Optimal Quantization for thepricing of American style options - PowerPoint PPT Presentation

Optimal Quantization for thepricing of American style options Gilles Pag` es gpa@ccr.jussieu.fr www.proba.jussieu.fr/pageperso/pages Univ. PARIS 6 (Labo. Proba. et Mod` eles Al´ eatoires, UMR 7599) Linz Special semester on Stochastics 18 th November 2008

1 Introduction to optimal quadratic Vector Quantization ? 1.1 What is (quadratic) Vector Quantization ? → ( R d , R ⊗ d ), | . | Euclidean norm, ⊲ Let X : (Ω , A , P ) − E | X | 2 < + ∞ . ⊲ When R d ← ( H, < . | . > ) separable Hilbert space ≡ Functional Quantization. . .. Example : If H = L 2 T := L 2 ([0 , T ] , dt ) a process X = ( X t ) t ∈ [0 ,T ] .

Discretization of the state/path space H = R d or L 2 ([0 , T ] , dt ) using ⊲ N -quantizer (or N -codebook) : Γ := { x 1 , . . . , x N } ⊂ R d . ⊲ Discretization by Γ-quantization X Γ : Ω → Γ := { x 1 , . . . , x N } . X � � X Γ := Proj Γ ( X ) � where Proj Γ denotes the projection on Γ following the nearest neighbour rule.

Fig. 1: A 2-dimensional 10-quantizer Γ = { x 1 , . . . , x 10 } and its Voronoi diagram. . .

X Γ and � X Γ ? What do we know about X − � 1.2 ⊲ Pointwise induced error : for every ω ∈ Ω, | X ( ω ) − � X Γ ( ω ) | = dist( X ( ω ) , Γ) = min 1 ≤ i ≤ N | X ( ω ) − x i | . ⊲ Mean quadratic induced error (or quadratic quantization error) : � � � e N ( X, Γ) := � X − � X Γ � 2 = E 1 ≤ i ≤ N | X − x i | 2 min . X Γ : weights associated to each x i : ⊲ Distribution of � X Γ = x i ) = P ( X ∈ C i (Γ)) , P ( � i = 1 , . . . , N where C i (Γ) denotes the Voronoi cell of x i (w.r.t. Γ) defined by � � ξ ∈ R d : | ξ − x i | = 1 ≤ j ≤ N | ξ − x j | C i (Γ) := min .

Fig. 2: Two N -quantizers related to N (0; I 2 ) of size N = 500. . . Which one is the best ?

1.3 Optimal (Quadratic) Quantization The quadratic distortion (squared quadratic quantization error) ( R d ) N − D X → R + : N � � Γ = ( x 1 , . . . , x N ) �− → � X − � X Γ � 2 1 ≤ i ≤ N | X − x i | 2 2 = E min is continuous [the quantization error is Lipschitz continuous !] for the (product topology on ( R d ) N ). One derives (Cuesta-Albertos & Matran (88), P¨ arna (90), P. (93)) by induction on N that D X N reaches a minimum at an (optimal) quantizer Γ ( N, ∗ ) of full size N (if card(supp( P )) ≥ N ). One derives X Γ ( N, ∗ ) � 2 e N ( X, R d ) := inf {� X − � X Γ � 2 , card(Γ) ≤ N, Γ ⊂ R d } = � X − �

X Γ ( N, ∗ ) � 2 = min {� X − Y � 2 , Y : Ω → R d , card( Y (Ω)) ≤ N } . � X − � Example ( N = 1) : Optimal 1-quantizer Γ = { E X } and e 2 ( X, R d ) = � X − E X � 2 . Extensions to the L r ( P ) -quantization of random 1.4 variables 0 < r ≤ ∞ → ( R d , | . | ) ⊲ X : (Ω , A , P ) − E | X | r < + ∞ (0 < r < + ∞ ) . ⊲ The N -level ( L r ( P ) , | . | )-quantization problem for X ∈ L r E ( P ) � � � X − � X Γ � r , Γ ⊂ E, card(Γ) ≤ N e r,N ( X, E ) := inf . Example ( N = 1, r = 1) : Optimal 1-quantizer Γ = { med( X ) } and e 1 ( X, H ) = � X − med( X ) � 1 .

