A Martingale Approach for Fractional Brownian Motions and Related - PowerPoint PPT Presentation

Introduction Heat equation Functional Itô formula Nonlinear extension A Martingale Approach for Fractional Brownian Motions and Related Path - Dependent PDEs Jianfeng ZHANG, University of Southern California Frederi VIENS , Michigan State University XXIII Escola Brasileira de Probabilidade Universidade de São Paulo, São Carlos, Brasil, 26/7/2019 Logo Jianfeng ZHANG (USC) Martingale Approach for fBM and PPDE

Introduction Heat equation Functional Itô formula Nonlinear extension Outline 1 Introduction 2 Heat equation 3 Functional Itô formula 4 Nonlinear extension Logo Jianfeng ZHANG (USC) Martingale Approach for fBM and PPDE

Introduction Heat equation Functional Itô formula Nonlinear extension The standard risk neutral pricing • Let S be an underlying asset price, l P a risk neutral measure : dS t = σ ( t , S t ) dB t • Let ξ = g ( S T ) be a payoff at T , then the price at t is : Y t = I E t [ ξ ] • In the above Markovian setting : Y t = u ( t , S t ) , ∂ t u + 1 2 σ 2 ( t , x ) ∂ 2 xx u = 0 , u ( T , x ) = g ( x ) . • In path dependent setting : σ = σ ( t , S · ) , ξ = g ( S · ) , then Y t = u ( t , S · ) , ∂ t u + 1 2 σ 2 ( t , ω ) ∂ 2 ωω u = 0 , u ( T , ω ) = g ( ω ) . Logo Jianfeng ZHANG (USC) Martingale Approach for fBM and PPDE

Introduction Heat equation Functional Itô formula Nonlinear extension Rough volatility model • Rough volatility : dS t = S t σ t dB t and σ is rough ⋄ See e.g. Gatheral-Jaisson-Rosenbaum (2014) • A natural model : σ driven by a fractional Brownian motion B H � F B , B H � � � • Goal : characterize Y t := I E ξ t ⋄ σ (hence B H ) can be observed ⋄ To focus on the main idea we will assume ξ is F B H T -measurable � F B H � � � and consider Y t = I E ξ t ⋄ Sone related recent works : El Euch-Rosenbaum (2017), Fouque-Hu (2017) Logo Jianfeng ZHANG (USC) Martingale Approach for fBM and PPDE

Introduction Heat equation Functional Itô formula Nonlinear extension Outline 1 Introduction 2 Heat equation 3 Functional Itô formula 4 Nonlinear extension Logo Jianfeng ZHANG (USC) Martingale Approach for fBM and PPDE

Introduction Heat equation Functional Itô formula Nonlinear extension Fractional Brownian Motion • Let B H be a fBM with 0 < H < 1 : ⋄ B H t − B H s ∼ Normal ( 0 , ( t − s ) 2 H ) ⋄ B H = B when H = 1 2 • Two main features : ⋄ B H is not Markovian ( H � = 1 2 ) ⋄ B H is not a semimartingale ( H < 1 2 ) � F B H � � g ( B H � • Our goal : characterize Y t := I E · ) t Logo Jianfeng ZHANG (USC) Martingale Approach for fBM and PPDE

Introduction Heat equation Functional Itô formula Nonlinear extension Heat equation in BM case • Let ξ := g ( B T ) and Y t := I E t [ g ( B T )] . • Denote � � � v ( t , x ) := I g ( x + B T − B t ) = R g ( y ) p ( T − t , y − x ) dy E I 2 π t e − x 2 1 2 t . where p ( t , x ) := √ • Heat equation : ∂ t p ( t , x ) − 1 2 ∂ xx p ( t , x ) = 0 ∂ t v ( t , x ) + 1 2 ∂ xx v ( t , x ) = 0 , v ( T , x ) = g ( x ) . • Y t = v ( t , B t ) , 0 ≤ t ≤ T Logo Jianfeng ZHANG (USC) Martingale Approach for fBM and PPDE

