Asymptotics for local volatility and Sabr models Peter Laurence - - PowerPoint PPT Presentation

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Asymptotics for local volatility and Sabr models

Peter Laurence

Facoltá di Statistica, Università di Roma I

Kolmogorov Equations in Finance and Physics, Modena, Italy

September 8, 2010

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results

Collaborators in work presented today

Gérard Ben Arous, Courant Institute Jim Gatheral, Baruch College Elton Hsu, Northwestern University Cheng Ouyang, Purdue University Tai-Ho Wang, Baruch College, CUNY

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Overview

Outline

1

Collaborators in various aspects of work presented today

2

Outline of Work to be discussed Overview

3

Background Models

Local Volatility Models Stochastic Volatility models

Methodology to be used Curvature

4

Our Approach and Results Local volatility Models revisited

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Overview

Outline of Results

Two lines: Contributions of a Theoretical nature Provide rigorous proofs of short time to maturity expansion formulas for i) call prices and ii) implied volatility in local volatility setting. Practical Nature New expansion formulas for call prices and implied volatility. I.e. expansion up to second order with optimal (in a certain sense)

• coefficients. Already order 1 more accurate for several models

tested than earlier expansions tested. σBS(t, T ) = σ0

BS(t) + σ(1) BS (t)(T − t) + σ(2) BS (t)(T − t)2

• second order coeffts

+o(T − t)2 Dimensionless parameters σ2(T − t) should be small. In practice expansion highly accurate even when this parameter is of order 1.

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Overview

Regimes

Historically, in mathematical finance, as time to maturity τ → 0,several regimes have been considered: ATM regime: S0 = K, τ → 0 τ = T − t Megvedev-Scaillet Regime: S0 − K = C√ τ τ → 0 C is constant Can be viewed as a close to the money regime. Standard heat kernel Regime (Henry-Labordère) S0, K fixed, τ → 0 In this case, when τ gets small, the deviation of call prices from their intrinsic value, decays exponentially (like τ3/2e− d2

2τ ). But the

rate of exponential decay is a key tool to derive asymptotic formulas for the implied volatility.

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Overview

Regimes

In practice, since the heat kernel leads to a representation for the transition probability density, one can try to use it to calculate all three regimes. Once can also show, using L ’Hospital’s rule, that limit of away from the money expansion for small time, yields ATM expansion. Similarly, introducing S0(τ) = K + C√τ, into the heat kernel expansion,

• ne can recover Megvedev-Scaillet results (diffusion case).

General Principle: Heat kernel expansion was devised to be accurate for small (dimensionless time σ2τ), but is often accurate for moderate values of σ2τ. To make expansion work for longer maturities, can combine heat kernel method with Levi Parametrix approach (A. Friedman). Use Heat kernel

• exp. as first iterate, rather traditional constant coefft solution.

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Outline

1

Collaborators in various aspects of work presented today

2

Outline of Work to be discussed Overview

3

Background Models

Local Volatility Models Stochastic Volatility models

Methodology to be used Curvature

4

Our Approach and Results Local volatility Models revisited

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Local volatility

The Local volatility model dSt = b(t)Stdt + a(St, t)dWt where {St}t≥0 is price process for the stock {Wt}t≥0 is a Brownian motion. Important contributions by Bruno

• Dupire. Still popular model today, in certain (e.g. French) banks.

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Sabr type models

Sabr Model in its original form (Hagan and Woodward, Hagan, Kumar,

Lesniewski and Woodward, Andreasen-Andersen) dFt = F β

t ytdW1t

dyt = αytdW2t < dW1t, dW2t >= ρdt Calibrates well to smile, but for only one maturity.

"Dynamic Sabr Model"

dFt = γ(t) C(Ft)ytdW1t dyt = ν(t) ytdW2t < dW1t, dW2t >= ρ(t)dt, time dependent parameters. Can be calibrated to implied volatility surface for several maturities.

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Heston

Heston Model dSt = µStdt +

• VtStdWt

dVt = κ(θ − Vt)dt + σ

• VtdZt

dWtdZt = ρdt where {St}t≥0 and {Vt}t≥0 are price and variance processes. {Wt}t≥0 and {Zt}t≥0 are Wiener processes with constant instantaneous correlation ρ. θ is long-run mean, κ is the rate of reversion and σ is volatility of variance.

