Introduction to dual quantization and first applications Gilles - - PowerPoint PPT Presentation

Introduction to dual quantization and first applications

Gilles Pag` es and Benedikt Wilbertz

LPMA-Universit´ e Pierre et Marie Curie

New advances in BSDE’s for financial engineering and applications October 25-28, 2010

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Introduction to Optimal Quantization History

What is Quantization?

Has its origin in the fields of signal processing in the late 1940’s Describes the discretization of a random signal and analyses the recovery of the original signal from the discrete one Examples: Pulse-Code-Modulation(PCM), JPEG-Compression Extensive Survey about the IEEE-History: [Gray/Neuhoff ’98] Mathematical Foundation of Quantization Theory: [Graf/Luschgy ’00]

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Introduction to Optimal Quantization Definition

Definition of Optimal Quantization

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Introduction to Optimal Quantization Definition

Definition of Optimal Quantization

Let X : (Ω, S, P) → (Rd, Bd) be a random vector with values in a normed space (Rd, ·) and EXp < ∞ for some p ∈ [1, ∞).

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Introduction to Optimal Quantization Definition

Definition of Optimal Quantization

Let X : (Ω, S, P) → (Rd, Bd) be a random vector with values in a normed space (Rd, ·) and EXp < ∞ for some p ∈ [1, ∞). The p-th quantization error for a grid Γ ⊂ Rd with size |Γ| ≤ n, n ∈ N is given by ep(X; Γ) = E dist(X, Γ)p = E min

x∈ΓX − xp.

(1)

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Introduction to Optimal Quantization Definition

Definition of Optimal Quantization

Let X : (Ω, S, P) → (Rd, Bd) be a random vector with values in a normed space (Rd, ·) and EXp < ∞ for some p ∈ [1, ∞). The p-th quantization error for a grid Γ ⊂ Rd with size |Γ| ≤ n, n ∈ N is given by ep(X; Γ) = E dist(X, Γ)p = E min

x∈ΓX − xp.

(1) The optimal quantization problem consists in minimizing (1) over all grids of size |Γ| ≤ n. We define the optimal quantization error of level n as ep

n(X) := inf

• E min

x∈ΓX − xp : Γ ⊂ Rd, |Γ| ≤ n

• .

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Introduction to Optimal Quantization Quantization Rates/Zador Theorem

Rates of Optimal Quantization

The sharp asymptotics for the optimal quantization error are known from Zador’s theorem, which reads in its final version as follows:

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Introduction to Optimal Quantization Quantization Rates/Zador Theorem

Rates of Optimal Quantization

The sharp asymptotics for the optimal quantization error are known from Zador’s theorem, which reads in its final version as follows: Theorem (Zador, Kiefer, Bucklew & Wise, Graf & Luschgy, cf. [Graf/Luschgy ’00]) Let X ∈ Lr(Rd), r > p and denote by ϕ the λd-density of the absolutely continuous part of PX. Then lim

n→∞ np/d · ep n(X) = Qp,· ·

• Rdϕd/(d+p) dλd

(d+p)/d where Qp,· = limn→∞ np/d · ep

n

• U([0, 1]d)
• .

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Introduction to Optimal Quantization Voronoi Quantizer

Voronoi-Quantization

Given a quantization grid Γ = {x1, x2, . . . , xn}, we can easily construct a discretization of the random vector X:

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Introduction to Optimal Quantization Voronoi Quantizer

Voronoi-Quantization

Given a quantization grid Γ = {x1, x2, . . . , xn}, we can easily construct a discretization of the random vector X: Let

• Ci(Γ)
• 1≤i≤n be a Voronoi partition of Rd generated by Γ, i.e.
• Ci(Γ)
• is a Borel partition of Rd satisfying

Ci(Γ) ⊂

• z ∈ Rd : z − xi ≤ min

1≤j≤nz − xj

• .

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Introduction to Optimal Quantization Voronoi Quantizer

Voronoi-Quantization

Given a quantization grid Γ = {x1, x2, . . . , xn}, we can easily construct a discretization of the random vector X: Let

• Ci(Γ)
• 1≤i≤n be a Voronoi partition of Rd generated by Γ, i.e.
• Ci(Γ)
• is a Borel partition of Rd satisfying

Ci(Γ) ⊂

• z ∈ Rd : z − xi ≤ min

1≤j≤nz − xj

• .

Let πΓ : Rd → Γ the Nearest Neighbor projection, z →

n

• i=1

xi1Ci(Γ)(z).

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Introduction to Optimal Quantization Voronoi Quantizer

Voronoi-Quantization

Given a quantization grid Γ = {x1, x2, . . . , xn}, we can easily construct a discretization of the random vector X: Let

• Ci(Γ)
• 1≤i≤n be a Voronoi partition of Rd generated by Γ, i.e.
• Ci(Γ)
• is a Borel partition of Rd satisfying

Ci(Γ) ⊂

• z ∈ Rd : z − xi ≤ min

1≤j≤nz − xj

• .

Let πΓ : Rd → Γ the Nearest Neighbor projection, z →

n

• i=1

xi1Ci(Γ)(z). ⇒ We define the Voronoi Quantization as

• XΓ = πΓ(X) =

n

• i=1

xi1Ci(Γ)(X).

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Introduction to Optimal Quantization Voronoi Quantizer

Voronoi-Quantization

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Introduction to Optimal Quantization Voronoi Quantizer

Voronoi-Quantization

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Introduction to Optimal Quantization Voronoi Quantizer

Voronoi-Quantization

X(ω)

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Introduction to Optimal Quantization Voronoi Quantizer

Voronoi-Quantization

X(ω)

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Introduction to Optimal Quantization Voronoi Quantizer

Voronoi-Quantization

One easily shows ep

n(X) = inf

• EX −

Xp :

• X ∈ Lp(Rd), |

X(Ω)| ≤ n

• .

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Introduction to Optimal Quantization Voronoi Quantizer

Voronoi-Quantization

One easily shows ep

n(X) = inf

• EX −

Xp :

• X ∈ Lp(Rd), |

X(Ω)| ≤ n

• .

⇒ The Voronoi Quantization XΓ provides an optimal Lp-mean discretization of X as soon as Γ is an optimal quantization grid for X.

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Introduction to Optimal Quantization Voronoi Quantizer

Voronoi-Quantization

One easily shows ep

n(X) = inf

• EX −

Xp :

• X ∈ Lp(Rd), |

X(Ω)| ≤ n

• .

