A regularity structure for rough volatility Christian Bayer Joint - PowerPoint PPT Presentation

Weierstrass Institute for Applied Analysis and Stochastics A regularity structure for rough volatility Christian Bayer Joint work with: P . Friz, P . Gassiat, J. Martin, B. Stemper Jim Gatheral’s 60’th birthday conference Mohrenstrasse 39 · 10117 Berlin · Germany · Tel. +49 30 20372 0 · www.wias-berlin.de October 14, 2017

Outline Rough volatility models 1 A minimal view on regularity structures 2 The simple regularity structure for rough volatility 3 4 The full regularity structure for rough volatility A regularity structure for rough volatility · October 14, 2017 · Page 2 (25)

Rough volatility models Some years ago, Jim Gatheral (et al.) kicked off exciting new √ 2 H | t − s | H − 1 / 2 1 t > s . development in rough volatility. Let K ( s , t ) ≔ Example (Rough Bergomi model) � t � � � � dS t = √ v t S t η � � ρ dW t + ρ dW ⊥ , v t = ξ ( t ) E W t , W t ≔ K ( s , t ) dW s t 0 � t f ( s , � W s ) dW s � 0 Example (Rough Heston model) � t � t c √ v s K ( s , t ) dW s ( a − bv s ) K ( s , t ) ds + dS t = . . . , v t = v 0 + 0 0 � t � t K ( s , t ) v ( Z s ) ds + K ( s , t ) u ( Z s ) dW s � Z t = z + 0 0 A regularity structure for rough volatility · October 14, 2017 · Page 3 (25)

Objectives Provide unified analytic (i.e., pathwise) framework for rough volatility models as above. ◮ Stratonovich version of rough volatility models ◮ Existence and uniqueness and stability of solutions ◮ Numerical approximation based on approximation of the driving Brownian motion W ◮ Large deviation principle for analyzing behaviour of implied volatility Requirements ◮ Smoothness of coefficient functions ◮ Structure adapted to Hurst index H – more detailed structure needed for H ≪ 1 2 A regularity structure for rough volatility · October 14, 2017 · Page 4 (25)

Numerical approximation Theorem W t ) � ρ dW t + ρ dW ⊥ � , W ε approximation (at scale ε ) of W . Let dS t = f ( � t 1. There is C ε = C ε ( t ) s.t. � S ε → S (in probability, on [0 , T ] ) with �� W ε � � W ⊥ ,ε � − ρ C ε f ′ �� W ε � 2 f 2 �� W ε � d − 1 S ε = f W ε + ρ ˙ � ρ ˙ . dt � T ε → 0 For H < 1 0 C ε ( t ) dt 2 , − − − → ∞ . � � � � ρ I − ρ 2 , K , ρ 2 V 2. Let Ψ ( I , V ) ≔ C BS S 0 exp 2 V and � T � T � T �� � C ε ( t ) f ′ �� � f 2 �� � � W ε dW ε W ε dt , V ε ≔ W ε I ε ≔ f t − dt . t t t 0 0 0 � I ε , V ε � Then E � ( S T − K ) + � = lim ε → 0 E Ψ ( � . A regularity structure for rough volatility · October 14, 2017 · Page 5 (25)

Numerical approximation Theorem W t ) � ρ dW t + ρ dW ⊥ � , W ε approximation (at scale ε ) of W . Let dS t = f ( � t 1. There is C ε = C ε ( t ) s.t. � S ε → S (in probability, on [0 , T ] ) with �� W ε � � W ⊥ ,ε � − ρ C ε f ′ �� W ε � 2 f 2 �� W ε � d − 1 S ε = f W ε + ρ ˙ � ρ ˙ . dt � T ε → 0 For H < 1 0 C ε ( t ) dt 2 , − − − → ∞ . � � � � ρ I − ρ 2 , K , ρ 2 V 2. Let Ψ ( I , V ) ≔ C BS S 0 exp 2 V and � T � T � T �� � C ε ( t ) f ′ �� � f 2 �� � � W ε dW ε W ε dt , V ε ≔ W ε I ε ≔ f t − dt . t t t 0 0 0 � I ε , V ε � Then E � ( S T − K ) + � = lim ε → 0 E Ψ ( � . A regularity structure for rough volatility · October 14, 2017 · Page 5 (25)

Outline Rough volatility models 1 A minimal view on regularity structures 2 The simple regularity structure for rough volatility 3 4 The full regularity structure for rough volatility A regularity structure for rough volatility · October 14, 2017 · Page 6 (25)

Polynomial regularity structure �� 1 , X , X 2 , . . . , X M �� � � X k � � � ◮ Model space T ≔ � ≔ k with degrees ◮ Describes jet of local expansions at any point ◮ Model ( Π , Γ ) . Π x : T → S ′ ( R ) local expansion around x ∈ R ◮ ( Π x X k )( z ) ≔ ( z − x ) k , z ∈ R ◮ Γ xy : T → T translates a “local expansion” around y to one around x , i.e., Π y = Π x Γ xy ◮ Canonical choice: Γ xy X k ≔ ( X + ( y − x ) 1 ) k ◮ Modelled distribution F : R → T is in D γ if it is “regular” in the sense that F ( x ) − Γ xy F ( y ) “small” at each level ◮ “Jets” of local expansions in terms of defining symbols ◮ Reconstruction operator R : D γ → S ′ ( R ) such that R F − Π x F ( x ) is “small” when tested against test functions centered in x ∈ R In this case all distributions are regular functions! A regularity structure for rough volatility · October 14, 2017 · Page 7 (25)

