.
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 7/4
λ-Sabr model (incorporating mean reversion dFt = νC(Ft)ytdW1t dyt = κ(θ − yt)dt + γytdW2t < dW1t, dW2t >= ρdt Introduced by Henry-Labordère (2005). dFt = C(Ft)yδ
t dW1t
dyt = κ(θ − yt)dt + yδ
t dW2t
< dW1t, dW2t >= ρdt The homogeneous"delta-geometry" introduced by Bourgade and Croissant (2005).
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 8/4
dFt = γ(t)C(Ft)yδ
t dW1t
dyt = κ(θ − yt)dt + w(yt) dW2t < dW1t, dW2t >= ρdt Generalized class of SABR models introduced by G. Ben Arous, P . Laurence, TH Wang (2008): Try to understand how the asymptotics depends on function w(y). Also, refine and generalize existing asymptotics. Osajima (2007): Approach to SABR models using Malliavin calculus. Easily quantifiable results only for original (lognormal) SABR models.
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 9/4
Three (loosely speaking) different approaches:
Key idea: Bypass fundamental solution and aim directly for the call option prices, using that at time zero,from Tanaka’s formula one can derive (Dupire 1996) Call( ¯ F, T) = ( ¯ F − K)+ + C2( ¯ F) T E
y2
T δ ¯ F (FT ) | F0 = ˆ
F; α0 = α
F − K)+ + C2( ¯ F) T
y2p( ˆ F, ˆ y, ¯ F, ¯ y, T)d¯ y
:= ( ¯ F − K)+ + C2( ¯ F) T
=
F, ˆ y, ¯ F, T) Function P above, in the backward variables, satisfies a backward Kolmogorov equation, final condition P(f, y, T) = y2δ ¯
F (f)
PT + 1 2 C2(f)y2Pff + C(f)αρyPαf + α2y2Pyy + drift terms = 0 subject to the final condition
2
delta function one variable.
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 10/4
Geometric/Analytic approach This approach was introduced by Lesniewski in a lecture at the Courant Institute. Uses the hyperbolic plane. Generalized by several papers: Introduction of McKean heat kernel (see page 50), ie. fundamental solution of heat equation in hyperbolic plane, plays a central role. Henry-Labordere (2005) introduces the use of the heat kernel expansion and the λ Sabr model Geometric/Stochastic approach Ground breaking work by Varadhan (1965) and then by Molchanov (1975)and Azencott (1981), Ben Arous (1989). This is followed by the work by Bourgade and Croissant (2005) who apply Molchanov’ s results to stochastic volatility models.
work of Bismut, Kusuoka, Malliavin, due to Takahashi (1999-2008) at al.(also Fournier-Lions et al. for calculation of Greeks) and, more recently, by Osajima (2007, 2008), based on work
Fourier Transform based methods.
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 11/4
Local volatility models: dSt = Stσ(St, t)dt + rSdt Dupire’s formula: From traded option prices to parametric form of σ(F, t). σ2 loc(S, t, K, T) =
∂C ∂T + rK ∂C ∂K 1 2 K2 ∂2C ∂K2
From local volatility to implied volatility and vice-versa (fully non-linear PDE) σ(log(
x
K ), T) = 2TI ∂I
∂T + I2
(1 − x
∂I ∂x
I )2 + TI ∂2I ∂x2 − 1 4 T 2I2 ∂2I ∂x2
, (take r =0) Approximate relationship ( Berestycki-Busca-Florent, QF 2002) as τ = T − t → 0: lim
τ→0 I(log( F
K ), τ) = 1 1
1 σ(s log( F
K ),0)ds
, uniformly as τ → 0. Ie. Implied volatility is the harmonic mean of local volatility, in small time limit.
