A generalized Dupire formula and a stable way to estimate it P. - - PowerPoint PPT Presentation

A generalized Dupire formula and a stable way to estimate it

• P. Mayer

mayer@opt.math.tugraz.at

Institute for Mathematics Graz University of Technology

RICAM Special Semester Concluding Workshop

Linz, 2.12.2008

Based on joint work with D. Belomestny

Outline

Introduction On the way to a formula Robust estimation

2.12.2008 P.Mayer 2 / 21

Outline

Introduction On the way to a formula Robust estimation

2.12.2008 P.Mayer Introduction 3 / 21

The well known Dupire model.

We all know: B-S-M model not capable of explaining vol smile. ⇒ Dupire, Derman & Kani ”engineered” market model, that is: ST = e(r−η)TeXT , with Xt = − t σ2(s, Ss) 2 ds + t σ(s, Ss)dWs. Other way to understand the above: Xt = − t σ2(s, Ss) 2 ds + ˜ W ( t σ2(s, Ss)ds) = Y ( t σ2(s, Ss)ds), where Y (t) = ˜ W (t) − t/2.

2.12.2008 P.Mayer Introduction 4 / 21

The well known Dupire model.

We all know: B-S-M model not capable of explaining vol smile. ⇒ Dupire, Derman & Kani ”engineered” market model, that is: ST = e(r−η)TeXT , with Xt = − t σ2(s, Ss) 2 ds + t σ(s, Ss)dWs. Other way to understand the above: Xt = − t σ2(s, Ss) 2 ds + ˜ W ( t σ2(s, Ss)ds) = Y ( t σ2(s, Ss)ds), where Y (t) = ˜ W (t) − t/2.

2.12.2008 P.Mayer Introduction 4 / 21

Drawbacks of the local volatility model.

√ often very steep volatility structure √ Problems when pricing path-dependent options √ No jumps possible

Possible improvement: L´ evy models Advantages:

√ jump-risk included √ fat tails √ skewed log-returns via asymmetric L´

evy measure

2.12.2008 P.Mayer Introduction 5 / 21

Drawbacks of the local volatility model.

√ often very steep volatility structure √ Problems when pricing path-dependent options √ No jumps possible

Possible improvement: L´ evy models Advantages:

√ jump-risk included √ fat tails √ skewed log-returns via asymmetric L´

evy measure

2.12.2008 P.Mayer Introduction 5 / 21

Why taking Brownian motion?

Carr et al.: local L´ evy market model: ST = e(r−η)TeXT . X time-changed L´ evy process: Xt = Y (A(t)), where A(T) = t

0 a0(St−, t) dt and Y given L´

evy process. Assumptions:

√ ∃δ > 0 s.t. L´

evy measure ν of Y fulfills

• R(yey(1+δ) − y)ν(dy) < ∞.

√ 0 < a ≤ a0 ≤ a < ∞ such that SDE solvable & pdf of S continuous.

Question: How to identify local speed function a0?

2.12.2008 P.Mayer Introduction 6 / 21

Why taking Brownian motion?

Carr et al.: local L´ evy market model: ST = e(r−η)TeXT . X time-changed L´ evy process: Xt = Y (A(t)), where A(T) = t

0 a0(St−, t) dt and Y given L´

evy process. Assumptions:

√ ∃δ > 0 s.t. L´

evy measure ν of Y fulfills

• R(yey(1+δ) − y)ν(dy) < ∞.

√ 0 < a ≤ a0 ≤ a < ∞ such that SDE solvable & pdf of S continuous.

Question: How to identify local speed function a0?

2.12.2008 P.Mayer Introduction 6 / 21

What is the task?

√ Wanted: local speed function a0. √ European options liquid ⇒ Usable for calibration.

Forward: a0 → C(a0)(K, T) := EQ

• e−rT(e(r−η)TeY (A(T)) − K)+

Model price of European option Inverse: ˆ a0 ← CM(K, T) Market price of European option Problems:

√ Existence of ˆ

a0?

√ Stability of ˆ

a0?

√ Computationally feasible algorithm to find ˆ

a0?

2.12.2008 P.Mayer Introduction 7 / 21

What is the task?