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Heston + local vol

The Heston Model with local vol dSt = µStdt +

• Vt σ(St, t) dWt

dVt = κ(θ − Vt)dt + ¯ σ

• VtdZt

< dWt, dZt >= ρdt where {St}t≥0 and {Vt}t≥0 are price and volatility processes {Wt}t≥0 and {Zt}t≥0 are Wiener processes with correlation ρ θ is long-run mean, κ is the rate of reversion and ¯ σ is volatility of variance. Andreasen and others.

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Lipton-Andersen Quadratic SV Model

Lipton-Andersen Model dS(t) = λ(t)

• z(t)
• b(t)S(t) + (1 − b(t))S0 + 1

2 c(t) S0 (S(t) − S0)2

• dWt

dz(t) = κ(1 − z(t))dt + η(t)

• z(t)dZ (t)

z(0) = 1 where < dW(t), dZ (t) >= ρdt Needs adjustment at the wings, since local martingale but not a martingale in general. Can be seen as special case of Heston-local vol model.

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Outline

1

Collaborators in various aspects of work presented today

2

Outline of Work to be discussed Overview

3

Background Models

Local Volatility Models Stochastic Volatility models

Methodology to be used Curvature

4

Our Approach and Results Local volatility Models revisited

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Methods

Passage from stochastic volatility model to local vol model: Gyongy-Dupire-Derman and Britten-Jones and Neuberger method for reducing the computation of call prices in stochastic volatility model to computation of an effective local volatility in a local volatility model.

Combine with:

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Methods

Passage from stochastic volatility model to local vol model: Gyongy-Dupire-Derman and Britten-Jones and Neuberger method for reducing the computation of call prices in stochastic volatility model to computation of an effective local volatility in a local volatility model.

Combine with:

Heat kernel method for the determination of transition probability density in the local and stochastic volatility models. This reduction requires knowledge of the corresponding Riemannian distance function and/or geodesics for the SV

• model. These were known in closed form in the λ-Sabr model. In

the case of the Heston model, the geodesics are known and, very recently (L. & Gulisashvili), obtained a very accurate fully analytic closed form expression for the Legendre transform of the distance function.

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Gyongy-Dupire: From stochastic volatility to local volatility

Stochastic-local volatility models: dFt = αtb(Ft)dW1t dαt = g(αt)dW2t F0 = F, α0 = α initial conditions < dW1t, dW2t >= ρdt Obtaining a local volatility model with same F marginals: “Equivalent” local volatility function is given by: σ2

loc(K, T ) = b2(K)E[α2 T | FT = K]

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Gyongy-Dupire: Effective parameters

More general result, giving rise to the concept of "mimicking": SV model dSt = c(St, νt, t)dt + b(St, t)g(ν(t), t)dW1t dνt = ζ(νt)dt + β(νt)dW2t < dW1t, dW2t >= ρdt S0 = S, ν0 = ν, yields the same marginal distributions with respect to the S variable as the following sde: dSt = γ(S, t)dt + σ(St, t)d ¯ Wt, S(0) = S where, effective parameters are σ2(K, T) = b2(K, T)E

• g2 | ST = K
• and

γ(K, T) = E [c | ST = K ]

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Laplace asymptotics

Local volatility Representation σ2(k, t) = ∞

0 y2p(t, (s0, y0), (k, y))dy

∞

0 p(t, (s0, y0), (k, y)dy

Now use p(t, (s0, y0), (k, y)) =

1 2πt e−

d2 R((s0,y0),(K,y)) 2t

f(K, y), where dR is the natural Riemannian distance, and f is derived from heat kernel expansion. Apply Laplace asymptotics to express (for small t) in terms of min ymin = argminy d2

R((s0, y0), (K, y))

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Varadhan Lemma

• Let p(t, x, y) be the fundamental solution corresponding to the

non-degenerate diffusion with infinitesimal generator: aij(x)pxixj, x ∈ Rn and the time homogeneous diffusion (Heat flow) on Rn. Hp = pt − Lxp = 0

• The main theorem concerning the small time behaviour of the

fundamental solution of this 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. • Note: “uniformity is key” to theorem being hard: holds also for distant points .