⇒ The Voronoi Quantization XΓ provides an optimal Lp-mean discretization of X as soon as Γ is an optimal quantization grid for X. A further characterization for the optimal quantization error is given by ep

n(X) = inf

• EX − f(X)p : f : Rd → Rd Borel mb, |f(Rd)| ≤ n
• ,

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Introduction to Optimal Quantization Voronoi Quantizer

Voronoi-Quantization

One easily shows ep

n(X) = inf

• EX −

Xp :

• X ∈ Lp(Rd), |

X(Ω)| ≤ n

• .

⇒ The Voronoi Quantization XΓ provides an optimal Lp-mean discretization of X as soon as Γ is an optimal quantization grid for X. A further characterization for the optimal quantization error is given by ep

n(X) = inf

• EX − f(X)p : f : Rd → Rd Borel mb, |f(Rd)| ≤ n
• ,

⇒ The Nearest Neighbor projection is the coding rule, which yields the smallest Lp-mean approximation error for X.

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Introduction to Optimal Quantization Optimal Quantizers

Figure: A Quantizer for N(0, I2) of size 500 in (R2, ·2).

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Quantization and Cubature A Cubature formulae

Quantization for Cubature

Application as Cubature formula Assume that we have access to the Voronoi-Cell weights wi(Γ) := P(X ∈ Ci(Γ)).

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Quantization and Cubature A Cubature formulae

Quantization for Cubature

Application as Cubature formula Assume that we have access to the Voronoi-Cell weights wi(Γ) := P(X ∈ Ci(Γ)). ⇒ The computation of EF( XΓ) for some Lipschitz continuous F : Rd → R becomes straightforward: EF( XΓ) = EF n

• i=1

xi1Ci(Γ)(X)

• =
• i=1

wi(Γ)F(xi).

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Quantization and Cubature A Cubature formulae

Quantization for Cubature

Application as Cubature formula Assume that we have access to the Voronoi-Cell weights wi(Γ) := P(X ∈ Ci(Γ)). ⇒ The computation of EF( XΓ) for some Lipschitz continuous F : Rd → R becomes straightforward: EF( XΓ) = EF n

• i=1

xi1Ci(Γ)(X)

• =
• i=1

wi(Γ)F(xi). As a first error estimate, we clearly have |EF(X) − EF( XΓ)| ≤ [F]Lip EX − XΓ.

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Quantization and Cubature A Cubature formulae

Quantization for Cubature

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Quantization and Cubature A Cubature formulae

Quantization for Cubature

Second order rate If F ∈ C1

Lip and the grid Γ is a stationary, i.e.

• XΓ = E(X|

XΓ), then a Taylor expansion yields |EF(X) − EF( XΓ)| ≤ [F ′]Lip · EX − XΓ2.

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Quantization and Cubature A Cubature formulae

Quantization for Cubature

Second order rate If F ∈ C1

Lip and the grid Γ is a stationary, i.e.

• XΓ = E(X|

XΓ), then a Taylor expansion yields |EF(X) − EF( XΓ)| ≤ [F ′]Lip · EX − XΓ2. Furthermore, if F is convex, then Jensen’s inequality implies for stationary Γ EF( XΓ) ≤ EF(X).

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Quantization and Cubature Applications

Further Applications

Applications for optimal quantization grids

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Quantization and Cubature Applications

Further Applications

Applications for optimal quantization grids Approximation of conditional expectations in non-linear problems by means of the Backward Dynamic Programming Principle

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Quantization and Cubature Applications

Further Applications

Applications for optimal quantization grids Approximation of conditional expectations in non-linear problems by means of the Backward Dynamic Programming Principle Obstacle Problems: Valuation of Bermudan and American options ([Bally/Pag` es ’03])

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Quantization and Cubature Applications

Further Applications

Applications for optimal quantization grids Approximation of conditional expectations in non-linear problems by means of the Backward Dynamic Programming Principle Obstacle Problems: Valuation of Bermudan and American options ([Bally/Pag` es ’03]) δ-Hedging for American options ([Bally/Pag` es/Printems ’05])

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Quantization and Cubature Applications

Further Applications

Applications for optimal quantization grids Approximation of conditional expectations in non-linear problems by means of the Backward Dynamic Programming Principle Obstacle Problems: Valuation of Bermudan and American options ([Bally/Pag` es ’03]) δ-Hedging for American options ([Bally/Pag` es/Printems ’05]) Optimal Stochastic Control problems, e.g. Pricing of Swing

• ptions ([Bronstein/Pag`

es/W. ’09] and [Bardou/Bouthemy/Pag` es ’09])

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Dual Quantization Idea

Dual Quantization

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Dual Quantization Idea

Dual Quantization

Idea Do not map X(ω) to its nearest neighbor, but split up the projection randomly between the “surrounding” neighbors of X(ω).

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Dual Quantization Idea

Dual Quantization

Idea Do not map X(ω) to its nearest neighbor, but split up the projection randomly between the “surrounding” neighbors of X(ω).

X(ω)

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Dual Quantization Idea

Dual Quantization

Idea Do not map X(ω) to its nearest neighbor, but split up the projection randomly between the “surrounding” neighbors of X(ω).

X(ω)

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Dual Quantization Idea

Dual Quantization

Idea Do not map X(ω) to its nearest neighbor, but split up the projection randomly between the “surrounding” neighbors of X(ω).

X(ω)

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Dual Quantization Idea

Dual Quantization

Idea Do not map X(ω) to its nearest neighbor, but split up the projection randomly between the “surrounding” neighbors of X(ω).

X(ω)

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Dual Quantization Idea

Dual Quantization

Idea Do not map X(ω) to its nearest neighbor, but split up the projection randomly between the “surrounding” neighbors of X(ω).

1 − λ λ λ X(ω)

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Dual Quantization Idea

Dual Quantization

X(ω)

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Dual Quantization Idea

Dual Quantization

X(ω)

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Dual Quantization Idea

Dual Quantization

X(ω)

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Dual Quantization Stationary Operators

Ideas behind Dual Quantization

Suppose that τ = {t1, . . . , td+1} ⊂ Rd spans a d-simplex in Rd, i.e. t1, . . . , td+1 are affinely independent.