⊲ Other examples : – Non-Euclidean norms on E = R d like ℓ p -norms, 1 ≤ p ≤ ∞ , etc. – dispersion of compactly supported distribution : r = ∞

1.5 Stationary Quantizers N is | . | -differentiable at N -quantizers Γ ∈ ( R d ) N of full ⊲ Distortion D X size : �� � � � ( x i − ξ ) P X ( dξ ) E ( x i − X ) 1 { b ∇ D X N (Γ) = 2 = 2 X Γ = x i } 1 ≤ i ≤ N C i (Γ) 1 ≤ i ≤ N ⊲ Definition : If Γ ⊂ ( R d ) N is a zero of ∇ D X N (Γ), then Γ is called a stationary quantizer (or self-consistent quantizer). � X Γ � X Γ = E � X | � ∇ D X N (Γ) = 0 ⇐ ⇒ since σ ( � X Γ ) = σ ( { X ∈ C i (Γ) } , i = 1 , . . . , N ) . ⊲ An optimal quadratic quantizer Γ is stationary First by-product : E X = E � X Γ .

1.6 Numerical Integration and conditional expectation (I) : cubature formulae Let F : ( R d ) N − → R be a functional and let Γ ⊂ R d be an N -quantizer. ⊲ If F is Lipschitz continuous, then for every r ∈ [1 , + ∞ ), � � � � � E ( F ( X ) | � X Γ ) − F ( � r ≤ [ F ] Lip � X − � X Γ ) X Γ � r � ⊲ If F is Lipschitz continuous, then (with r = 1) � � � � � E F ( X ) − E F ( � � ≤ [ F ] Lip � X − � X Γ � 1 ≤ [ F ] Lip � X − � X Γ ) X Γ � 2 . Hence the cubature formula since : N � E ( F ( � X Γ )) = F ( x i ) P ( � X = x i ) i =1 In fact � � � � � X − � � E F ( X ) − E F ( � X Γ � 1 = X Γ ) sup � . [ F ] Lip ≤ 1

⊲ Assume F is C 1 on R d , DF is Lipschitz continuous and the quantizer Γ is a stationary. Taylor expansion yields F ( X ) = F ( � X Γ ) + DF ( � X Γ ) . ( X − � X Γ ) + ( DF ( � X Γ ) − DF ( ζ )) . ( X − � X Γ ) ζ ∈ ( X, � X Γ ), so that � � X Γ � � X Γ � � F ( X ) | � − F ( � DF ( � X Γ ) . ( X − � X Γ ) | � X Γ ) � E E − | �� � X Γ � 2 � � � X − � | � X Γ [ DF ] Lip E ≤ �

. . .so that � � � � � X Γ � � X Γ � � � F ( X ) | � − F ( � DF ( � X Γ ) . ( X − � X Γ ) | � � X Γ ) � E E − � � � �� � � =0 �� � X Γ � 2 � � � X − � | � X Γ [ DF ] Lip E ≤ � since � � � � X Γ | � DF ( � X Γ ) . ( X − � DF ( � X Γ ) . E ( X − � X Γ ) X Γ ) E = E = 0 . so that � � � � X Γ | 2 | � � E ( F ( X ) | � X Γ ) − F ( � � ≤ [ DF ] Lip E ( | X − � X Γ ) X Γ )

⊲ As a consequence for conditional expectation � � � � � E ( F ( X ) | � X Γ ) − F ( � ≤ [ DF ] Lip � X − � X Γ ) X Γ � 2 � 2 r r ⊲ Hence the cubature formulas for numerical integration � � � � � E F ( X ) − E F ( � � ≤ [ DF ] Lip � X − � X Γ ) X Γ � 2 2