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Varadhan’s lemma generalizations

Varadhan’s lemma also holds for certain strongly degenerate but hypoelliptic diffusions L = 1 2

n

∑

i=1

X 2

i + X0

and if generating condition is satisfied Lie(X1, X2, . . . , Xn)(x) = Rn ∀x ∈ Rn then, again lim

t→0 log pt(x, y) = 1

2d2(x, y) Léandre, Deuschel, Taylor

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Varadhan’s lemma for non-complete manifolds

We will see below, that due to the degeneracy of diffusions in math-finance on the boundary, the associated Riemannian manifold is often not complete. What can one say for non-complete Riemannian manifolds?

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Varadhan’s lemma for non-complete manifolds

We will see below, that due to the degeneracy of diffusions in math-finance on the boundary, the associated Riemannian manifold is often not complete. What can one say for non-complete Riemannian manifolds? Azencott: Varadhan’s lemma continues to hold as long as d(x, y) ≤ max(d(x, ∞), d(y, ∞))

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Varadhan’s lemma for non-complete manifolds

We will see below, that due to the degeneracy of diffusions in math-finance on the boundary, the associated Riemannian manifold is often not complete. What can one say for non-complete Riemannian manifolds? Azencott: Varadhan’s lemma continues to hold as long as d(x, y) ≤ max(d(x, ∞), d(y, ∞)) Hsu improves this condition to d(x, y) ≤ d(x, ∞) + d(y, ∞) (which is larger than

max(d(x, ∞), d(y, ∞)))

Sufficient but not known to be necessary condition for Varadhan’s lemma to hold.

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Heat kernel Series solution for fundamental solution

Seek solution of backward heat equation in y, τ in the form: Heat Kernel Series: Time homogeneous case F(y, x, τ) =

• g(x)

(2πτ)n/2

• ∆(x, y)P(x, y)e− d2(x,y)

2τ

+∞

∑

j=0

Uj(x, y)τn, τ → 0

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Heat kernel Series solution for fundamental solution

Seek solution of backward heat equation in y, τ in the form: Heat Kernel Series: Time homogeneous case F(y, x, τ) =

• g(x)

(2πτ)n/2

• ∆(x, y)P(x, y)e− d2(x,y)

2τ

+∞

∑

j=0

Uj(x, y)τn, τ → 0 where, d(x, y) is the geodesic distance between x and y, i.e., minimizer of the functional

1

0 gij

d ¯ xi dt d ¯ xj dt dt ¯ x(0) = x ¯ x(1) = y, where g(x) = det(gij) and where g = a−1, here a = {aij} is principal part of elliptic operator aij

∂2 ∂xi ∂xj

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Heat kernel ct’d

fτ −

1 2aij

∂2 ∂xi∂xj f − bi ∂ ∂xi f = 0 Solution in the form :

• g(x)

(2πτ)n/2

• ∆(x, y)P(x, y)e− d2(x,y)

2τ

+∞

∑

j=0

aj(x, y)τn, 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

• C(x,y)<A,dl>R

A is constructed from PDE, using two ingredients: diffusion matrix and the drift b, i.e. Ai = bi − det(g)−1/2 ∂ ∂xj

• det(g)1/2gij

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Time inhomogeneous case

Suppose the coefficients of the diffusion and/or drift depend explicitly on time. How does the heat expansion change? Time inhomogeneous case F(y, t, x, T ) =

• g(x, T)

(2π(T − t)n/2

• ∆(x, y, t)P(x, y, t)e− d2(x,y,t)

2(T−t) ×

{

+∞

∑

j=0

Uj(x, y, t)(T − t)j}, as T − t → 0 satisfies the backward Kolmogorov equation in the variables (y, t).

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Finding the coefficients in the heat kernel expansion

Zero-th order coefficient can be solved for in closed form only when we know the distance function in closed form. This is why Sabr model succeeds since in Sabr model Riemannian distance is diffeomorphic image of distance in the hyperbolic plane ( In formula below f β−1 is local vol, i.e.dft = f βytdWt). distance in diffeormorphic image of hyperbolic plane d(X, Y) = arccosh   1 + x

X 1 f β du22

− 2ρ(y − Y) x

X 1 f β du + (y − Y)2

2(1 − ρ2)yY    Coefficients in the heat equation satisfy the so-called transport equations, i.e. ordinary equations along the geodesics connecting points y and x. Cannot usually solve these in closed form but can Taylor expand for y close to x.