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Dual Quantization Stationary Operators

Ideas behind Dual Quantization

Suppose that τ = {t1, . . . , td+1} ⊂ Rd spans a d-simplex in Rd, i.e. t1, . . . , td+1 are affinely independent. Moreover, let U ∼ U[0, 1] be defined on some exogenous probability space (Ω0, S0, P0). Denoting by λ(ξ) the barycentric coordinate of ξ ∈ conv{τ}, we define a dual quantization operator J U

τ : conv{τ} → τ as

ξ →

d+1

• i=1

ti1i−1

• j=1

λj(ξ) ≤ U <

i

• j=1

λj(ξ)

.

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Dual Quantization Stationary Operators

Ideas behind Dual Quantization

Suppose that τ = {t1, . . . , td+1} ⊂ Rd spans a d-simplex in Rd, i.e. t1, . . . , td+1 are affinely independent. Moreover, let U ∼ U[0, 1] be defined on some exogenous probability space (Ω0, S0, P0). Denoting by λ(ξ) the barycentric coordinate of ξ ∈ conv{τ}, we define a dual quantization operator J U

τ : conv{τ} → τ as

ξ →

d+1

• i=1

ti1i−1

• j=1

λj(ξ) ≤ U <

i

• j=1

λj(ξ)

. This operator satisfies a mean preserving property: E0

• J U

τ (ξ)

• =

d+1

• i=1

λi(ξ) · ti = ξ, ∀ξ ∈ conv{τ}. (2)

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Dual Quantization Stationary Operators

Ideas behind Dual Quantization

Suppose that τ = {t1, . . . , td+1} ⊂ Rd spans a d-simplex in Rd, i.e. t1, . . . , td+1 are affinely independent. Moreover, let U ∼ U[0, 1] be defined on some exogenous probability space (Ω0, S0, P0). Denoting by λ(ξ) the barycentric coordinate of ξ ∈ conv{τ}, we define a dual quantization operator J U

τ : conv{τ} → τ as

ξ →

d+1

• i=1

ti1i−1

• j=1

λj(ξ) ≤ U <

i

• j=1

λj(ξ)

. This operator satisfies a mean preserving property: E0

• J U

τ (ξ)

• =

d+1

• i=1

λi(ξ) · ti = ξ, ∀ξ ∈ conv{τ}. (2) Similarly, we can construct such an operator for any triangulation on a grid Γ = {x1, . . . , xn}, so that (2) holds for any ξ ∈ conv{Γ}.

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Dual Quantization Stationary Operators

Stationarity

Motivated by this observation, we call a random splitting operator JΓ : Ω0 × Rd → Γ for a grid Γ ⊂ Rd intrinsic stationary, if E0

• JΓ(ξ)
• = ξ,

∀ξ ∈ conv{Γ}.

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Dual Quantization Stationary Operators

Stationarity

Motivated by this observation, we call a random splitting operator JΓ : Ω0 × Rd → Γ for a grid Γ ⊂ Rd intrinsic stationary, if E0

• JΓ(ξ)
• = ξ,

∀ξ ∈ conv{Γ}. The deeper meaning of this definition is revealed by the following Proposition. Proposition JΓ is intrinsic stationary, if and only if it satisfies the stationarity condition EP⊗P0

• JΓ(Y )|Y
• = Y

for any r.v. Y : (Ω, S, P) → (Rd, Bd) with supp(PY ) ⊂ conv{Γ}.

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Dual Quantization Stationary Operators

Stationarity

Motivated by this observation, we call a random splitting operator JΓ : Ω0 × Rd → Γ for a grid Γ ⊂ Rd intrinsic stationary, if E0

• JΓ(ξ)
• = ξ,

∀ξ ∈ conv{Γ}. The deeper meaning of this definition is revealed by the following Proposition. Proposition JΓ is intrinsic stationary, if and only if it satisfies the stationarity condition EP⊗P0

• JΓ(Y )|Y
• = Y

for any r.v. Y : (Ω, S, P) → (Rd, Bd) with supp(PY ) ⊂ conv{Γ}. Note that this kind of stationarity now is very robust, since it holds by construction for any r.v. Y with support in Γ.

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Dual Quantization Stationary Operators

Stationarity II

As in the case of regular quantization, this kind of stationarity also yields a second order bound.

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Dual Quantization Stationary Operators

Stationarity II

As in the case of regular quantization, this kind of stationarity also yields a second order bound. Proposition (a) Let F ∈ C1

Lip, Γ ⊂ Rd and JΓ be intrinsic stationary. Then it holds

for any r.v. Y ∈ L2(P) with supp(PY ) ⊂ conv{Γ}, |EF(Y ) − EF(JΓ(Y )| ≤ [F ′]Lip · EY − JΓ(Y )2.

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Dual Quantization Stationary Operators

Stationarity II

As in the case of regular quantization, this kind of stationarity also yields a second order bound. Proposition (a) Let F ∈ C1

Lip, Γ ⊂ Rd and JΓ be intrinsic stationary. Then it holds

for any r.v. Y ∈ L2(P) with supp(PY ) ⊂ conv{Γ}, |EF(Y ) − EF(JΓ(Y )| ≤ [F ′]Lip · EY − JΓ(Y )2. (b) If F is convex, then Jensen’s inequality implies EF(JΓ(X)) ≥ EF(X)

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Dual Quantization Definition Dual Quantization

Question What is the best approximation, which can be achieved by an intrinsic stationary operator JΓ for a given grid Γ of size n ∈ N?

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Dual Quantization Definition Dual Quantization

Question What is the best approximation, which can be achieved by an intrinsic stationary operator JΓ for a given grid Γ of size n ∈ N? Problem: The grid Γ gives raise to many possible triangulations.

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Dual Quantization Definition Dual Quantization

Question What is the best approximation, which can be achieved by an intrinsic stationary operator JΓ for a given grid Γ of size n ∈ N? Problem: The grid Γ gives raise to many possible triangulations. We aim at selecting the triangulation with the lowest p-inertia i.e. to solve ∀ ξ ∈ conv(Γ), F p(ξ; Γ) = min

λ∈Rn n

• i=1

λi ξ − xip

s.t. [ x1 ... xn 1 ... 1 ]λ= ξ 1

• , λ≥0

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Dual Quantization Definition Dual Quantization

Question What is the best approximation, which can be achieved by an intrinsic stationary operator JΓ for a given grid Γ of size n ∈ N? Problem: The grid Γ gives raise to many possible triangulations. We aim at selecting the triangulation with the lowest p-inertia i.e. to solve ∀ ξ ∈ conv(Γ), F p(ξ; Γ) = min

λ∈Rn n

• i=1

λi ξ − xip

s.t. [ x1 ... xn 1 ... 1 ]λ= ξ 1

• , λ≥0

For every ξ ∈ conv(Γ) we choose the best “triangle” in Γ which contains ξ.