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 12/4
loc(K, T) = b2(K)E[α2 T | FT = K]
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 13/4
One can actually show a more general result Gyongi, giving rise to the concept of
dSt = c(St, νt, t)dt + b(St, t)g(ν(t), t)dW1t dνt = ζ(νt)dt + β(νt)dW2tdt < dW1t, dW2t >= ρdt S(0) = S, ν(0) = ν, yields the same marginal distributions with respect to the S variable as the following sde: dSt = σ(St, t)d ¯ Wt + γ(S, t)dt, S(0) = S where, the effective parameters are: σ2(K, T) = b(K, T)E
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 14/4
Proof: Dupire and Derman and Kani
Following is heuristic. See Klebaner for rigour. From Breeden-Litzenberger (assume r = 0, for simplicity). ∂2C(F, t, K, T) ∂K2 = E[δ(FT − K)] d(Ft − K)+ = 1[K,+∞)(Ft)dFt + 1 2 α2
t b2(Ft)δ(Ft − K)dt
(FT − K)+ = T 1[K,+∞)(Ft)dFt + 1 2 T α2
t b2(Ft)δ(Ft − K)dt
= T 1[K,+∞)(Ft)dFt + b2(K) 1 2 T α2
t δ(Ft − K)dt
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 15/4
Taking expectations: C(K, T) = 1 2b2(K)E T α2
t δ(Ft − K)dt
∂C(K, T) ∂T = 1 2b2(K)E
T & FT = K
∂T = 1 2b2(K)E
T | FT = K
∂C ∂T = 1 2 b2(K)E
T | FT = K
∂2C ∂K2 , to conclude that →
∂C ∂T ∂2C ∂K2
= 1 2 b2(K)E
T | FT = K
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 16/4
PDE view: From stochastic volatility model to local volatility Consider a stochastic volatility model: dFt = b(Ft)FtytdW1t, dyt = ytc(yt)dW2t, F(0) = ¯ F, y(0) = ¯ y Recall Gyongi formula: There exist a local volatility model (ie. a model with one less state variable) dFt = σ(Ft, t)FtdW1t which has the same marginals with respect to the Ft process, given by σ2(F, t) = E
t y2 | Ft = F, F0 = ¯
F, y0 = ¯ y
F, ¯ y, F, y, t) be the corresponding fundamental solution: Then in PDE language we have (σ2)
¯ F ,¯ y(F, T) = F 2b2(F)
F, ¯ y, F, y, T)dy
F, ¯ y, F, y, T)dy So, if we knew the Fundamental solution in closed form or in quasi-closed form, for small t, we could recover the asymptotic value of the local volatilityand then the implied volatility, as a function
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 17/4
Two approaches at this stage to obtaining the call option prices In the RHS of formula: P( ¯ F, ¯ y, t) =
F, ¯ y, F, y, T)dy insert an expansion F = F0 + tF1 + . . . valid asymptotically for small σ2T, into the above formula. Approach II, mentioned earlier is to look directly for a solution of the equation satisfied by P in the backward variables: PT + 1 2C2( ¯ F)y2P ¯
F ¯ F + C( ¯
F)αρyPy ¯
F + α2y2Pyy + drift terms = 0
subject to the final condition P( ¯ F, y, T) = y2δK( ¯ F) delta function one variable and, again, seek an expansion: P = P0 + tP1 + t2P2 . . .
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 18/4
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 19/4
Small Time limit for parabolic problems: Where did it really begin?
infinitesimal generator: aij(x)pxixj , x ∈ Rn and the time homogeneous diffusion (Heat flow) on Rn. Hp = pt − Lxp = 0
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 20/4
Small Time limit for parabolic problems: Where did it really begin?
infinitesimal generator: aij(x)pxixj , x ∈ Rn and the time homogeneous diffusion (Heat flow) on Rn. Hp = pt − Lxp = 0
equation is due to Varadhan: lim
t→0 4t log(pt) = −d2(x, y),
holds uniformly for x, y in compact subsets of Rn.
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 20/4
Small Time limit for parabolic problems: Where did it really begin?
infinitesimal generator: aij(x)pxixj , x ∈ Rn and the time homogeneous diffusion (Heat flow) on Rn. Hp = pt − Lxp = 0
equation is due to Varadhan: lim
t→0 4t log(pt) = −d2(x, y),
holds uniformly for x, y in compact subsets of Rn. d(x, y) is the Riemannian distance, associated to {gij}, inverse of {aij}, ds2 = gijdsidsj.