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Heat kernel coefficients one or higher

One way to get around the inability to solve the transport equations explicitly proposed by Henry-Labordère: Use on diagonal (say first order) heat kernel coefficients: U1(x, x) Approximate off diagonal heat kernel coefficient U1(x, y) by U1(x, y) = U1(x + y 2 , x + y 2 ) This method works quite well when we are near the diagonal, i.e., y close to x.

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Outline

1

Collaborators in various aspects of work presented today

2

Outline of Work to be discussed Overview

3

Background Models

Local Volatility Models Stochastic Volatility models

Methodology to be used Curvature

4

Our Approach and Results Local volatility Models revisited

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Influence of curvature

• G. Ben Arous, P

.L.

Theorem Consider the SV model dxt = b(xt)ytdW1t + µxdt dyt = γyq

t dW2t + µydt

< dW1t, dW2t >= ρdt where ρ and γ are constants. Then The (Gaussian) curvature of the Riemannian metric naturally associated to the problem is independent of the factor b(x) and independent of the correlation and

• f the drift.

The curvature is equal to (q − 2)y2(q−1) Thus The curvature is identically zero if and only if q = 2 , ie. in the quadratic case, and is negative when q < 1.

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

influence of curvature II: (q − 2)y2(q−1)

When q = 1, the curvature is constant. This is the original Sabr model. When q = 0 i.e. Heston model, the curvature is negative and it blows up at y = 0. In fact the curvature blows up at y = 0 as soon as q < 1. In this case it can be shown that the associated Riemannian manifold is not complete. Note: The sign and size of the curvature is important in the heat kernel asymptotic approach to the heat kernel. Here is why: On Riemannian manifolds of negative Riemannian curvature, the cut locus is empty.

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

Hyperbolic Space

H : ds2 = 1 y2 (dx2 + dy2) Space of constant negative Gaussian curvature Gc equal to −1: Gc = 1 2H ∂ ∂u F EH ∂E ∂v − 1 H ∂G ∂u

• + ∂

∂v 2 H ∂F ∂u − 1 H ∂E ∂v − F EH ∂E ∂u

• where ds2 = Edx2 + 2Fdxdy + Gdy2, &H =

√ EG − F 2 and where, in the case of hyperbolic plane E = G = 1

y2 ,

F = 0

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

geodesics

2 p γ ρ γ through p parallels to θ Geodesics in the hyperbolic plane x y y > 0 H

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Models Methodology to be used Curvature

geodesics Heston: Richard Hamilton’s observation

Metric; ds2 = dx2+dy2

2y

Geodesics: x = s − sin s, y = 1 − cos s.

(Loading cycloid.mov)

Geodesics are dilates and translates of standard cycloid. Only very recently, do we know Legendre transform of distance function in closed form.

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Local volatility Models revisited

PDE

If 0 < C1 < σ < C2 : Berestycki,Busca, Florent (2002, 2004) The implied volatility lies in W 1,2,p for all 1 < p < ∞ and satisfies, in the viscosity sense, the equation 2τφφτ + φ2 − σ2(x, τ)(1 − x φx φ )2 −σ2(x, τ)τφφxx + 1 4σ2(x, τ)τ2φ2φ2

x = 0

where x = log( Serτ

K ). Also, short time limit

lim

τ→0 φ(x, τ) =

1 1

ds σ(sx,0)

Note that BBF require the diffusions to be non-degenerate,i.e. σ(x, τ) > C > 0.

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Local volatility Models revisited

Outline

1

Collaborators in various aspects of work presented today

2

Outline of Work to be discussed Overview

3

Background Models

Local Volatility Models Stochastic Volatility models

Methodology to be used Curvature

4

Our Approach and Results Local volatility Models revisited

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Local volatility Models revisited

Local Volatility Models revisited: Motivations

Highly accurate approximations for transition density in local volatility models of interest because Local volatility model of independent interest. Asymptotics for local volatility models when combined with Gyongy projection technique, provide highly accurate asymptotics for stochastic volatility models (two factor). Asymptotics can be fully justified even in the context of degenerate diffusions, by virtue of the so-called principle of not feeling the boundary.