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Dual Quantization Definition Dual Quantization

Question What is the best approximation, which can be achieved by an intrinsic stationary operator JΓ for a given grid Γ of size n ∈ N? Problem: The grid Γ gives raise to many possible triangulations. We aim at selecting the triangulation with the lowest p-inertia i.e. to solve ∀ ξ ∈ conv(Γ), F p(ξ; Γ) = min

λ∈Rn n

• i=1

λi ξ − xip

s.t. [ x1 ... xn 1 ... 1 ]λ= ξ 1

• , λ≥0

For every ξ ∈ conv(Γ) we choose the best “triangle” in Γ which contains ξ. ⊲ The optimal p-th dual quantization error is then defined as dp

n(X) = inf

• E F p(X; Γ) : Γ ⊂ Rd, |Γ| ≤ n
• .

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Dual Quantization Definition Dual Quantization

Optimality regions for F p(ξ; Γ)

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Dual Quantization Definition Dual Quantization

Optimality regions for F p(ξ; Γ)

⊲ To design the optimal dual quantization operator matching F p(ξ; Γ), we need optimality regions, counterparts of the Voronoi regions for regular quantization.

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Dual Quantization Definition Dual Quantization

Optimality regions for F p(ξ; Γ)

⊲ To design the optimal dual quantization operator matching F p(ξ; Γ), we need optimality regions, counterparts of the Voronoi regions for regular quantization. ⊲ (λi)1≤i≤n → minλ∈Rn n

i=1 λi ξ − xip s.t. [ x1 ... xn 1 ... 1 ]λ= ξ 1

• , λ≥0

atteins a minimum (at least) at an extremal n-tuple λ∗(ξ) of the convex constraint set.

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Dual Quantization Definition Dual Quantization

Optimality regions for F p(ξ; Γ)

⊲ To design the optimal dual quantization operator matching F p(ξ; Γ), we need optimality regions, counterparts of the Voronoi regions for regular quantization. ⊲ (λi)1≤i≤n → minλ∈Rn n

i=1 λi ξ − xip s.t. [ x1 ... xn 1 ... 1 ]λ= ξ 1

• , λ≥0

atteins a minimum (at least) at an extremal n-tuple λ∗(ξ) of the convex constraint set. Therefore, I∗(ξ) := {i : λ∗

i (ξ) > 0} defines an affinely independent

family (xi)i∈I∗(ξ) which can be completed into a Γ-valued affine basis.

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Dual Quantization Definition Dual Quantization

Optimality regions for F p(ξ; Γ)

⊲ To design the optimal dual quantization operator matching F p(ξ; Γ), we need optimality regions, counterparts of the Voronoi regions for regular quantization. ⊲ (λi)1≤i≤n → minλ∈Rn n

i=1 λi ξ − xip s.t. [ x1 ... xn 1 ... 1 ]λ= ξ 1

• , λ≥0

atteins a minimum (at least) at an extremal n-tuple λ∗(ξ) of the convex constraint set. Therefore, I∗(ξ) := {i : λ∗

i (ξ) > 0} defines an affinely independent

family (xi)i∈I∗(ξ) which can be completed into a Γ-valued affine basis. DI(Γ) =

• ξ ∈ Rd : ∃I∗(ξ) ⊂ I},

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Dual Quantization Definition Dual Quantization PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 19 / 49

Dual Quantization Definition Dual Quantization

• r equivalently in term of linear programming

DI(Γ) =

• ξ ∈ Rd : λI = A−1

I

ξ

1

• ≥ 0 and
• i∈I

λI

i ξ − xip = F p(ξ; Γ)

• ,

where I ∈ I(Γ) =

• J ⊂ {1, . . . , n} : |J| = d + 1, rk(AJ) = d + 1
• and AI denotes the submatrix of

x1 ... xn

1 ... 1

• whose columns are given by

I.

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 19 / 49

Dual Quantization Properties of Dual Quantization

Quadratic Euclidean case

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 20 / 49

Dual Quantization Properties of Dual Quantization

Quadratic Euclidean case

In the case · = |·|2 and p = 2,

• ptimality regions are to Delaunay “triangles” in Γ,

i.e. the sphere spanned by such a d-simplex contains no further point in its interior.

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 20 / 49

Dual Quantization Properties of Dual Quantization

Quadratic Euclidean case

In the case · = |·|2 and p = 2,

• ptimality regions are to Delaunay “triangles” in Γ,

i.e. the sphere spanned by such a d-simplex contains no further point in its interior. The following theorem is an extention of an important theorem by Rajan ([Rajan ’91]). Theorem Let · = |·|2, p = 2, and Γ = {x1, . . . , xn} ⊂ Rd with aff. dim{Γ} = d. (a) If I ∈ I(Γ) defines a Delaunay triangle (or d-simplex), then λI = A−1

I

ξ

1

• provides a solution to F p(ξ; Γ) for every ξ ∈ conv{xj : j ∈ I} i.e.

DI(Γ) = conv{xj : j ∈ I}.

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 20 / 49

Dual Quantization Properties of Dual Quantization

Quadratic Euclidean case

In the case · = |·|2 and p = 2,

• ptimality regions are to Delaunay “triangles” in Γ,

i.e. the sphere spanned by such a d-simplex contains no further point in its interior. The following theorem is an extention of an important theorem by Rajan ([Rajan ’91]). Theorem Let · = |·|2, p = 2, and Γ = {x1, . . . , xn} ⊂ Rd with aff. dim{Γ} = d. (a) If I ∈ I(Γ) defines a Delaunay triangle (or d-simplex), then λI = A−1

I

ξ

1

• provides a solution to F p(ξ; Γ) for every ξ ∈ conv{xj : j ∈ I} i.e.