Introduction Heat equation Functional Itô formula Nonlinear extension A Heat equation for fBM • Let ξ := g ( B H E t [ g ( B H T ) and Y t := I T )] . • Denote � g ( x + B H T − B H � � v ( t , x ) := I E t ) = R g ( y ) p H ( T − t , y − x ) dy I x 2 2 π t H e − 2 t 2 H . 1 where p H ( t , x ) := √ • Heat equation : ∂ t v ( t , x ) + Ht 2 H − 1 ∂ xx v ( t , x ) = 0 , v ( T , x ) = g ( x ) . • Y 0 = v ( 0 , 0 ) Logo Jianfeng ZHANG (USC) Martingale Approach for fBM and PPDE

Introduction Heat equation Functional Itô formula Nonlinear extension A heat equation for fBM • Let ξ := g ( B H E t [ g ( B H T ) and Y t := I T )] . � g ( x + B H T − B H � • Denote v ( t , x ) := I E t ) • Heat equation : ∂ t v ( t , x ) + Ht 2 H − 1 ∂ xx v ( t , x ) = 0 , v ( T , x ) = g ( x ) . • Y 0 = v ( 0 , B H 0 ) , Y T = v ( T , B H T ) • However, v ( t , B H t ) is not a martingale : Y t � = v ( t , B H t ) for 0 < t < T . Logo Jianfeng ZHANG (USC) Martingale Approach for fBM and PPDE

Introduction Heat equation Functional Itô formula Nonlinear extension A crucial representation of fBM � t • Representation : B H t = 0 K ( t , r ) dW r F B H = l F W ⋄ l F := l ⋄ K ( t , r ) ∼ ( t − r ) 2 H − 1 , which blows up at t = r when H < 1 2 • Decomposition : � T � t � T B H T = K ( T , r ) dW r = K ( T , r ) dW r + K ( T , r ) dW r 0 0 t � t ⋄ 0 K ( T , r ) dW r is F t -measurable � T ⋄ t K ( T , r ) dW r is independent of F t ⋄ The previous decomposition B H T = B H t + [ B H T − B H t ] does not Logo satisfy this property Jianfeng ZHANG (USC) Martingale Approach for fBM and PPDE

Introduction Heat equation Functional Itô formula Nonlinear extension An alternative heat equation • Let ξ := g ( B H T ) and �� t � T � �� Y t = I K ( T , r ) dW r + K ( T , r ) dW r E t g 0 t � T � �� � • Denote v ( t , x ) := I x + t K ( T , r ) dW r E g � t � � • Then Y t = v t , 0 K ( T , r ) dW r , 0 ≤ t ≤ T � t � � • Note : v t , 0 K ( T , r ) dW r is a martingale • Heat equation : ∂ t v ( t , x ) + 1 2 K 2 ( T , t ) ∂ xx v ( t , x ) = 0 , v ( T , x ) = g ( x ) . Logo Jianfeng ZHANG (USC) Martingale Approach for fBM and PPDE

Introduction Heat equation Functional Itô formula Nonlinear extension A closer look � t • Θ t E t [ B H T := 0 K ( T , r ) dW r = I T ] is F t -measurable ⋄ Θ t T is the forward variance and is observable in market • Three ways to express Y t : Y t = v 1 ( t , B H t ∧· ) = v 2 ( t , W t ∧· ) = v ( t , Θ t T ) ⋄ B H is not a semimartingale ⋄ W is a martingale (of course) but v 2 is not continuous ⋄ v has desired regularity and t �→ Θ t T is a martingale Logo Jianfeng ZHANG (USC) Martingale Approach for fBM and PPDE