Quantized approximation of E ( F ( X ) | Y ) 1.7 → R d and F : R d → R a Borel functional. ⊲ Let X , Y (Ω , A , P ) − Y Γ ′ are (Voronoi) quantizations . X Γ and � X = � � Y = � Let ⊲ Natural idea E ( F ( X ) | Y ) ≈ E ( F ( � X ) | � Y ).To what extend ? E ( F ( X ) | Y ) = ϕ F ( Y ) . ⊲ In a Feller Markovian framework : regularity of F � regularity ϕ F E ( F ( X ) | Y ) − E ( F ( � X ) | � Y ) = E ( F ( X ) | Y ) − E ( F ( X ) | � Y )+ E ( F ( X ) − F ( � X ) | � Y ) so that, using that conditional expectation is an L 2 -contraction and � Y is σ ( Y )-measurable, E ( E ( F ( � X ) | Y ) | � � E ( F ( X ) | Y ) − Y ) � 2 � ϕ F ( Y ) − E ( F ( X ) | � Y ) � 2 + � F ( X ) − F ( � ≤ X ) � 2 � ϕ F ( Y ) − E ( ϕ F ( Y ) | � Y ) � 2 + � F ( X ) − F ( � X ) � 2 = � ϕ F ( Y ) − ϕ F ( � Y ) � 2 + � F ( X ) − F ( � ≤ X ) � 2 .

The last inequality follows from the very definition of conditional expectation given � Y � E ( F ( X ) | Y ) − E ( F ( � X ) | � Y ) � 2 ≤ [ F ] Lip � X − � X � 2 + [ ϕ F ] Lip � Y − � Y � 2 . ⊲ Non-quadratic case the above inequality remains valid provided [ ϕ F ] Lip is replaced by 2[ ϕ F ] Lip . ⊲ These are the ingredients for the proofs of both theorems for – Bermuda options (orders 0 & 1). – Swing options

Vector Quantization rate ( H = R d ) 1.8 ⊲ Theorem ( a ) Asymptotic (Zador, Kiefer, Bucklew & Wise, Graf & Luschgy al., from 1963 to 2000) ⊥ Let X ∈ L r + ( P ) and P X ( dξ ) = ϕ ( ξ ) dξ + ν ( dξ ). Then �� � 1 d + 1 r d × N − 1 e N,r ( X, R d ) ∼ � d +2 ( u ) du J 2 ,d × N → + ∞ . R d ϕ as d ( b ) Non asymptotic (Pierce Lemma) (Luschgy-P.(2005) Let d ≥ 1. Let r, δ > 0. There exists a universal constant C d,r,δ ∈ (0 , ∞ ) e N,r ( X, R d ) ≤ C d,r,δ � X � r + δ N − 1 ∀ N ≥ 1 , d ⊲ The true value of � J r,d is unknown for d ≥ 3 but (Euclidean norm) � � d d � J r,d ∼ 2 πe ≈ as d → + ∞ . 17 , 08

Conclusions : • For every N the same rate as with “naive” product-grids for the U ([0 , 1] d ) distribution with N = m d points + the best constant • No escape from “The curse of dimensionality” . . . • Equalization of local inertia (see Comm. in Statist. , S.Delattre-J.C. Fort-G. P., 2004)

2 Numerical optimization of the grids : Gaussian and non-Gaussian vectors The case of normal distribution N (0; I d ) on R d 2.1 ⊲ As concerns Gaussian N (0 , I d ) Already quantized for you (see J. Printems-G.P., MCMA 2003).

⊲ For d = 1 up to 10 and N = 1 ≤ N ≤ 5 000, new grid files available including( L 1 & L 2 -distortion, local L 1 & L 2 -pseudo-inertia, etc). on Download at our WEBSITE : www.quantize.maths-fi.com

2.2 The 1 -dimension. . . ⊲ Theorem (Kiefer (82), LLoyd (82), Lamberton-P. (90)) H = R . If P X ( dξ ) = ϕ ( ξ ) dξ with log ϕ concave, then there is exactly one stationary quantizer. Hence argmin D X N = { Γ ( N ) } . ∀ N ≥ 1 , Examples : The normal distribution, the gamma distributions, etc. 2 = x i +1 + x i ⊲ Voronoi cells : C i (Γ) = [ x i − 1 2 , x i + 1 2 [, x i + 1 . 2   � x i + 1 2 ( x i − ξ ) ϕ ( ξ ) dξ ⊲ Gradient : ∇ D X   N (Γ) = 2 x i − 1 2 1 ≤ i ≤ N � x � x Hessian : D 2 ( D X N )(Γ) = . . . . . . only involves 0 ϕ ( ξ ) dξ and 0 ξϕ ( ξ ) dξ