DI(Γ) = conv{xj : j ∈ I}. ˚

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 20 / 49

Dual Quantization Properties of Dual Quantization

Optimal dual quantization operator

For a Γ = {x1, . . . , xn} ⊂ Rd with aff. dim{Γ} = d,

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 21 / 49

Dual Quantization Properties of Dual Quantization

Optimal dual quantization operator

For a Γ = {x1, . . . , xn} ⊂ Rd with aff. dim{Γ} = d, choose a Borel partition (CI(Γ))I∈I(Γ) of conv{Γ} such that CI(Γ) ⊂ DI(Γ),

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 21 / 49

Dual Quantization Properties of Dual Quantization

Optimal dual quantization operator

For a Γ = {x1, . . . , xn} ⊂ Rd with aff. dim{Γ} = d, choose a Borel partition (CI(Γ))I∈I(Γ) of conv{Γ} such that CI(Γ) ⊂ DI(Γ), let U ∼ U[0, 1] on (Ω0, S0, P0).

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 21 / 49

Dual Quantization Properties of Dual Quantization

Optimal dual quantization operator

For a Γ = {x1, . . . , xn} ⊂ Rd with aff. dim{Γ} = d, choose a Borel partition (CI(Γ))I∈I(Γ) of conv{Γ} such that CI(Γ) ⊂ DI(Γ), let U ∼ U[0, 1] on (Ω0, S0, P0). The optimal dual quantization operator J ∗

Γ is defined as

J ∗

Γ(ξ) =

• I∈I(Γ)

k

• i=1

xi · 1i−1

• j=1

λI

j(ξ) ≤ U< i

• j=1

λI

j(ξ)

• 1CI(Γ)(ξ).

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 21 / 49

Dual Quantization Properties of Dual Quantization

Optimal dual quantization operator

For a Γ = {x1, . . . , xn} ⊂ Rd with aff. dim{Γ} = d, choose a Borel partition (CI(Γ))I∈I(Γ) of conv{Γ} such that CI(Γ) ⊂ DI(Γ), let U ∼ U[0, 1] on (Ω0, S0, P0). The optimal dual quantization operator J ∗

Γ is defined as

J ∗

Γ(ξ) =

• I∈I(Γ)

k

• i=1

xi · 1i−1

• j=1

λI

j(ξ) ≤ U< i

• j=1

λI

j(ξ)

• 1CI(Γ)(ξ).

One easily checks that this operator is intrinsic stationary.

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 21 / 49

Dual Quantization Properties of Dual Quantization

Equivalence of optimal dual quantization

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 22 / 49

Dual Quantization Properties of Dual Quantization

Equivalence of optimal dual quantization

The operator J ∗

Γ then leads to the following characterizations of the

• ptimal dual quantization error:

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 22 / 49

Dual Quantization Properties of Dual Quantization

Equivalence of optimal dual quantization

The operator J ∗

Γ then leads to the following characterizations of the

• ptimal dual quantization error:

Theorem ([Pag` es/W. ’10a]) Let X ∈ Lp(P) and n ∈ N. Then dp

n(X) =

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 22 / 49

Dual Quantization Properties of Dual Quantization

Equivalence of optimal dual quantization

The operator J ∗

Γ then leads to the following characterizations of the

• ptimal dual quantization error:

Theorem ([Pag` es/W. ’10a]) Let X ∈ Lp(P) and n ∈ N. Then dp

n(X) = inf

• EX − JΓ(X)p : JΓ : Ω0 × Rd → Γ is intrinsic stationary,

supp(PX) ⊂ conv{Γ}, |Γ| ≤ n

• PAG`

ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 22 / 49

Dual Quantization Properties of Dual Quantization

Equivalence of optimal dual quantization

The operator J ∗

Γ then leads to the following characterizations of the

• ptimal dual quantization error:

Theorem ([Pag` es/W. ’10a]) Let X ∈ Lp(P) and n ∈ N. Then dp

n(X) = inf

• EX − JΓ(X)p : JΓ : Ω0 × Rd → Γ is intrinsic stationary,

supp(PX) ⊂ conv{Γ}, |Γ| ≤ n

• = inf
• EX −

Y p : Y is a r.v. on (Ω × Ω0, S ⊗ S0, P ⊗ P0), | Y (Ω × Ω0)| ≤ n, E( Y |X) = X

• .

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 22 / 49

Dual Quantization Unbounded support

Extension to unbounded support

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 23 / 49

Dual Quantization Unbounded support

Extension to unbounded support

Since it is not possible to obtain intrinsic stationarity for ξ / ∈ conv{Γ}, we have to limit the claim for stationarity to a subset of supp(PX) in

• rder to extend the dual quantization problem to distributions with

unbounded support.

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 23 / 49

Dual Quantization Unbounded support

Extension to unbounded support

Since it is not possible to obtain intrinsic stationarity for ξ / ∈ conv{Γ}, we have to limit the claim for stationarity to a subset of supp(PX) in

• rder to extend the dual quantization problem to distributions with

unbounded support. We therefore drop the requirement supp(PX) ⊂ conv{Γ} in above theorem

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 23 / 49

Dual Quantization Unbounded support

Extension to unbounded support

Since it is not possible to obtain intrinsic stationarity for ξ / ∈ conv{Γ}, we have to limit the claim for stationarity to a subset of supp(PX) in

• rder to extend the dual quantization problem to distributions with

unbounded support. We therefore drop the requirement supp(PX) ⊂ conv{Γ} in above theorem and set ¯ dp

n(X) = inf

• EX − JΓ(X)p : JΓ is intrinsic stationary, |Γ| ≤ n
• .

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 23 / 49

Dual Quantization Unbounded support

Extension to unbounded support

Since it is not possible to obtain intrinsic stationarity for ξ / ∈ conv{Γ}, we have to limit the claim for stationarity to a subset of supp(PX) in

• rder to extend the dual quantization problem to distributions with

unbounded support. We therefore drop the requirement supp(PX) ⊂ conv{Γ} in above theorem and set ¯ dp

n(X) = inf

• EX − JΓ(X)p : JΓ is intrinsic stationary, |Γ| ≤ n
• .

This means that we use a Nearest Neighbor projection beyond conv{Γ} while preserving stationarity in the interior of conv{Γ}.

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 23 / 49

Dual Quantization Existence

Existence of optimal dual quantizers

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 24 / 49

Dual Quantization Existence

Existence of optimal dual quantizers

Theorem ([Pag` es/W. ’10a]) (a) Let p > 1 and assume that PX has a compact support. Then for every n ≥ d + 1 optimal dual quantizers actually exist, i.e. the dual quantization problem dp

n(X) attains its infimum. Moreover, dp n(X) is

(strictly) decreasing to 0 as n → ∞, if it does not vanish.

