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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
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.
Tool:
Asymptotic Analysis: K = exT ,T →∞,
using a Large Deviation Principle (LDP).
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
Implied Volatility
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.
Tool:
Asymptotic Analysis: K = exT ,T →∞,
using a Large Deviation Principle (LDP).
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
Implied Volatility
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.
Tool:
Asymptotic Analysis: K = exT ,T →∞,
using a Large Deviation Principle (LDP).
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
Implied Volatility Surface
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
1 Affine Stochastic Volatility Models
2 Large deviations and option prices
3 Examples
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
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
EQ[
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.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
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
EQ[
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.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
Affine Stochastic Volatility Models (2)
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.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
Affine Stochastic Volatility Models (2)
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.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
Affine Stochastic Volatility Models (2)
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.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
Affine Stochastic Volatility Models (2)
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.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
Affine Stochastic Volatility Models (3)
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 .
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
Affine Stochastic Volatility Models (4)
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
.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
Affine Stochastic Volatility Models (4)
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
.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
Affine Stochastic Volatility Models (4)
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
.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
Affine Stochastic Volatility Models (4)
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
.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
Example: Heston Model
Heston in SDE form
dXt = −Vt
2dt +
√Vt dW 1
t ,
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) +
ζ2
2w 2 − λw + uwρζ.
Moreover we have χ(u) = ρζu − λ.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
Example: Heston Model
Heston in SDE form
dXt = −Vt
2dt +
√Vt dW 1
t ,
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) +
ζ2
2w 2 − λw + uwρζ.
Moreover we have χ(u) = ρζu − λ.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
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.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
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.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
1 Affine Stochastic Volatility Models
2 Large deviations and option prices
3 Examples
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
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→∞
1
tlogP (Zt ∈ B) ≤
lim supt→∞
1
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.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
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→∞
1
tlogP (Zt ∈ B) ≤
lim supt→∞
1
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.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
The Gartner-Ellis theorem
Assumption A.1: For all u ∈ R, define
Λz(u) := limt→∞
t−1 logE(
eutZt
)= 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.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
The Gartner-Ellis theorem
Assumption A.1: For all u ∈ R, define
Λz(u) := limt→∞
t−1 logE(
eutZt
)= 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.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
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.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
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
R
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 →∞.
Lemma
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 →∞.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
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
R
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 →∞.
Lemma
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 →∞.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
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)
where
σ∞(x) =√
2[sgn(Λ′(1)− x)
√Λ∗(x)− x + sgn(x − Λ′(0))
√Λ∗(x)
],
and Λ∗(x) = supu∈R(xu − Λ(u)).
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
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) =√
2[√
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.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
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) =√
2[√
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.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
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) =√
2[√
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.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
1 Affine Stochastic Volatility Models
2 Large deviations and option prices
3 Examples
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
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) =
ω1
2
(1 + ω2ρx +
√(ω2x + ρ)2 + 1− ρ2
).
In the BNS-Model we obtain
Λ(u) = λκ
(u2
2λ+ u
(ρ− 1
2λ
))− uλκ(ρ).
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
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) =
ω1
2
(1 + ω2ρx +
√(ω2x + ρ)2 + 1− ρ2
).
In the BNS-Model we obtain
Λ(u) = λκ
(u2
2λ+ u
(ρ− 1
2λ
))− uλκ(ρ).
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
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) =
ω1
2
(1 + ω2ρx +
√(ω2x + ρ)2 + 1− ρ2
).
In the BNS-Model we obtain
Λ(u) = λκ
(u2
2λ+ u
(ρ− 1
2λ
))− uλκ(ρ).
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
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) =
ω1
2
(1 + ω2ρx +
√(ω2x + ρ)2 + 1− ρ2
).
In the BNS-Model we obtain
Λ(u) = λκ
(u2
2λ+ u
(ρ− 1
2λ
))− uλκ(ρ).
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
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.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
Thank you for your attention!
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models
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.
Martin Keller-Ressel Asymptotic Implied Volatility for Affine Models