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Local volatility Models revisited

Heat Kernel coefficients time inhomogeneous case

Heat kernel coefficients in time inhomogeneous case satisfy transport equations, i.e., first order, inhomogeneous ordinary differential equations along the geodesics associated to the natural Riemannian metric. In one-D can be integrated exactly. For example:

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Local volatility Models revisited

Heat Kernel coefficients time inhomogeneous case

Heat kernel coefficients in time inhomogeneous case satisfy transport equations, i.e., first order, inhomogeneous ordinary differential equations along the geodesics associated to the natural Riemannian metric. In one-D can be integrated exactly. For example:

Coefficients u0(s, K, t) = exp

• −

s

K

1 d(K, η, t)

• − 1

2 +

a2 2 (d2)ηη + b(d2)η

2 + (d2)t 2

• dη

a(η,t)

• ,

L = 1

2 a(s, t) ∂2 ∂s2 + b(s, t) ∂ ∂S + c(s, t).

d(s, K) =

K

s

1 a(u, t) du

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Local volatility Models revisited

Example:Driftless one dimensional case

Lu = 1 2a2(y, t)uyy Heat kernel coefficients in closed form:

Coefficients: 1 D case u0 = exp(1 2 log a(y, t) a(x, t) ) × exp y

x

• 1

a(˜ y, t)

˜

y x

at a2(u, t)du

• d ˜

y

• =
• a(y, t)
• a(x, t)

exp

• −

y

x

b(y, t) a(y, t)2

• exp

y

x

1 a(˜ y, t) )

˜

y x

at a2(u, t)dud ˜ y

• Peter Laurence

Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Local volatility Models revisited

Heat Kernel coefficients 2

b = c = 0 in PDE & a independent of time, obtain the following integral:

• ff diagonal: time homogeneous case

u1(x, y) 1 4 U0 1 y

x 1 a(u)du

y

x (a¯ y ¯ y − 1

2 a2

¯ y

a )d ¯ y = 1 4 1 y

x 1 a(u)

• harmonic mean of volatility

a(y)

• a(x)
• ay(y) − ay(x) − 1

2

y

x

a2

¯ y

a

• d ¯

y

• (1)

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Local volatility Models revisited

Solving for implied volatility

Use Dupire-Derman-Kani to express Call Prices in the form:

Call price asymptotics Call(y, x, t, T) = (y − x)+ + 1 2E[

T

t

• a21yt=x
• ]

= (y − x)+ + 1 2

T

t

a2(x, u)

• 1

a((x, T) 1 (4π(u − t)1/2 e−

d2 4(u−t) [u0(x, y, t)

+(u − t)u1(x, y, t) + (u − t)2u2(x, y, t) + . . .)

• du

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Local volatility Models revisited

Call Prices: Illustration in time homogeneous case

Grouping powers of T − t = ¯ T, this leads to expressions for call prices of the form

Call Prices C(y, x, ¯ T) = (y − x)+ + 1 2 √ 2π (U0(x, y)u0(x, y, ¯ T) + . . .) = (y − x)+ + 1 2 √ 2π

n

∑

i=0

ui(x, y)

heat kernel coefficients

Ui(x, y, ¯ T) where Ui(x, y, t) =

¯

T 0 (

√ v)2i−1e− ω2

v dv

ω = 1 √ 2

y

x

1 a(u)du

Peter Laurence Asymptotics for local volatility and Sabr models

Collaborators in various aspects of work presented today Outline of Work to be discussed Background Our Approach and Results Local volatility Models revisited

matching

Recall ω = y

x 1 a(u,t)du ∼ "d(x, y, t)".

In Black-Scholes setting ¯ ω = log( y

x )

σ2

BS

. In regime ω2

¯ T >> 1, the auxiliary function

U1 (expressible in terms of erfc (complimentary error function)) admits asymptotic expansions U0(ω, ¯ T ) =∼ ¯ T 3/2 ω2(t) e− ¯

ω2 ¯ T , U1 ∼

¯ T 5/2 ω2(t) e− ¯

ω2 ¯ T

This leads to following matching: Matching

Black Scholes price

• σBS

√xye− ¯

ω2 ¯ T ¯

T 3/2 1 ¯ ω2 + . . . =

local vol price

• a(x, t)a(y, t)e− ω2(t)

¯ T

¯ T 3/2 1 ω2(t) + . . .