√ Wanted: local speed function a0. √ European options liquid ⇒ Usable for calibration.

Forward: a0 → C(a0)(K, T) := EQ

• e−rT(e(r−η)TeY (A(T)) − K)+

Model price of European option Inverse: ˆ a0 ← CM(K, T) Market price of European option Problems:

√ Existence of ˆ

a0?

√ Stability of ˆ

a0?

√ Computationally feasible algorithm to find ˆ

a0?

2.12.2008 P.Mayer Introduction 7 / 21

What is the task?

√ Wanted: local speed function a0. √ European options liquid ⇒ Usable for calibration.

Forward: a0 → C(a0)(K, T) := EQ

• e−rT(e(r−η)TeY (A(T)) − K)+

Model price of European option Inverse: ˆ a0 ← CM(K, T) Market price of European option Problems:

√ Existence of ˆ

a0?

√ Stability of ˆ

a0?

√ Computationally feasible algorithm to find ˆ

a0?

2.12.2008 P.Mayer Introduction 7 / 21

Outline

Introduction On the way to a formula Robust estimation

2.12.2008 P.Mayer On the way to a formula 8 / 21

Getting lsf and call prices in touch.

Using the Tanaka-Meyer formula and taking expectation: erT C(K, T) = (S0 − K)+ + E T (r − η)St−1{St−>K} dt

• +E

T a0(St−, t)σ2 2 δ(St− − K)S2

t− dt

• +E

T 1{St−>K}a0(St−, t) log(

K St− )

−∞

(K − St−ex) ν(dx) dt

• +E

  T 1{St−≤K}a0(St−, t) ∞

log(

K St− )

(St−ex − K) ν(dx) dt   . ⇒ implicit equation for a0.

2.12.2008 P.Mayer On the way to a formula 9 / 21

Can we get a more explicit relation?

Define

√ γ(k, T) = eηT C(ek+(r−η)T, T) √ a(y, T) = a0(ey+(r−η)T, T) √ ψ . . . double exponential tail of L´

evy measure

ψ(z) =  R z

−∞ (ez − ex) ν(dx)

for z < 0 R ∞

z

(ex − ez) ν(dx) for z > 0.

Then ...after some calculations...: F(γT(., T)) F(ψ) + σ2 = F

• (γkk(., T) − γk(., T))a(., T)
• .
• cf. Carr et al. (2004) and Dupire formula:

a = F−1(

1 F(ψ)+σ2 ) ∗ γT

γkk − γk .

2.12.2008 P.Mayer On the way to a formula 10 / 21

Can we get a more explicit relation?

Define

√ γ(k, T) = eηT C(ek+(r−η)T, T) √ a(y, T) = a0(ey+(r−η)T, T) √ ψ . . . double exponential tail of L´

evy measure

ψ(z) =  R z

−∞ (ez − ex) ν(dx)

for z < 0 R ∞

z

(ex − ez) ν(dx) for z > 0.

Then ...after some calculations...: F(γT(., T)) F(ψ) + σ2 = F

• (γkk(., T) − γk(., T))a(., T)
• .
• cf. Carr et al. (2004) and Dupire formula:

a = F−1(

1 F(ψ)+σ2 ) ∗ γT

γkk − γk .

2.12.2008 P.Mayer On the way to a formula 10 / 21

Can we get a more explicit relation?

Define

√ γ(k, T) = eηT C(ek+(r−η)T, T) √ a(y, T) = a0(ey+(r−η)T, T) √ ψ . . . double exponential tail of L´

evy measure

ψ(z) =  R z

−∞ (ez − ex) ν(dx)

for z < 0 R ∞

z

(ex − ez) ν(dx) for z > 0.

Then ...after some calculations...: F(γT(., T)) F(ψ) + σ2 = F

• (γkk(., T) − γk(., T))a(., T)
• .
• cf. Carr et al. (2004) and Dupire formula:

a = F−1(

1 F(ψ)+σ2 ) ∗ γT

γkk − γk .

2.12.2008 P.Mayer On the way to a formula 10 / 21

How does F(ψ) look like?