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 20/4
{gij} the inverse of diffusion matrix {aij}, the Riemannian distance d(x, y) is defined by: d(x, y) = inf
¯ z(·):z(0)=x,z(1)=y
1 gij ˙ zi ˙ zjdt
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 21/4
{gij} the inverse of diffusion matrix {aij}, the Riemannian distance d(x, y) is defined by: d(x, y) = inf
¯ z(·):z(0)=x,z(1)=y
1 gij ˙ zi ˙ zjdt Γ(x, y), the square of the Riemannian Distance satisfies the Hamilton-Jacobi equation aijΓxiΓxj = 4Γ Inside a so-called “normal neighborhood” (Milnor (1969)) around a given point x0 the solution is C∞.
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 21/4
{gij} the inverse of diffusion matrix {aij}, the Riemannian distance d(x, y) is defined by: d(x, y) = inf
¯ z(·):z(0)=x,z(1)=y
1 gij ˙ zi ˙ zjdt Γ(x, y), the square of the Riemannian Distance satisfies the Hamilton-Jacobi equation aijΓxiΓxj = 4Γ Inside a so-called “normal neighborhood” (Milnor (1969)) around a given point x0 the solution is C∞. A notion of solution in the large requires viscosity solutions framework.
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 21/4
{gij} the inverse of diffusion matrix {aij}, the Riemannian distance d(x, y) is defined by: d(x, y) = inf
¯ z(·):z(0)=x,z(1)=y
1 gij ˙ zi ˙ zjdt Γ(x, y), the square of the Riemannian Distance satisfies the Hamilton-Jacobi equation aijΓxiΓxj = 4Γ Inside a so-called “normal neighborhood” (Milnor (1969)) around a given point x0 the solution is C∞. A notion of solution in the large requires viscosity solutions framework. To get intuition concerning Varadhan’s theorem, suppose that we had an analogue of fundamental solution for Euclidean heat equation, then we would have pτ ∼ 1 (4πτ)n/2 e− d2(x,y)
4τ
, τ → 0 Heston model, original SABR model have negative curvature, so don’t need to worry about normal neighborhood and cut-locus.
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 21/4
4τ
τ→0 4τ log(pτ) = −d2(x, y),
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 22/4
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 23/4
Hadamard Portraits http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Hadamard.html
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Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 25/4
Local Volatility Models dFt = Ftσ(Ft) , ut − 1
2 F 2σ2(F)uF F = 0.
d(F1, F2) = F2
F1
1 Fσ(F)dF
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 26/4
Local Volatility Models dFt = Ftσ(Ft) , ut − 1
2 F 2σ2(F)uF F = 0.
d(F1, F2) = F2
F1
1 Fσ(F)dF SABR stochastic (alpha-beta-rho) volatility model, with β = 0, in normalized form: dxt = − 1 2y2
t dt + ytdW1t, dyt = ytdW2t,
< dW1t, dW2t >= 0
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 26/4
Local Volatility Models dFt = Ftσ(Ft) , ut − 1
2 F 2σ2(F)uF F = 0.
d(F1, F2) = F2
F1
1 Fσ(F)dF SABR stochastic (alpha-beta-rho) volatility model, with β = 0, in normalized form: dxt = − 1 2y2
t dt + ytdW1t, dyt = ytdW2t,
< dW1t, dW2t >= 0 Think of x = log F. ds2 = 1 y2 (dx2 + dy2), y ≥ 0 We recognize this as the distance in hyperbolic plane in the Poincar´
e model.
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 26/4
1 y2,
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 27/4
2 p γ ρ γ through p parallels to θ Geodesics in the hyperbolic plane x y y > 0 H
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 28/4
Geodesics in the Poincaré upper half plane model of hyperbolic space
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 29/4
Setting z = (x, y), we have one can then go on to show that: d(z1, z2) = cosh−1
2y1y2
Setting z = (x, y), we have one can then go on to show that: d(z1, z2) = cosh−1
2y1y2
The mean reverting Heston model, in its traditional form (with < dW1t, dW2t >= ρdt is: d ft = ft √ V dW1t dVt = λ(Vt − ¯ V )dt + η √ V dW2t Let x = 1
2 σ log f − a2 2 , y = 1 2 V . Associated Riemnannian metric (non-constant negative
curvature, infinite curvature at y = 0) and ds2 = 4 η2 1 y (dx2 + dy2) = 4y η2
ds2
H2
hyperbolic metric So Heston model, in the new coordinates is in the same conformal class as hyperbolic plane.