Polynomial regularity structure �� 1 , X , X 2 , . . . , X M �� � � X k � � � ◮ Model space T ≔ � ≔ k with degrees ◮ Describes jet of local expansions at any point ◮ Model ( Π , Γ ) . Π x : T → S ′ ( R ) local expansion around x ∈ R ◮ ( Π x X k )( z ) ≔ ( z − x ) k , z ∈ R ◮ Γ xy : T → T translates a “local expansion” around y to one around x , i.e., Π y = Π x Γ xy ◮ Canonical choice: Γ xy X k ≔ ( X + ( y − x ) 1 ) k ◮ Modelled distribution F : R → T is in D γ if it is “regular” in the sense that F ( x ) − Γ xy F ( y ) “small” at each level ◮ “Jets” of local expansions in terms of defining symbols ◮ Reconstruction operator R : D γ → S ′ ( R ) such that R F − Π x F ( x ) is “small” when tested against test functions centered in x ∈ R In this case all distributions are regular functions! A regularity structure for rough volatility · October 14, 2017 · Page 7 (25)

Polynomial regularity structure �� 1 , X , X 2 , . . . , X M �� � � X k � � � ◮ Model space T ≔ � ≔ k with degrees ◮ Describes jet of local expansions at any point ◮ Model ( Π , Γ ) . Π x : T → S ′ ( R ) local expansion around x ∈ R ◮ ( Π x X k )( z ) ≔ ( z − x ) k , z ∈ R ◮ Γ xy : T → T translates a “local expansion” around y to one around x , i.e., Π y = Π x Γ xy ◮ Canonical choice: Γ xy X k ≔ ( X + ( y − x ) 1 ) k ◮ Modelled distribution F : R → T is in D γ if it is “regular” in the sense that F ( x ) − Γ xy F ( y ) “small” at each level ◮ “Jets” of local expansions in terms of defining symbols ◮ Reconstruction operator R : D γ → S ′ ( R ) such that R F − Π x F ( x ) is “small” when tested against test functions centered in x ∈ R In this case all distributions are regular functions! A regularity structure for rough volatility · October 14, 2017 · Page 7 (25)

Polynomial regularity structure �� 1 , X , X 2 , . . . , X M �� � � X k � � � ◮ Model space T ≔ � ≔ k with degrees ◮ Describes jet of local expansions at any point ◮ Model ( Π , Γ ) . Π x : T → S ′ ( R ) local expansion around x ∈ R ◮ ( Π x X k )( z ) ≔ ( z − x ) k , z ∈ R ◮ Γ xy : T → T translates a “local expansion” around y to one around x , i.e., Π y = Π x Γ xy ◮ Canonical choice: Γ xy X k ≔ ( X + ( y − x ) 1 ) k ◮ Modelled distribution F : R → T is in D γ if it is “regular” in the sense that F ( x ) − Γ xy F ( y ) “small” at each level ◮ “Jets” of local expansions in terms of defining symbols ◮ Reconstruction operator R : D γ → S ′ ( R ) such that R F − Π x F ( x ) is “small” when tested against test functions centered in x ∈ R In this case all distributions are regular functions! A regularity structure for rough volatility · October 14, 2017 · Page 7 (25)

Modelled distributions ◮ For a degree β and τ ∈ T , let | τ | k be the modulus of the coefficient X β ◮ Modelled distributions: F ∈ D γ K for K > 0 iff � � � � � F ( x ) − Γ xy F ( y ) � β � F � D γ sup | F ( x ) | β + sup K ≔ | x − y | γ − β β<γ, | x |≤ K β<γ, | x | , | y |≤ K , x � y ◮ Example: f ∈ C α ( R ) (in the Lipschitz sense), then ⌊ α ⌋ � 1 k ! f ( k ) ( x ) X k ∈ D α F : x �→ K . k = 0 A regularity structure for rough volatility · October 14, 2017 · Page 8 (25)

Modelled distributions ◮ For a degree β and τ ∈ T , let | τ | k be the modulus of the coefficient X β ◮ Modelled distributions: F ∈ D γ K for K > 0 iff � � � � � F ( x ) − Γ xy F ( y ) � β � F � D γ sup | F ( x ) | β + sup K ≔ | x − y | γ − β β<γ, | x |≤ K β<γ, | x | , | y |≤ K , x � y ◮ Example: f ∈ C α ( R ) (in the Lipschitz sense), then ⌊ α ⌋ � 1 k ! f ( k ) ( x ) X k ∈ D α F : x �→ K . k = 0 A regularity structure for rough volatility · October 14, 2017 · Page 8 (25)

Reconstruction For ϕ ∈ C M c (compactly supported in a fixed set), let � z − x � x ( z ) ≔ 1 ϕ λ λϕ , λ > 0 , x ∈ R λ Theorem and definition Reconstruction operator R : D γ → S ′ ( R ) defined by the property that � � � � � R F ( ϕ λ x ) − ( Π x F ( x ))( ϕ λ � � λ γ ∀ x : x ) In the polynomial regularity structure, with F ∈ D γ constructed from f ∈ C γ , we get R F = f . A regularity structure for rough volatility · October 14, 2017 · Page 9 (25)