Exponential Lévy Model Method and Known Results Asymptotic Normality and Confidence Sets

Confidence Sets in Nonparametric Calibration ofExponential Lévy Models

Jakob Söhl

Institut für MathematikHumboldt-Universität zu Berlin

Haindorf SeminarFebruary 10, 2012

Exponential Lévy Model Method and Known Results Asymptotic Normality and Confidence Sets

Outline

Exponential Lévy Model

Method and Known Results

Asymptotic Normality and Confidence Sets

Exponential Lévy Model Method and Known Results Asymptotic Normality and Confidence Sets

Exponential Lévy Model

Exponential Lévy Model (Merton 1976)

Let (e−rtSt , t ≥ 0) be a martingale on a filtered probability space(Ω,F ,Q, (Ft)), where r ≥ 0 is the riskless interest rate.

Let St = S0ert+Xt with a Lévy process Xt for t ≥ 0,

where S0 > 0 is the present value.

Nonparametric estimation of the Lévy triplet (σ2, γ, ν) from option data(Cont & Tankov 2004)

Aim: Confidence intervals and confidence sets

Restriction: Let (Xt) have finite intensity and an absolutely continuousjump measure.

Exponential Lévy Model Method and Known Results Asymptotic Normality and Confidence Sets

Lévy Processes

A Lévy process (Xt , t ≥ 0) is astochastically continuous processwith independent and stationaryincrements, X0 = 0.

Lévy processes are characterized by their Lévy triplets (σ2, γ, ν), withvolatility σ ≥ 0, drift γ ∈ R, jump measure ν and intensity λ = ν(R).

Lévy-Khintchine representation:

ϕT (u) := E[e iuXT ] = exp(T

(−σ

2u2

2+ iγu +

∫ ∞−∞

(e iux − 1)ν(x)dx))

.

Exponential Lévy Model Method and Known Results Asymptotic Normality and Confidence Sets

Observations

C(K ,T ) := Prices of European call options,P(K ,T ) := Prices of European put options, K strike, T maturity.

Put-call parity: C(K ,T )− P(K ,T ) = S0 − e−rTK .

Substitute K by x := log(K/S0)− rT .

Define the option function O by:

O(x) :={

S−10 C(x ,T ), x ≥ 0,S−10 P(x ,T ), x < 0.

Observations:

Oj = O(xj) + �j ,

�j independent, E[�j ] = 0 and supj E[�4j ]

Exponential Lévy Model Method and Known Results Asymptotic Normality and Confidence Sets

Description of the Method

Putting µ(x) := exν(x) and FO(u) :=∫∞−∞ e

iuxO(x)dx , from an optionpricing formula (Carr & Madan 1999) and from the Lévy-Khintchinerepresentation follows:

ψ(u) :=1

Tlog(

1 + iu(1 + iu)FO(u))

= −σ2

2u2 + i(σ2 + γ)u + (σ2/2 + γ − λ) + Fµ(u).

Method (Belomestny & Reiß 2006):

1. Interpolate the data (Oj) to obtain a function O�(x).2. Calculate ψ�(u) with FO� instead of FO.3. Determine σ̂2, γ̂, λ̂ from the coefficients of the quadratic

polynomial. F µ̂ is given by the remainder.

Exponential Lévy Model Method and Known Results Asymptotic Normality and Confidence Sets

Definition of σ̂2

ψ�(u) is an empirical version of

ψ(u) = −σ2

2u2 + i(σ2 + γ)u +

(σ2/2 + γ − λ

)+ Fµ(u).

Let wUσ be a weight function such that∫ U−U

−u2

2wUσ (u)du = 1,

∫ U−U

wUσ (u)du = 0, wUσ (u) = U

−3wσ(u/U).

Regularization by spectral cut-off for |u| > U:

σ̂2 :=

∫ U−U

Re(ψ�(u))wUσ (u)du,

= σ2 +

∫ U−U

Re(Fµ(u))wUσ (u)du︸ ︷︷ ︸approximation error

+

∫ U−U

Re(ψ�(u)− ψ(u))wUσ (u)du︸ ︷︷ ︸stochastic error

.

Exponential Lévy Model Method and Known Results Asymptotic Normality and Confidence Sets

Known Results

For these estimators holds (Belomestny & Reiß 2006):

• The Lévy triplet is estimated consistently.

• In general the rates are logarithmic. If σ = 0 is known, the rates arepolynomial.

• The rates depend on the smoothness s of µ.

• The rates are optimal in the minimax sense.

Exponential Lévy Model Method and Known Results Asymptotic Normality and Confidence Sets

Real Data: DAX options

Figure: Estimated Jump Densities by DAX options May 2008

Exponential Lévy Model Method and Known Results Asymptotic Normality and Confidence Sets

Continuous Observations

It is easier to work in the Gaussian white noise model, where O isobserved continuously:

dO�(x) = O(x)dx + � δ(x)dW (x),

with Brownian motion W , δ ∈ L2(R) and � > 0.

There is an asymptotic equivalence between nonparametric regressionand the Gaussian white noise model.