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 30/4
Setting z = (x, y), we have one can then go on to show that: d(z1, z2) = cosh−1
2y1y2
The mean reverting Heston model, in its traditional form (with < dW1t, dW2t >= ρdt is: d ft = ft √ V dW1t dVt = λ(Vt − ¯ V )dt + η √ V dW2t Let x = 1
2 σ log f − a2 2 , y = 1 2 V . Associated Riemnannian metric (non-constant negative
curvature, infinite curvature at y = 0) and ds2 = 4 η2 1 y (dx2 + dy2) = 4y η2
ds2
H2
hyperbolic metric So Heston model, in the new coordinates is in the same conformal class as hyperbolic plane.
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 30/4
x − x0 = ρ ξ (y1 − y0) ∓ 1 ξ √y1
1y1
C1 − √y0
1y0
C1 + 1 C2
1(1 − ρ2) arcsin
√y1
1(1 − ρ2)y0 − √y0
1(1 − ρ2)y1
where, to determine the geodesic passing through two fixed points (x0, y0) and (x1, y1), we solve the above implicit equation for C1((x0, y0), (x1, y1)) and then plug back into formula for the distance:
Geod((x0,y0),(x1,y1)) gijdxidxj The first step can be done in an approximate fashion
using Taylor series expansions (or numerically). Martin Forde made first attempt to determine the geodesics in Heston model, in the case ρ = 0.
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 31/4
Seek solution in the form of a series: F(x, y, τ) =
(2πτ)n/2
2τ
+∞
un(x, y)τ n, τ → 0
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 32/4
Seek solution in the form of a series: F(x, y, τ) =
(2πτ)n/2
2τ
+∞
un(x, y)τ n, τ → 0 where, d(x, y) is the geodesic distance between x and y, i.e., minimizer of the functional 1 gij d¯ xi dt d¯ xj dt dt ¯ x(0) = x ¯ x(1) = y, where recall: g = a−1, where a = {aij} is principle part of elliptic operator aij ∂2 ∂xi∂xj
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 32/4
fτ − aij ∂2 ∂xi∂xj f − bi ∂ ∂xi f = 0
Solution in the form :
(4πτ)n/2
4τ
+∞
an(x, y)τ n, τ → 0 ∆(x, y) = |g(x)|−1/2det ∂ d2
2
∂x∂y |g(y)|−1/2 Van-Vleck-DeWitt determinant P = exponential of work done by field A, e
A is constructed from PDE, using two ingredients: principle part and from the drift b, i.e. Ai = bi − det(g)−1/2 ∂ ∂xj
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 33/4
(4πτ)n/2
4τ
+∞
un(x, y)τ n, τ → 0
Characterization of heat kernel coefficients
Obtained via a recursive scheme: u0(x, y) = 1 (1 + 1 k
)∇i)uk = P−1∆−1/2LS∆1/2Puk−1
∇i = ∂i + Ai .
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 34/4
(4πτ)n/2
4τ
+∞
an(x, y)τ n, τ → 0 Laborious calculations by a host of mathematicians and physicists characterize the
u0(x, x) = 1 u1(x, x) = P = 1 6 Scalar Curvature + gij(AiAj − bjAi − ∂ ∂xj Ai)
u2(x, x) = 1 180
+ 1 2 a2
1 + 1
2|R|2 + 1 30 ∆BelR + 1 6 Q
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 35/4
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 36/4
Adjustments in analytic/diff.geometric approach PDE approach due to Minakshisundaram-Pleijel( See Berger-Gauduchon-Mazet). Idea: construct a parametrix (defin. of "parametrix" on next slide), via a series in two stages (assume Laplace-Beltrami for simplicity) Stage 1): Geometric Stage , for close points: Essentially same as above-mentioned asymptotic ansatz. Fundamental Solution F = 1 (4πτ)n/2 e−d2(x,y)/4τ
∞
ui(x, y)τ n
Stage 2) To define globally, ie. for distant points, 1) truncate series for any k > n
2 (using
geometrically determined coefficients for n < k/2) and 2) use cut-off function away from the diagonal: I.e: Let ρ be smooth cut-off with ρ(0, ǫ/4) = 1, ρ(ǫ/2, ∞) = 0. Consider Hk = ρ(d(x, y)) 1 (4πτ)n/2 e−d2(x,y)/4τ
k
ui(x, y)τ n .