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 24 / 49

Dual Quantization Existence

Existence of optimal dual quantizers

Theorem ([Pag` es/W. ’10a]) (a) Let p > 1 and assume that PX has a compact support. Then for every n ≥ d + 1 optimal dual quantizers actually exist, i.e. the dual quantization problem dp

n(X) attains its infimum. Moreover, dp n(X) is

(strictly) decreasing to 0 as n → ∞, if it does not vanish. (b) Let p > 1 and assume that the distribution PX is strongly

• continuous. Then also optimal quantizers for ¯

dp

n(X) exists and ¯

dp

n(X)

is (strictly) decreasing to 0 as n → ∞, if it does not vanish.

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 24 / 49

Dual Quantization Asymptotics

Asymptotic behavior

Theorem ([Pag` es/W. ’10b]) (a) Let X ∈ Lp+(Rd) and denote by ϕ the λd-density of the absolutely continuous part of PX.

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 25 / 49

Dual Quantization Asymptotics

Asymptotic behavior

Theorem ([Pag` es/W. ’10b]) (a) Let X ∈ Lp+(Rd) and denote by ϕ the λd-density of the absolutely continuous part of PX. Then lim

n→∞ np/d · ¯

dp

n(X) = Qd,p,· ·

• Rd ϕd/(d+p) dλd

d+p

d

where Qd,p,· = lim

n→∞ np/d · dp n

• U([0, 1]d)
• .

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 25 / 49

Dual Quantization Asymptotics

Asymptotic behavior

Theorem ([Pag` es/W. ’10b]) (a) Let X ∈ Lp+(Rd) and denote by ϕ the λd-density of the absolutely continuous part of PX. Then lim

n→∞ np/d · ¯

dp

n(X) = Qd,p,· ·

• Rd ϕd/(d+p) dλd

d+p

d

where Qd,p,· = lim

n→∞ np/d · dp n

• U([0, 1]d)
• .

(b) If d = 1, Qd,p,· = 2p+1

p+2 lim n→∞ np/d · ep n

• U([0, 1])
• . If d ≥ 2, ???

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 25 / 49

Dual Quantization Asymptotics

Asymptotic behavior

Sketch of the proof

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 26 / 49

Dual Quantization Asymptotics

Asymptotic behavior

Sketch of the proof Prove existence of the limit for U([0, 1]d)

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 26 / 49

Dual Quantization Asymptotics

Asymptotic behavior

Sketch of the proof Prove existence of the limit for U([0, 1]d) Derive upper and lower bounds for piecewise constant densities (with compact support) on hypercubes

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 26 / 49

Dual Quantization Asymptotics

Asymptotic behavior

Sketch of the proof Prove existence of the limit for U([0, 1]d) Derive upper and lower bounds for piecewise constant densities (with compact support) on hypercubes Use Differentiation of measure to cover the general case (still compact support)

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 26 / 49

Dual Quantization Asymptotics

Asymptotic behavior

Sketch of the proof Prove existence of the limit for U([0, 1]d) Derive upper and lower bounds for piecewise constant densities (with compact support) on hypercubes Use Differentiation of measure to cover the general case (still compact support) Random dual quantization argument (so-called extended Pierce Lemma) to get the unbounded case.

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 26 / 49

Dual Quantization Numerical computations

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 −0,05 0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00 1,05

Figure: Dual Quantization for U([0, 1]2) and n = 8

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Dual Quantization Numerical computations

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 −0,05 0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00 1,05

Figure: Dual Quantization for U([0, 1]2) and n = 12

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Dual Quantization Numerical computations

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 −0,05 0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00 1,05

Figure: Dual Quantization for U([0, 1]2) and n = 13

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Dual Quantization Numerical computations

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 −0,05 0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,45 0,50 0,55 0,60 0,65 0,70 0,75 0,80 0,85 0,90 0,95 1,00 1,05

Figure: Dual Quantization for U([0, 1]2) and n = 16

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Dual Quantization Numerical computations

−3,5 −3,0 −2,5 −2,0 −1,5 −1,0 −0,5 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 −3,50 −3,25 −3,00 −2,75 −2,50 −2,25 −2,00 −1,75 −1,50 −1,25 −1,00 −0,75 −0,50 −0,25 0,00 0,25 0,50 0,75 1,00 1,25 1,50 1,75 2,00 2,25 2,50 2,75 3,00 3,25 3,50

Figure: Dual Quantization for N(0, I2) and N = 250

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 31 / 49

Dual Quantization Numerical computations

Figure: Joint Dual Quantization of the BM and its supremum, N = 250

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Applications Problem description

Numerical Applications

Pricing of Early Exercise Options:

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 33 / 49

Applications Problem description

Numerical Applications

Pricing of Early Exercise Options: Using a Backward-Dynamic-Programming principle for the valuation of early exercise options with underlying Markov dynamics (Xk)1≤k≤N the numerical challenge in this approach consists in the approximation

• f conditional expectations

E

• vk+1(Xk+1)|Xk
• .

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 33 / 49

Applications Problem description

Numerical Applications

Pricing of Early Exercise Options: Using a Backward-Dynamic-Programming principle for the valuation of early exercise options with underlying Markov dynamics (Xk)1≤k≤N the numerical challenge in this approach consists in the approximation

• f conditional expectations

E

• vk+1(Xk+1)|Xk
• .

As in the case of Quantization for numerical cubature, we may replace the Markov chain (Xk) by a Quantization ( Xk), so that the the computation of E

• f(Xk+1)|Xk
• becomes straightforward as

E(f( ˆ Xk+1)| ˆ Xk = xk

i ) = nk+1

• j=1

f(xk+1

j

)πk

ij,

with transition probabilities πk

ij = P( ˆ

Xk+1 = xk+1

j

| ˆ Xk = xk

i ).

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 33 / 49

Applications Problem description

Numerical Applications

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 34 / 49

Applications Problem description

Numerical Applications

For the approximation error the following result can be derived. Proposition If the mappings f : Rd → R and Φf,k : Rd → R, x → E

• f(Xk+1)|Xk = x
• are Lipschitz, then it holds

E(f(Xk+1)|Xk) − E(f( ˆ Xk+1)| ˆ Xk)p ≤ [Φf,k]Lip · Xk − ˆ Xkp + [f]Lip · Xk+1 − ˆ Xk+1p.

