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Large Deviations andStochastic Volatility with Jumps:

Asymptotic Implied Volatility for Affine Models

Martin Keller-ResselTU Berlin

with A. Jaquier and A. Mijatovic (Imperial College London)

SIAM conference on Financial Mathematics, Minneapolis, MNJuly 10, 2012

Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models

Implied Volatility

For strike K ≥ 0 and time-to-maturity T > 0 implied volatility isthe quantity σimp(K ,T ) ≥ 0 that solves

CBlack-Scholes(K ,T ;σimp(K ,T )) = e−rTEQ [(ST − K )+]

Goal: Understand qualitatively how the stochastic model for Sdetermines σimp(K ,T ) and thus the shape of the implied volatilitysurface.

Asymptotic Analysis: K = exT ,T →∞,

using a Large Deviation Principle (LDP).

Implied Volatility

Implied Volatility Surface

1 Affine Stochastic Volatility Models

2 Large deviations and option prices

3 Examples

Affine Stochastic Volatility Models (1)

Xt . . . log-price-processVt . . . a latent factor (or factors), such as stochastic variance orstochastic arrival rate of jumps.St := exp(Xt) . . . price-process. We assume it is a true martingaleunder the pricing measure Q.For simplicity we assume zero interest rate r = 0.

Definition

We call (X ,V ) an affine stochastic volatility model, if (X ,V ) is astochastically continuous, conservative and time-homogeneousMarkov process, such that

euXt+wVt

∣∣∣X0 = x ,V0 = v]

= eux exp (φ(t, u,w) + vψ(t, u,w))

for all (u,w) ∈ C where the expectation is finite.

Xt . . . log-price-processVt . . . a latent factor (or factors), such as stochastic variance orstochastic arrival rate of jumps.St := exp(Xt) . . . price-process. We assume it is a true martingaleunder the pricing measure Q.For simplicity we assume zero interest rate r = 0.

Definition

We call (X ,V ) an affine stochastic volatility model, if (X ,V ) is astochastically continuous, conservative and time-homogeneousMarkov process, such that

euXt+wVt

∣∣∣X0 = x ,V0 = v]

= eux exp (φ(t, u,w) + vψ(t, u,w))

for all (u,w) ∈ C where the expectation is finite.

We can prove that φ and ψ are differentiable in t, and thusthat (Xt ,Vt)t≥0 is a regular affine process in the sense ofDuffie et al. [2003].

Implies in particular that (Xt ,Vt)t≥0 is a semi-martingale withabsolutely continuous characteristics.

The class of ASVMs such defined, includes many importantstochastic volatility models: the Heston model with andwithout added jumps, the models of Bates [1996, 2000] andthe Barndorff-Nielsen-Shephard (BNS) model.

Exponential-Levy models and the Black-Scholes model can betreated as ‘degenerate’ ASVMs.

Define

F (u,w) =∂

∂tφ(t, u,w)

∣∣∣∣t=0

R(u,w) =∂

∂tψ(t, u,w)

∣∣∣∣t=0

The functions φ and ψ satisfy. . .

Generalized Riccati Equations

∂tφ(t, u,w) = F (u, ψ(t, u,w)), φ(0, u,w) = 0

∂tψ(t, u,w) = R(u, ψ(t, u,w)), ψ(0, u,w) = w .

F and R are functions of Levy-Khintchine form

We call F (u,w), R(u,w) the functional characteristics of themodel.

The martingale condition on exp (Xt) implies that

F (0, 0) = R(0, 0) = F (1, 0) = R(1, 0) = 0 .

We also define χ(u) = ∂∂w R(u,w)

∣∣w=0

F (0, 0) = R(0, 0) = F (1, 0) = R(1, 0) = 0 .

∣∣w=0

F (0, 0) = R(0, 0) = F (1, 0) = R(1, 0) = 0 .

∣∣w=0

F (0, 0) = R(0, 0) = F (1, 0) = R(1, 0) = 0 .

∣∣w=0

Example: Heston Model

Heston in SDE form

dXt = −Vt

√Vt dW 1

dVt = −λ(Vt − θ) dt + ζ√

Vt dW 2t ,

where W 1,W 2 are BMs with correlation ρ ∈ (−1, 1), and ζ, λ, θ > 0

Functional Characteristics of the Heston Model

F (u,w) = λθw ,

R(u,w) =1

2(u2 − u) +

2w 2 − λw + uwρζ.

Moreover we have χ(u) = ρζu − λ.

Example: Heston Model

Heston in SDE form

dXt = −Vt

√Vt dW 1

dVt = −λ(Vt − θ) dt + ζ√

Vt dW 2t ,

where W 1,W 2 are BMs with correlation ρ ∈ (−1, 1), and ζ, λ, θ > 0

Functional Characteristics of the Heston Model

F (u,w) = λθw ,

R(u,w) =1

2(u2 − u) +

2w 2 − λw + uwρζ.