Denoting ˆ k(ω) = ∞

−∞

• ey i ω − 1 − y i ω
• ν(dy),

pure-jump term part in the L´ evy-Khintchine formula associated to the L´ evy density ν with the identity as ”‘truncation function”’. Fourier transform of ψ: F(ψ)(ω) = ˆ k(− i −ω) + (i ω − 1)ˆ k(− i) √ 2π i ω(i ω − 1) .

2.12.2008 P.Mayer On the way to a formula 11 / 21

How does F(ψ) look like?

Denoting ˆ k(ω) = ∞

−∞

• ey i ω − 1 − y i ω
• ν(dy),

pure-jump term part in the L´ evy-Khintchine formula associated to the L´ evy density ν with the identity as ”‘truncation function”’. Fourier transform of ψ: F(ψ)(ω) = ˆ k(− i −ω) + (i ω − 1)ˆ k(− i) √ 2π i ω(i ω − 1) .

2.12.2008 P.Mayer On the way to a formula 11 / 21

Some facts for pure jump processes.

For pure jump-processes:

√ F(ψ)(ω) = 0 so division justified. √ for the asymptotic

Proposition

The asymptotic of the Fourier transformed double exponential tail can be calculated as follows:

• 1

F(ψ)(ω)

• = O
• |ω|min(1,2−β)

for |ω| → ∞, where β = sup{β : |x|−β = o(ν(R/[−x, x])) for x → 0}.

2.12.2008 P.Mayer On the way to a formula 12 / 21

An example

Y tempered stable process ν(x) = e−λ+x

x1+α+

for x>0 eλ−x

|x|1+α−

for x<0. Then for α± = 0, 1 and for simplicity λ = λ− = λ+ and α = α+ = α− ˆ k(ω) = 2λαΓ(−α) ((1 − iω/λ)α − 1 + αiω/λ) . and

F(ψ)(ω) = ˆ k(− i −ω) + (i ω − 1)ˆ k(− i) √ 2π i ω(i ω − 1) = 2λαΓ(−α) √ 2π i ω(i ω − 1) n (1 − 1/λ + i ω/λ)α − (1 − 1/λ)α − i ω (1 − (1 − 1/λ)α)

• .

Hence: 1/F(ψ)(ω) ∼ 1 for ω → 0, |1/F(ψ)(ω)| ∼ |ω|min(2−α,1) for |ω| → ∞

2.12.2008 P.Mayer On the way to a formula 13 / 21

Outline

Introduction On the way to a formula Robust estimation

2.12.2008 P.Mayer Robust estimation 14 / 21

Why is the talk not over yet?

Given data: γM

n = γ(kn, Tn) + σnεn,

n = 1, . . . , N, εn iid random variables with zero mean and bounded variance. Note: a = F−1(

1 F(ψ)+σ2 ) ∗ γT

γkk − γk unstable, if γ substituted by noisy observations. ⇒ Ill-posed problem ⇒ Regularized estimation needed!

2.12.2008 P.Mayer Robust estimation 15 / 21

Why is the talk not over yet?

Given data: γM

n = γ(kn, Tn) + σnεn,

n = 1, . . . , N, εn iid random variables with zero mean and bounded variance. Note: a = F−1(

1 F(ψ)+σ2 ) ∗ γT

γkk − γk unstable, if γ substituted by noisy observations. ⇒ Ill-posed problem ⇒ Regularized estimation needed!

2.12.2008 P.Mayer Robust estimation 15 / 21

What can happen when problems are ill-posed?

Calibration Problem: Calculate (deterministic) short rate from given bond-prices (ansatz: piecewise constant, noiselevel < 1%) yearly observations used

5 10 15 20 25 0.04 0.05 0.06 0.07 0.08

2.12.2008 P.Mayer Robust estimation 16 / 21

What can happen when problems are ill-posed?

Calibration Problem: Calculate (deterministic) short rate from given bond-prices (ansatz: piecewise constant, noiselevel < 1%) half-yearly observations used

5 10 15 20 25 0.04 0.05 0.06 0.07 0.08

2.12.2008 P.Mayer Robust estimation 16 / 21

What can happen when problems are ill-posed?