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Applications Swing options

Valuation of Swing options

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 35 / 49

Applications Swing options

Valuation of Swing options

Swing options - A common contract in energy markets The right to buy every day a certain quantity of gas/electricity for a given price, where the bought quantity has to respect certain daily and global constraints.

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 35 / 49

Applications Swing options

Valuation of Swing options

Swing options - A common contract in energy markets The right to buy every day a certain quantity of gas/electricity for a given price, where the bought quantity has to respect certain daily and global constraints. The fair premium of such an contract leads to a stochastic control problem (SCP) esssup

• E

n−1

• k=0

qkvk(Xk)|F0

• , qk : (Ω, Fk) → [0, 1], ¯

qn ∈ [Qmin, Qmax]

• for ¯

qk := k−1

l=0 ql.

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 35 / 49

Applications Swing options

Backward Dynamic Programming Principle

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 36 / 49

Applications Swing options

Backward Dynamic Programming Principle

It was shown in [Bardou/Bouthemy/Pag` es ’09] that (SCP) can be solved by the Backward Dynamic Programming Principle with bang-bang control, i.e we set

P n

n ≡ 0

P n

k (Qk) = max

• xvk(Xk) + E(P n

k+1(χn−k−1(Qk, x))|Xk), x ∈ {0, 1} ∩ In−k−1 Qk

• with admissible set IM

Qk := [(Qk min − M)+ ∧ 1, Qk max ∧ 1] and

χM(Qk, x) :=

• (Qk

min − x)+, (Qk max − x) ∧ M

• .

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 36 / 49

Applications Swing options

Backward Dynamic Programming Principle

It was shown in [Bardou/Bouthemy/Pag` es ’09] that (SCP) can be solved by the Backward Dynamic Programming Principle with bang-bang control, i.e we set

P n

n ≡ 0

P n

k (Qk) = max

• xvk(Xk) + E(P n

k+1(χn−k−1(Qk, x))|Xk), x ∈ {0, 1} ∩ In−k−1 Qk

• with admissible set IM

Qk := [(Qk min − M)+ ∧ 1, Qk max ∧ 1] and

χM(Qk, x) :=

• (Qk

min − x)+, (Qk max − x) ∧ M

• .

Then P n

0 (Qmin, Qmax) is a solution to (SCP).

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Applications Swing options

Backward Dynamic Programming Principle

Using the Quantization ( Xk) we define an approximation of (Pk) as

ˆ P n

n ≡ 0

ˆ P n

k (Qk) = max

• xvk( ˆ

Xk) + E( ˆ P n

k+1(χn−k−1(Qk, x))| ˆ

Xk), x ∈ {0, 1} ∩ In−k−1

Qk

• PAG`

ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 37 / 49

Applications Swing options

Backward Dynamic Programming Principle

Using the Quantization ( Xk) we define an approximation of (Pk) as

ˆ P n

n ≡ 0

ˆ P n

k (Qk) = max

• xvk( ˆ

Xk) + E( ˆ P n

k+1(χn−k−1(Qk, x))| ˆ

Xk), x ∈ {0, 1} ∩ In−k−1

Qk

• Under the same assumptions on (Xk) and f = vk as in the above

Proposition about the approximation power of E(f( ˆ Xk+1)| ˆ Xk) one gets |P n

0 (Q) − ˆ

P n

0 (Q)| ≤ C n−1

• k=0

EXk − ˆ Xk for any reasonable initial global constraints Q = (Qmin, Qmax) (see [Bardou/Bouthemy/Pag` es ’10]).

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Applications Swing options

Example: Gaussian 2-factor model

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 38 / 49

Applications Swing options

Example: Gaussian 2-factor model

In this model, the dynamics of the underlying are given as St = s0 exp

• σ1

t e−α1(t−s)dW 1

s + σ2

t e−α2(t−s)dW 2

s − 1

2µt

• for Brownian Motions W 1 and W 2 with some correlation parameter ρ.

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 38 / 49

Applications Swing options

Example: Gaussian 2-factor model

In this model, the dynamics of the underlying are given as St = s0 exp

• σ1

t e−α1(t−s)dW 1

s + σ2

t e−α2(t−s)dW 2

s − 1

2µt

• for Brownian Motions W 1 and W 2 with some correlation parameter ρ.

For a time-step parameter ∆t we consider the 2-dimensional Markov process Xk = k∆t e−α1(k∆t−s)dW 1

s ,

k∆t e−α2(k∆t−s)dW 2

s

• .

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 38 / 49

Applications Swing options

Numerical Results

Example Gaussian 2-factor with parameters s0 = 20, α1 = 1.11, α2 = 5.4, σ1 = 0.36, σ2 = 0.21, ρ = −0.11 and n = 30 exercise days for the swing contract. Results in the Benchmark case of a Call-Strip, i.e. the global consumption constraints are (Qmin, Qmax) = (0, n).

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Applications Swing options

−3,0 −2,5 −2,0 −1,5 −1,0 −0,5 0,0 0,5 1,0 1,5 2,0 2,5 3,0 −2,00 −1,75 −1,50 −1,25 −1,00 −0,75 −0,50 −0,25 0,00 0,25 0,50 0,75 1,00 1,25 1,50 1,75 2,00

Figure: Triangulation for Xn and N = 250 in Gaussian 2-factor model

Picture grid

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Applications Swing options

Swing option: #exercise days: 30, K = 5.0

regular Quantization dual Quantization ref value

2 5 5 0 7 5 100 125 150 175 200 225 250 quantizer size 2.665 2.670 2.675 2.680 2.685 2.690 2.695 2.700 2.705 2.710 2.715 2.720 2.725 2.730 2.735 premium

−0,0125 −0,0100 −0,0075 −0,0050 −0,0025 0,0000 0,0025 0,0050 0,0075 0,0100 0,0125

rel Deviation

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Applications Swing options

Swing option: #exercise days: 30, K = 15.0

regular Quantization dual Quantization ref value

2 5 5 0 7 5 100 125 150 175 200 225 250 quantizer size 900 905 910 915 920 925 930 935 940 945 premium

−0,0250 −0,0225 −0,0200 −0,0175 −0,0150 −0,0125 −0,0100 −0,0075 −0,0050 −0,0025 0,0000 0,0025 0,0050 0,0075 0,0100 0,0125 0,0150 0,0175 0,0200 0,0225 0,0250

rel Deviation

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 42 / 49

Applications Bermudan options

Bermudan options

In the same way we use the BDP-Principle for the valuation of Bermudan options: BDP for Bermudan options

• Vn = ϕtn(

Xn)

• Vk = max
• ϕtk(

Xk); E

• Vk+1
• Xk
• , 0 ≤ k ≤ n − 1,

so that V0 yields an approximation to the Bermudan option premium V0 = esssup{Eϕ(Xτ) : τ is {t0, . . . , tn}-valued stopping time}.