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Matching continued: transcendent matching vs algebraic matching

(Transcendental matching) Exponential contributions on both sides must balance: ⇒zero-th order exponents of exponentials must match (Algebraic Matching) Once zero-th order exponents match, match like powers of ¯ T on both sides. Results Transcendental matching leads to Berestycki-Busca-Florent formula, in the time homogeneous case and to a slightly different formula in time inhomogeneous case.

Peter Laurence Asymptotics for local volatility and Sabr models

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Implied volatility expansion

Expansion σBS(S0, K, t, T ) = σ(0)

BS (t) +

dσBS dT |T=t

σ(1)

BS (t) (T − t) + . . .

(Zero-th order "generalized BBF") zero order σ(0)

BS (t) =

log( S0

K )

S0

K 1 a(u,t)du

Note, to recover optimal expansions, volatility a(u, ·) needs to be evaluated at t, not at T, in time inhomogeneous case.

Peter Laurence Asymptotics for local volatility and Sabr models

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Implied volatility expansion

first order time homogeneous, r = 0 σ(1)(S0, K ) here σ(0)= BBF = log  

• a(K ))a(S0))
• S0K log(S0/K )

S0 K 1 a(u) du   + r S0 K 1 (σ(0))2u du − r S0 K u a2(u) du ( S0 K 1 a(u) du)3 log(S0/K ) Peter Laurence Asymptotics for local volatility and Sabr models

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Implied volatility expansion

first order time homogeneous, r = 0 σ(1)(S0, K ) here σ(0)= BBF = log  

• a(K ))a(S0))
• S0K log(S0/K )

S0 K 1 a(u) du   + r S0 K 1 (σ(0))2u du − r S0 K u a2(u) du ( S0 K 1 a(u) du)3 log(S0/K ) ATM ⇒ σBS,1 = 2 3K (at + au1) + 1 12 ( a(K , t) K )3 Compare with Hagan-Woodward formula (for r = 0, with Sav = S0+K 2 ) σ(0)            1 +

• a2(Sav )

24

• 2 a′′(Sav )

a(Sav ) − ( a′(Sav ) a(Sav ) )2 + 1 S2 av

• σ(1)(S0,K )

T            Peter Laurence Asymptotics for local volatility and Sabr models

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Comparisons

Henry-Labordère refinement of Hagan-Woodward Labordere refinement σ(0)

• 1 +
• 1

24(σ(0))2 + a2(S0) 4

• (a′′(S0)

a(S0) ) − 1 2(a′(S0) a(S0) )2

• T
• (Analytical comparison): Labordère’s and Hagan et al.’s σ1

involves derivatives of the volatility. Optimal σ1 does not ⇒ Better stability properties. Labordère σ1 involves U1(x, y), first order term in heat kernel

• expansion. Optimal σ1 involves only U0. In fact this pattern holds

throughout: Optimal σi involves only Ui−1

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Numerical comparison:

Performance in time H. CEV model:dS = σ √S S0 dZ

0.5 1.0 1.5 2.0 0.0e+00 5.0e−06 1.0e−05 1.5e−05 2.0e−05

Sqrt CEV model: HL in green, GHLOW in blue

Strike Implied vol difference (approx. − exact) Expiry = 1 year σ = 0.2

Figure: Comparison CEV, β = 1

2 σ = .2, S0 = 1

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Numerical comparison

Performance in Andersen model: dS = σ

• ψ S + (1 − ψ) S0 + γ

2 (S−S0)2 S0

• dZ

0.5 1.0 1.5 2.0 −8e−05 −4e−05 0e+00 4e−05

Andersen quadratic model: HL in green, GHLOW in blue

Strike Implied vol difference (approx. − exact) Expiry = 1 year σ = 0.2 ψ = −0.5 γ = 0.1