Exponential Lévy Model Method and Known Results Asymptotic Normality and Confidence Sets

Theorem (Asymptotic Normality for σ = 0)

If

• �U(�)5/2 → 0 as �→ 0 (small variance),• �U(�)(2s+5)/2 →∞ as �→ 0 (undersmoothing),

then

1

�

U(�)−1/2(γ̂ − γ)U(�)−3/2(λ̂− λ)

U(�)−5/2(µ̂(x1)− µ(x1))...

U(�)−5/2(µ̂(xn)− µ(xn))

d−→

d(0)∫ 1

0u2wγ(u)dV0(u)

d(0)∫ 1

0u2wλ(u)dW0(u)

d(x1)∫ 1

0u2wµ(u)dWx1 (u)/2π

...

d(xn)∫ 1

0u2wµ(u)dWxn(u)/2π

where V0,W0,Wx1 . . . ,Wxn are independent Brownian motions andd(x) := 2

√π δ(x + Tγ)T−1 exp(T (λ− γ)).

Exponential Lévy Model Method and Known Results Asymptotic Normality and Confidence Sets

Theorem (Asymptotic Normality for σ > 0)

If

• �U(�)2√

log(U(�)) exp(Tσ2U(�)2/2)→ 0 as �→ 0 (small variance),• �U(�)s+1 exp(Tσ2U(�)2/2)→∞ as �→ 0 (undersmoothing),

then

1

�eTσ2U(�)2/2

U(�)2(σ̂2 − σ2)

U(�)(γ̂ − γ)(λ̂− λ)

U(�)−1(µ̂(x)− µ(x))

−

d wσ(1)W�d wγ(1)V�d wλ(1)W�d wµ(1)Z�(x)/2π

P−→ 0,where W� and V� are normal random variables,(

W�V�

)d−→ N(0, I2),

Z�(x) := cos(U(�)x)W� + sin(U(�)x)V�,

d :=√

2 ‖δ‖L2(R)σ−2T−2 exp(T (λ− γ − σ2/2)).

Exponential Lévy Model Method and Known Results Asymptotic Normality and Confidence Sets

Strategy of Proof I

The stochastic errors involve the difference:

ψ�(u)− ψ(u) =1

Tlog(

1 +� iu(1 + iu)

ϕT (u − i)

∫ ∞−∞

e iuxδ(x)dW (x)),

Linearization:

L�(u) :=� iu(1 + iu)

TϕT (u − i)

∫ ∞−∞

e iuxδ(x)dW (x), Gaussian process,

R�(u) := ψ�(u)− ψ(u)− L�(u), remainder term.

PropositionE[supu∈[−U,U] |L�(u)|

]. �U2

√log(U) exp(Tσ2U2/2) as U →∞.

Exponential Lévy Model Method and Known Results Asymptotic Normality and Confidence Sets

Strategy of Proof II

Stochastic error:∫ 1−1

Re(ψ�(Uu)− ψ(Uu))wσ(u)du

=

∫ 1−1

Re(L�(Uu))wσ(u)du︸ ︷︷ ︸Normal random variable

+

∫ 1−1

Re(R�(Uu))wσ(u)du.

• Derive the asymptotic distribution of the first term.Behaves differently for σ = 0 and σ > 0.

• Second term negligible by Taylor expansion and the bound on thesupremum of the Gaussian process L�.

Exponential Lévy Model Method and Known Results Asymptotic Normality and Confidence Sets

Confidence Sets

For σ = 0 and for σ > 0 known:

Confidence intervals:

lim�→0

infT

P(ρ ∈ Iρ,�) = 1− α

for all ρ ∈ {γ, λ, ν(x)|x ∈ R} from quantiles of the normal distribution.

Confidence sets:

lim�→0

infT

P((γ, λ)> ∈ A�) = 1− α

from quantiles of the chi-square distribution.

Exponential Lévy Model Method and Known Results Asymptotic Normality and Confidence Sets

Confidence Intervals

Figure: True Lévy density with pointwise 95% confidence intervals and 100estimated Lévy densities from a Monte Carlo simulation

Exponential Lévy Model Method and Known Results Asymptotic Normality and Confidence Sets

Conclusion

• Derived joint asymptotic distribution in a nonlinear ill-posed inverseproblem.

• For σ = 0:• variance depends on the noise level δ locally• variance depends on the whole weight function w�• asymptotically independent

• For σ > 0:• variance depends on δ globally• variance depends on w� only through w�(1)• covariances do not converge

• Construction of confidence intervals and confidence sets.

Thank you for your attention!

Exponential Lévy Model Method and Known Results Asymptotic Normality and Confidence Sets

Conclusion

• Derived joint asymptotic distribution in a nonlinear ill-posed inverseproblem.

• For σ = 0:• variance depends on the noise level δ locally• variance depends on the whole weight function w�• asymptotically independent

• For σ > 0:• variance depends on δ globally• variance depends on w� only through w�(1)• covariances do not converge

• Construction of confidence intervals and confidence sets.

Thank you for your attention!

Exponential Lévy Model Method and Known Results Asymptotic Normality and Confidence Sets

Fitted Option Functions

Figure: Option data and fitted option functions, May 29, 2008

Exponential Lévy ModelMethod and Known ResultsAsymptotic Normality and Confidence Sets