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 43 / 49

Applications Bermudan options

Numerical Results

Example 2-asset Black-Scholes model with s1

0 = s2 0 = 40, r = 0.05, σ1 = 0.2, σ2 = 0.3, ρ = 0.5, K = 40,

for a put on the min, i.e. payoff ϕ(S1

t , S2 t ) = (K − min{S1 t , S2 t })+.

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 44 / 49

Applications Bermudan options

Numerical Results

Example 2-asset Black-Scholes model with s1

0 = s2 0 = 40, r = 0.05, σ1 = 0.2, σ2 = 0.3, ρ = 0.5, K = 40,

for a put on the min, i.e. payoff ϕ(S1

t , S2 t ) = (K − min{S1 t , S2 t })+.

As underlying Markov process we have chosen a 2-dimensional Brownian Motion with correlation ρ.

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 44 / 49

Applications Bermudan options

Numerical Results

Example 2-asset Black-Scholes model with s1

0 = s2 0 = 40, r = 0.05, σ1 = 0.2, σ2 = 0.3, ρ = 0.5, K = 40,

for a put on the min, i.e. payoff ϕ(S1

t , S2 t ) = (K − min{S1 t , S2 t })+.

As underlying Markov process we have chosen a 2-dimensional Brownian Motion with correlation ρ. Reference values were computed using a Boyle-Evnine-Gibbs tree with 10.000 timesteps.

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 44 / 49

Applications Bermudan options

Martingale Adjustment

Bermudan option: #exercise days: 10

regular + martgl adj dual + martgl adj european ref value american ref value bermudan ref value

5 0 100 150 200 250 300 350 400 450 500 Quantizer size 3,750 3,775 3,800 3,825 3,850 3,875 3,900 3,925 3,950 3,975

−3,0 −2,5 −2,0 −1,5 −1,0 −0,5 0,0 0,5 1,0 1,5 2,0 2,5

rel Deviation in %

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Applications Bermudan options

Martingale Adjustment

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 46 / 49

Applications Bermudan options

Conclusion / Outlook

Conclusion / Summary

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 47 / 49

Applications Bermudan options

Conclusion / Outlook

Conclusion / Summary Interesting and challenging extention of regular Quantization

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 47 / 49

Applications Bermudan options

Conclusion / Outlook

Conclusion / Summary Interesting and challenging extention of regular Quantization Provides a stationarity, which holds independently of the choice of the quantization grid

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 47 / 49

Applications Bermudan options

Conclusion / Outlook

Conclusion / Summary Interesting and challenging extention of regular Quantization Provides a stationarity, which holds independently of the choice of the quantization grid Represented in the Euclidean case by the dual concept of Voronoi tesselations: the Delaunay triangulation

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 47 / 49

Applications Bermudan options

Conclusion / Outlook

Conclusion / Summary Interesting and challenging extention of regular Quantization Provides a stationarity, which holds independently of the choice of the quantization grid Represented in the Euclidean case by the dual concept of Voronoi tesselations: the Delaunay triangulation Yields very promising results in first numerical applications

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 47 / 49

Applications Bermudan options

Conclusion / Outlook

Conclusion / Summary Interesting and challenging extention of regular Quantization Provides a stationarity, which holds independently of the choice of the quantization grid Represented in the Euclidean case by the dual concept of Voronoi tesselations: the Delaunay triangulation Yields very promising results in first numerical applications Further applications in optimal grid generation, adaptive grid refinements possible

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 47 / 49

Applications Bermudan options

Conclusion / Outlook

Conclusion / Summary Interesting and challenging extention of regular Quantization Provides a stationarity, which holds independently of the choice of the quantization grid Represented in the Euclidean case by the dual concept of Voronoi tesselations: the Delaunay triangulation Yields very promising results in first numerical applications Further applications in optimal grid generation, adaptive grid refinements possible Application to 3-factor models, etc.

PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 47 / 49

References

References I

• O. Bardou, S. Bouthemy, and G Pag`

es. Optimal Quantization for the Pricing of Swing Options. Applied Mathematical Finance, 16(2):183–217, 2009.

• O. Bardou, S. Bouthemy, and G Pag`

es. When are Swing options bang-bang? International Journal of Theoretical and Applied Finance (IJTAF), 13(06):867–899, 2010.

• V. Bally and G. Pag`

es. Error analysis of the optimal quantization algorithm for obstacle problems. Stochastic Processes and their Applications, 106:1–40(40), July 2003.

• V. Bally, G. Pag`

es, and J. Printems. A quantization tree method for pricing and hedging multidimensional american options. Mathematical Finance, 15:119–168(50), January 2005.

• A. L. Bronstein, G. Pag`

es, and B. Wilbertz. How to speed up the quantization tree algorithm with an application to swing options. to appear in Quantitative Finance, 2009.

• S. Graf and H. Luschgy.

Foundations of Quantization for Probability Distributions. Lecture Notes in Mathematics n01730. Springer, Berlin, 2000. R.M. Gray and D.L. Neuhoff. Quantization. IEEE Trans. Inform., 44:2325–2383, 1998. PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 48 / 49

References

References II

• G. Pag`

es and B. Wilbertz. Intrinsic stationarity for vector quantization: Foundation of dual quantization. Work in progress, 2010.

• G. Pag`

es and B. Wilbertz. Sharp rate for the dual quantization problem. Work in progress, 2010.

• V. T. Rajan.

Optimality of the delaunay triangulation in Rd. In SCG ’91: Proceedings of the seventh annual symposium on Computational geometry, pages 357–363, New York, NY, USA, 1991. ACM. PAG` ES/WILBERTZ (LPMA-UPMC) Dual Quantization Tamerza 49 / 49