Calibration Problem: Calculate (deterministic) short rate from given bond-prices (ansatz: piecewise constant, noiselevel < 1%) monthly observations used

5 10 15 20 25 0.04 0.05 0.06 0.07 0.08

2.12.2008 P.Mayer Robust estimation 16 / 21

Estimating the term structure

∃ γT,N(k, T) estimator for γT(k, T) with γT,N(k, T) − γT(k, T) = E1(k, T) + E2(k, T), where E1 stochastic error with E(E1) = 0 and E2 = E(γT,N − γT). ∃ a sequence KN with KN → ∞ as N → ∞ such that

∆N := sup

t∈[0,Tmax]

Z KN

−KN

|E2(k, t)| dk and ΣN := sup

t∈[0,Tmax]

Z KN

−KN

Z KN

−KN

| Cov(E1(k, t), E1(k′, t))| dk dk′

satisfy Error: ∆N → 0, ∆N e−δKN, ΣN → 0, N → ∞.

2.12.2008 P.Mayer Robust estimation 17 / 21

Estimating the term structure

∃ γT,N(k, T) estimator for γT(k, T) with γT,N(k, T) − γT(k, T) = E1(k, T) + E2(k, T), where E1 stochastic error with E(E1) = 0 and E2 = E(γT,N − γT). ∃ a sequence KN with KN → ∞ as N → ∞ such that

∆N := sup

t∈[0,Tmax]

Z KN

−KN

|E2(k, t)| dk and ΣN := sup

t∈[0,Tmax]

Z KN

−KN

Z KN

−KN

| Cov(E1(k, t), E1(k′, t))| dk dk′

satisfy Error: ∆N → 0, ∆N e−δKN, ΣN → 0, N → ∞.

2.12.2008 P.Mayer Robust estimation 17 / 21

Regularizing the estimation: Spectral cut-off

F((γkk − γk)a)T,N = F(ˆ γT,N) F(ψ) + σ2 Kh with e.g. Kh(x) = max(1 − h|x|, 0) and ˆ γT,N a capped estimator. Denote ρ(x) = (γkk − γk)a(x).

Theorem (Asymptotic)

For h → 0 (and β = 2 for σ2 > 0)

E » F −1 „ F(ˆ γT,N) F(ψ) + σ2 Kh « (x) − ρ(x) –2 ∆2

N(1/h)min(4,4−2β) +ΣN(1/h)min(3,5−2β) +h8/3.

Note: F−1 F(ˆ γT,N) F(ψ) + σ2 Kh

• = F−1
• Kh

F(ψ) + σ2

• ∗ ˆ

γT,N

2.12.2008 P.Mayer Robust estimation 18 / 21

Regularizing the estimation: Spectral cut-off

F((γkk − γk)a)T,N = F(ˆ γT,N) F(ψ) + σ2 Kh with e.g. Kh(x) = max(1 − h|x|, 0) and ˆ γT,N a capped estimator. Denote ρ(x) = (γkk − γk)a(x).

Theorem (Asymptotic)

For h → 0 (and β = 2 for σ2 > 0)

E » F −1 „ F(ˆ γT,N) F(ψ) + σ2 Kh « (x) − ρ(x) –2 ∆2

N(1/h)min(4,4−2β) +ΣN(1/h)min(3,5−2β) +h8/3.

Note: F−1 F(ˆ γT,N) F(ψ) + σ2 Kh

• = F−1
• Kh

F(ψ) + σ2

• ∗ ˆ

γT,N

2.12.2008 P.Mayer Robust estimation 18 / 21

Summary and numerical implementation

3 steps of the method:

√ First estimate time derivative by statistical method √ Use cut-off to robustly estimate Fourier-transform of ρ √ If local speed function explicitly needed: use stable method to

estimate pricing density Numerical implementation nearly explicitly, i.e. neither PIDE nor

• ptimization has to be performed.

2.12.2008 P.Mayer Robust estimation 19 / 21

Conclusions

√ Generalized time changed model. √ Local L´

evy instead of local volatility model.

√ “Dupire formula” for time-change of general L´

evy process.

√ Calibration using formula can be robustified. 2.12.2008 P.Mayer Robust estimation 20 / 21

Thank you for your attention!

2.12.2008 P.Mayer Robust estimation 21 / 21