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 37/4
Can show that this is parametrix ie. (∂τ − LS)
(4πτ)n/2 e− d2(x,y)
4τ
k
ui(x, y)τ i
2 )e−d2(x,y)/2tGk
where Gk is smooth. Use Levy parametrix idea (iterated convolution) to push error off to infinity. Ie. Fundamental Solution = Hk + Hk ∗ F where F ∗ G(x, y, t) = t
F(x, z, τ)G(z, y, t − τ)dV (z) and where, letting L = ∂t − LS, F =
∞
(LHk)∗l Iterated (infinitely) convolution.
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 38/4
Can give precise estimates for the remainder. If k > n
2 then :
Fundamental Solution = e
−d2(x,y) 2t
(2πt)n/2 ρ(d(x, y))
k
tjuj(x, y)
+O(tk+1) Molchanov approach: Use transition probability for close points and arrive at distant points by piecing together using Chapman-Kolmogorov equations. Evans-Fleming-Soner-Souganidis: Viscosity solution approach. (To our knowledge) Applied mainly to Exit time problems, so far. Perhaps, right question practically minded math-finance researchers?: which of the above adjustments is most efficient while remaining highly accurate?
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 39/4
.L., TH Wang Theorem 1 Consider the SV model dxt = b(xt)ytdW1t + µdt dyt = γyq+1
t
dW2t + νdt < dW1t, dW2t >= ρdt where ρ and γ are constants. Then The curvature of the Riemannian metric naturally associated to the problem is independent of the factor b(x) and independent of the correlation and of the drift. The curvature is equal to (q − 1)y2q Thus The curvature is identically zero if and only if q = 1 , ie. in the quadratic case, and is negative when q < 1.
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 40/4
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 41/4
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 42/4
4t
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 43/4
Recall that when q ≤ 0 (as in Heston model, where q = −1), the curvature blows up at y = 0. But leading order term in order 1, ie. U1 coefficient in heat kernel expansion, is : U1(x, x) = 1 3 K + Q ∼ C y2 + Q in Heston model ie., we have F(x, y) = 1 4πt e−d2/4t (U0(x, y) + U1(x, y)t + · · · ) so, we see that U1 term in expansion cannot be accurate for y with y2t = O(1). Needs adjustment: boundary layer, otherwise need to take t tremendously small, close to y = 0.
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 44/4
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 45/4
First Order system of linear equations To determine the symmetries, need to find infinitesimal generators of vector fields: V = ξ ∂ ∂x + η ∂ ∂y
+ φ ∂ ∂u change of dependent variable These satisfy an overdetermined system of first order partial differential equations, such as five equations Aφx + Bφy = ξu Bφx + Cφy = ηu 2ACξx + 2BCξy − 2ABηx − 2ACηy + (ACx − AxC)ξ + · · · = 0 . . . . . . . . . . . .
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 46/4
Various approaches to short time asymptotics exist in the literaure. A detailed comparison of the efficiency and accuracies of these is an important and open problem. For the heat kernel approach, we are essentially at the beginning of exploring it’s potential. This is because so far it has only been considered in detail in cases where the distance function is known in closed form. Optimally combining the parametrix approach with the geometric approach is an issue to be explored in depth in the future. Influence of off-diagonal corrections on asymptotic implied volatility formulas is an open problem. Asymptotics taking into account Dirichlet (barrier options) and Neumann boundary conditions unexplored so far.
Implied Volatility, Fundamental solutions, asymptotic analysis and symmetry methods, Linz, Ricam kick-off Workshop,September 11, 2008 – p. 47/4