Moreover we have χ(u) = ρζu − λ.

Example: Barndorff-Nielsen-Shephard Model

Barndorff-Nielsen-Shephard (BNS) Model in SDE form

dXt = (δ − 1

2Vt)dt +

√Vt dWt + ρ dJλt ,

dVt = −λVt dt + dJλt ,

where λ > 0, ρ < 0 and (Jt)t≥0 is a Levy subordinator with theLevy measure ν.

Functional Characteristics of BNS Model

F (u,w) = λκ(w + ρu)− uλκ(ρ),

R(u,w) =1

2(u2 − u)− λw .

where κ(u) is the cgf of J.

Example: Barndorff-Nielsen-Shephard Model

Barndorff-Nielsen-Shephard (BNS) Model in SDE form

dXt = (δ − 1

2Vt)dt +

√Vt dWt + ρ dJλt ,

dVt = −λVt dt + dJλt ,

where λ > 0, ρ < 0 and (Jt)t≥0 is a Levy subordinator with theLevy measure ν.

Functional Characteristics of BNS Model

F (u,w) = λκ(w + ρu)− uλκ(ρ),

R(u,w) =1

2(u2 − u)− λw .

where κ(u) is the cgf of J.

3 Examples

Large deviations theory

Definition

The family of random variables (Zt)t≥0 satisfies a large deviationsprinciple (LDP) with the ‘good rate function’ Λ∗ if for every Borelmeasurable set B in R,

− infx∈Bo

Λ∗(x) ≤ lim inft→∞

tlogP (Zt ∈ B) ≤

lim supt→∞

tlogP (Zt ∈ B) ≤ − inf

x∈BΛ∗(x).

Continuous rate function

If Λ∗ is continuous on B, then a large deviation principle implies that

P (Zt ∈ B) ∼ exp

(−t inf

x∈BΛ∗(x)

)for large t.

Large deviations theory

Definition

The family of random variables (Zt)t≥0 satisfies a large deviationsprinciple (LDP) with the ‘good rate function’ Λ∗ if for every Borelmeasurable set B in R,

− infx∈Bo

Λ∗(x) ≤ lim inft→∞

tlogP (Zt ∈ B) ≤

lim supt→∞

tlogP (Zt ∈ B) ≤ − inf

x∈BΛ∗(x).

Continuous rate function

If Λ∗ is continuous on B, then a large deviation principle implies that

P (Zt ∈ B) ∼ exp

(−t inf

x∈BΛ∗(x)

)for large t.

The Gartner-Ellis theorem

Assumption A.1: For all u ∈ R, define

Λz(u) := limt→∞

t−1 logE(

)= lim

t→∞t−1Λz

t (ut)

as an extended real number. Denote DΛz := {u ∈ R : Λz(u) <∞}and assume that

(i) the origin belongs to D◦Λz ;(ii) Λz is essentially smooth, i.e. Λz is differentiable throughoutDo

Λz and is steep at the boundaries.

Theorem (Gartner-Ellis)

Under Assumption A.1, the family of random variables (Zt)t≥0

satisfies the LDP with rate function (Λz)∗, defined as theFenchel-Legendre transform of Λz ,

(Λz)∗ (x) := supu∈R{ux − Λz(u)}, for all x ∈ R.

The Gartner-Ellis theorem

Assumption A.1: For all u ∈ R, define

Λz(u) := limt→∞

t−1 logE(

)= lim

t→∞t−1Λz

t (ut)

as an extended real number. Denote DΛz := {u ∈ R : Λz(u) <∞}and assume that

(i) the origin belongs to D◦Λz ;(ii) Λz is essentially smooth, i.e. Λz is differentiable throughoutDo

Λz and is steep at the boundaries.

Theorem (Gartner-Ellis)

Under Assumption A.1, the family of random variables (Zt)t≥0

satisfies the LDP with rate function (Λz)∗, defined as theFenchel-Legendre transform of Λz ,

(Λz)∗ (x) := supu∈R{ux − Λz(u)}, for all x ∈ R.

From LDP to option prices

Theorem (Option price asymptotics)

Let x be a fixed real number.

If (Xt/t)t≥1 satisfies a LDP under Q with good rate function Λ∗,the asymptotic behaviour of a put option with strike ext reads

limt→∞

t−1 logE[(

ext − eXt)

{x − Λ∗ (x) if x ≤ Λ′ (0) ,x if x > Λ′ (0) .

Analogous results can be obtained for call options using a measurechange to the share measure.

By comparing to the Black-Scholes model, the results can betransferred to implied volatility asymptotics.

LDP for affine models

Definition: We say that the function R explodes at the boundaryif limn→∞ R (un,wn) =∞ for any sequence {(un,wn)}n∈N ∈ Do

converging to a boundary point of DoR .

Theorem

Let (X ,V ) be an ASVM with χ(0) < 0 and χ(1) < 0 and assumethat F is not identically null. If R explodes at the boundary, F issteep and {(0, 0), (1, 0)} ∈ Do

F , then a LDP holds for Xt/t ast →∞.