Figure: Comparison Andersen-Lipton quadratic model, ψ = −.5, γ = 1

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Tables

Performance in CEV model:dS = σ √S S0 dZ

Strike ∆σHL ∆σGHLOW σexact σHL σGHLOW 0.50 2.1E-05 1.3E-06 23.68% 23.68% 23.68% 0.75 3.5E-06 8.0E-07 21.48% 21.48% 21.48% 1.00 5.6E-07 1.1E-07 20.01% 20.01% 20.01% 1.25 1.5E-06 4.2E-07 18.91% 18.91% 18.91% 1.50 3.4E-06 3.3E-07 18.05% 18.05% 18.05% 1.75 5.5E-06 2.7E-07 17.34% 17.34% 17.34% 2.00 7.3E-06 2.3E-07 16.74% 16.74% 16.74%

Table: CEV, β = 1/2, σ = .2, T = 1, S0 = 1, ψ = −.5, γ = .1

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Tables

Performance in Lipton-Andersen model: dS = σ

• ψ S + (1 − ψ) S0 + γ

2 (S−S0)2 S0

• dZ

Strike ∆σHL ∆σGHLOW σexact σHL σGHLOW 0.50 8.8E-05

• 1.0E-05

31.29% 31.28% 31.29% 0.75

• 3.4E-05
• 3.0E-06

24.51% 24.50% 24.51% 1.00

• 1.1E-06
• 1.1E-07

20.03% 20.03% 20.03% 1.25 2.0E-05

• 4.3E-07

16.75% 16.75% 16.75% 1.50 3.3E-05

• 1.8E-07

14.18% 14.19% 14.18% 1.75 4.1E-05

• 7.6E-08

12.09% 12.09% 12.09% 2.00 4.6E-05

• 3.2E-08

10.32% 10.33% 10.32%

Table: Quadratic model, σ = .2, T = 1, S0 = 1, ψ = −.5, γ = 1

Peter Laurence Asymptotics for local volatility and Sabr models

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Numerical comparison: add σ2

Performance in CEV model:dS = σ √S S0 dZ

0.5 1.0 1.5 2.0 0.0e+00 5.0e-06 1.0e-05 1.5e-05 2.0e-05 Strike Approximation error

CEV approximation errors: Expiry = 1 year, ! = 0.2

H-L !1 !2

Figure: Comparison CEV, β = 1

2 σ = .2, S0 = 1

Peter Laurence Asymptotics for local volatility and Sabr models

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Numerical comparison: add σ2

Performance in Andersen model: dS = σ

• ψ S + (1 − ψ) S0 + γ

2 (S−S0)2 S0

• dZ

0.5 1.0 1.5 2.0

• 8e-05
• 4e-05

0e+00 4e-05 Strike Approximation error

AQ approximation errors: Expiry = 1 year, ! = 0.2, ! = !0.5, ! = 0.1

H-L !1 !2

Figure: Comparison Andersen-Lipton quadratic model, ψ = −.5, γ = 1

Peter Laurence Asymptotics for local volatility and Sabr models

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Numerical comparison Tables: add σ2 in CEV

Strikes ∆σHL ∆σ1 ∆σ2 σexact 0.50 2.12e-05 1.31e-06 1.98e-08 0.2368 0.75 3.46e-06 7.98e-07 9.87e-09 0.2148 1.00 5.68e-07 5.68e-07 6.03e-09 0.2001 1.25 1.52e-06 4.21e-07 4.08e-09 0.1891 1.50 3.45e-06 3.33e-07 2.96e-09 0.1805 1.75 5.45e-06 2.73e-07 2.18e-09 0.1734 2.00 7.27e-06 2.29e-07 1.70e-09 0.1674

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Numerical comparison Tables: add σ2 in Lipton-Andersen

Strikes ∆σHL ∆σ1 ∆σ2 σexact 0.50

• 8.83e-05
• 1.04e-05
• 1.08e-07

0.3129 0.75

• 3.42e-05
• 3.05e-06
• 1.94e-08

0.2451 1.00

• 2.14e-06
• 1.09e-06
• 4.58e-09

0.2003 1.25 1.99e-05

• 4.31e-07
• 1.30e-09

0.1675 1.50 3.32e-05

• 1.80e-07
• 3.92e-10

0.1418 1.75 4.13e-05

• 7.59e-08
• 5.28e-11

0.1209 2.00 4.56e-05

• 3.16e-08

9.57e-12 0.1032

Peter Laurence Asymptotics for local volatility and Sabr models

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Add time dependence CEV and Quadratic

time dept CEV Model dft = e−λtσ √ fdWt time dept quadratic model dft = e−λtσ

• ψf + (1 − ψ) + γ

2 (f − 1)2 dWt

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Numerical comparison: Performance in CEV model time dept:

Figure 3: Implied volatility approximations in the CEV model with the pa- rameters of Section 1.2.1 for two expirations: τ = 0.25 on the left and τ = 1.0

• n the right. The solid line is exact implied volatility, the dashed line is our

approximation to first order in τ = (T − t) (with only σ1 and not σ2) and the dotted line is our approximation to second order in τ (including σ2).