Under the same assumptions, if either of the following conditionsholds:

(i) m and µ have exponential moments of all orders;

(ii) (X ,V ) is a diffusion;

then a LDP holds for Xt/t as t →∞.

LDP for affine models

Definition: We say that the function R explodes at the boundaryif limn→∞ R (un,wn) =∞ for any sequence {(un,wn)}n∈N ∈ Do

converging to a boundary point of DoR .

Theorem

Let (X ,V ) be an ASVM with χ(0) < 0 and χ(1) < 0 and assumethat F is not identically null. If R explodes at the boundary, F issteep and {(0, 0), (1, 0)} ∈ Do

F , then a LDP holds for Xt/t ast →∞.

Under the same assumptions, if either of the following conditionsholds:

(i) m and µ have exponential moments of all orders;

(ii) (X ,V ) is a diffusion;

then a LDP holds for Xt/t as t →∞.

Implied Volatility in ASVMs

Theorem (Implied Volatility Asymptotics for ASVMs)

Let (X ,V ) be an affine stochastic volatility model with functionalcharacteristics F (u,w) and R(u,w) satisfying the assumptionsfrom above.Let Λ(u) = F (u,w(u)) where w(u) is the solution of

R(u,w(u)) = 0.

Thenlimt→∞

σimp(t, ext) = σ∞(x)

σ∞(x) =√

2[sgn(Λ′(1)− x)

√Λ∗(x)− x + sgn(x − Λ′(0))

√Λ∗(x)

and Λ∗(x) = supu∈R(xu − Λ(u)).

Implied Volatility in ASVMs (2)

Corollary

Under the assumptions from above 0 ∈ (Λ′(0),Λ′(1)) and for allx ∈ [Λ′(0),Λ′(1)] it holds that

limt→∞

σimp(t, ext) =√

Λ∗(x)− x +√

Λ∗(x)].

Corollary

Let (X ,V ) be a non-degenerate affine stochastic volatility processthat satisfies the assumptions from above. Then there exists aLevy process Y , such that the limiting smiles of the models eX andeY are identical.

In the Heston model, the corresponding Levy model is theNormal-Inverse-Gaussian (NIG) model.

Corollary

limt→∞

σimp(t, ext) =√

Λ∗(x)− x +√

Λ∗(x)].

Corollary

limt→∞

σimp(t, ext) =√

Λ∗(x)− x +√

Λ∗(x)].

Corollary

3 Examples

Example: Heston model & BNS model

In the Heston model the rate function is Legendre transform of

Λ(u) = −λθζ2

(χ(u) +

√∆(u)

)where ∆(u) = χ(u)2−ζ2(u2−u).

The limiting volatility σ∞(x) can be explictly computed and

coincides - after reparameterization - with the SVIparameterization of Jim Gatheral:

σ2Heston(x) =

(1 + ω2ρx +

√(ω2x + ρ)2 + 1− ρ2

In the BNS-Model we obtain

Λ(u) = λκ

2λ+ u

(ρ− 1

))− uλκ(ρ).

Λ(u) = −λθζ2

(χ(u) +

√∆(u)

)where ∆(u) = χ(u)2−ζ2(u2−u).

σ2Heston(x) =

(1 + ω2ρx +

√(ω2x + ρ)2 + 1− ρ2

Λ(u) = λκ

2λ+ u

(ρ− 1

))− uλκ(ρ).

Λ(u) = −λθζ2

(χ(u) +

√∆(u)

)where ∆(u) = χ(u)2−ζ2(u2−u).

σ2Heston(x) =

(1 + ω2ρx +

√(ω2x + ρ)2 + 1− ρ2

Λ(u) = λκ

2λ+ u

(ρ− 1

))− uλκ(ρ).

Λ(u) = −λθζ2

(χ(u) +

√∆(u)

)where ∆(u) = χ(u)2−ζ2(u2−u).

σ2Heston(x) =

(1 + ω2ρx +

√(ω2x + ρ)2 + 1− ρ2

Λ(u) = λκ

2λ+ u

(ρ− 1

))− uλκ(ρ).

Numerical Illustration: BNS Model

Γ-BNS model with a = 1.4338, b = 11.6641, v0 = 0.0145,γ = 0.5783, (Schoutens)Solid line: asymptotic smile. Dotted and dashed: 5, 10 and 20years generated smile.

Thank you for your attention!

David S. Bates. Jump and stochastic volatility: exchange rate processesimplicit in Deutsche Mark options. The Review of Financial Studies, 9:69–107, 1996.

David S. Bates. Post-’87 crash fears in the S&P 500 futures option market.Journal of Econometrics, 94:181–238, 2000.

D. Duffie, D. Filipovic, and W. Schachermayer. Affine processes andapplications in finance. The Annals of Applied Probability, 13(3):984–1053,2003.

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