0.7 0.8 0.9 1.0 1.1 1.2 1.3 0.08 0.10 0.12 0.14 0.16 0.18 0.20 Strike Implied volatility 0.5 1.0 1.5 2.0 0.08 0.10 0.12 0.14 0.16 0.18 0.20 Strike Implied volatility

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Numerical comparison: performance in quadratic case time dept

Figure 4: Implied volatility approximations in the Andersen Quadratic model with the parameters of Section 1.2.2 for two expirations: τ = 0.25 on the left and τ = 1.0 on the right. The solid line is exact implied volatility, the dashed line is our approximation to first order in τ = (T − t) (with only σ1 and not σ2) and the dotted line is our approximation to second order in τ (including σ2).

0.7 0.8 0.9 1.0 1.1 1.2 1.3 0.08 0.10 0.12 0.14 0.16 0.18 0.20 Strike Implied volatility 0.5 1.0 1.5 2.0 0.08 0.10 0.12 0.14 0.16 0.18 0.20 Strike Implied volatility

Figure: Comparison Lipton-Andersen quadratic model, ψ = −.5, γ = 1

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Numerical comparison: performance in CEV case case time dept, T = .25

Table: CEV model implied volatility errors in the time-dependent case with T = 0.25 for various strike prices in the Henry-Labordère (HL) approximation and our first and second order approximations respectively. The exact volatility in the last column is obtained by inverting the closed-form expression for the option price in the CEV model.

Strikes ∆σHL ∆σ1 ∆σ2 σexact 0.70

• 2.63e-03
• 2.63e-03

2.05e-04 0.1937 0.80

• 2.55e-03
• 2.55e-03

1.94e-04 0.1875 0.90

• 2.48e-03
• 2.48e-03

1.88e-04 0.1821 1.00

• 2.41e-03
• 2.41e-03

1.83e-04 0.1774 1.10

• 2.36e-03
• 2.36e-03

1.79e-04 0.1732 1.20

• 2.30e-03
• 2.31e-03

1.75e-04 0.1695 1.30

• 2.26e-03
• 2.26e-03

1.72e-04 0.1660

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Numerical comparison: performance in quadratic model case case time dept, T = .25

Table: Quadratic model implied volatility errors in the time-dependent cases with T = 1.0 for various strike prices in the Henry-Labordère (HL) approximation and our first and second order approximations respectively. The exact volatility in the last column is obtained by inverting the closed-form expression for the option price in the quadratic model.

Strikes ∆σHL ∆σ1 ∆σ2 σexact 0.50

• 4.83e-02
• 4.82e-02

1.49e-02 0.2053 0.75

• 3.81e-02
• 3.81e-02

1.20e-02 0.1609 1.00

• 3.13e-02
• 3.16e-02

9.92e-03 0.1316 1.25

• 2.62e-02
• 2.62e-02

8.36e-03 0.1101 1.50

• 2.22e-02
• 2.22e-02

7.12e-03 0.0932 1.75

• 1.89e-02
• 1.90e-02

6.09e-03 0.0795 2.00

• 1.62e-02
• 1.62e-02

5.21e-03 0.0679

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Conclusion

Heat kernel expansion can be used to obtain highly accurate implied volatility. Enhanced accuracy of implied volatility expansions due to correct matching after use of off-diagonal heat kernel coefficients. Proper expansions involve regimes: No single expansion is best for all regimes! Optimal expansions for three factor models a challenge for the future. Terms in the expansions correspond to derivatives as function of final time for fixed spot and this can be established rigorously!

Peter Laurence Asymptotics for local volatility and Sabr models

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global estimates

Global Upper bounds for heat kernel Theorem (M, g, µ) be a connected, complete,

Peter Laurence Asymptotics for local volatility and Sabr models