Introduction Heat equation Functional Itô formula Nonlinear extension An extension � T � � g ( B H t f ( s , B H • Denote Y t := I E t T ) + s ) ds . • By previous computation : � T E t [ g ( B H E t [ f ( s , B H Y t = I T )] + t I s )] ds � T E t [ B H E t [ B H = v ( T , g ; t , I T ]) + t v ( s , f ( s , · ); t , I s ]) ds E t [ B H = u ( t , { I s ] } t ≤ s ≤ T ) • Note : u is path dependent ⋄ If H = 1 2 , I E t [ B s ] = B t , so Y t = u ( t , B t ) is state dependent ⋄ In more general cases, � � t , { B H E t [ B H Y t = u s } 0 ≤ s ≤ t ⊗ t { I s ] } t ≤ s ≤ T . Logo Jianfeng ZHANG (USC) Martingale Approach for fBM and PPDE

Introduction Heat equation Functional Itô formula Nonlinear extension Outline 1 Introduction 2 Heat equation 3 Functional Itô formula 4 Nonlinear extension Logo Jianfeng ZHANG (USC) Martingale Approach for fBM and PPDE

Introduction Heat equation Functional Itô formula Nonlinear extension The canonical setup • Recall � � t , { B H E t [ B H Y t = u s } 0 ≤ s ≤ t ⊗ t { I s ] } t ≤ s ≤ T . D 0 ([ 0 , t )) , and θ ∈ C 0 ([ t , T ]) , define : • For t ∈ [ 0 , T ] , ω ∈ l ( ω ⊗ t θ ) s := ω s 1 [ 0 , t ) ( s ) + θ s 1 [ t , T ] ( s ) , 0 ≤ s ≤ T . • The canonical space : � � D 0 ([ 0 , t )) , θ ∈ C 0 ([ t , T ]) Λ := ( t , ω ⊗ t θ ) : t ∈ [ 0 , T ] , ω ∈ l ; � � ( t , ω ⊗ t θ ) ∈ Λ : ω ∈ C 0 ([ 0 , t ]) , ω 0 = 0 , θ t = ω t Λ 0 := . Logo Jianfeng ZHANG (USC) Martingale Approach for fBM and PPDE

Introduction Heat equation Functional Itô formula Nonlinear extension Continuous mapping • Recall � � D 0 ([ 0 , t )) , θ ∈ C 0 ([ t , T ]) Λ := ( t , ω ⊗ t θ ) : t ∈ [ 0 , T ] , ω ∈ l . • The metric : d (( t , ω ⊗ t θ ) , ( t ′ , ω ′ ⊗ t ′ θ ′ )) | t − t ′ | + sup 0 ≤ s ≤ T | ( ω ⊗ t θ ) s − ( ω ′ ⊗ t ′ θ ′ ) s | . � := • C 0 (Λ) : continuous mapping u : Λ → I R • C 0 b (Λ) : bounded u ∈ C 0 (Λ) Logo Jianfeng ZHANG (USC) Martingale Approach for fBM and PPDE

Introduction Heat equation Functional Itô formula Nonlinear extension Path derivatives • Time derivative : u ( t + δ, ω ⊗ t θ ) − u ( t , ω ⊗ t θ ) ∂ t u ( t , ω ⊗ t θ ) := lim . δ δ ↓ 0 ⋄ ∂ t u is the right time derivative ! • First order spatial derivative : Fréchet derivative with respect to θ 1 � � � ∂ θ u ( t , ω ⊗ t θ ) , η � := lim u ( t , ω ⊗ t ( θ + εη )) − u ( t , ω ⊗ t θ ) , ε ε → 0 for all ( t , ω ⊗ t θ ) ∈ Λ , η ∈ C 0 ([ t , T ]) . Logo Jianfeng ZHANG (USC) Martingale Approach for fBM and PPDE

Introduction Heat equation Functional Itô formula Nonlinear extension Path derivatives (cont) • Second order spatial derivative : bilinear operator on C 0 ([ t , T ]) : � ∂ 2 θθ u ( t , ω ⊗ t θ ) , ( η 1 , η 2 ) � � � := lim ε → 0 1 � ∂ θ u ( t , ω ⊗ t ( θ + εη 1 )) , η 2 � − � ∂ θ u ( t , ω ⊗ t θ ) , η 2 � . ε for all ( t , ω ⊗ t θ ) ∈ Λ , η 1 , η 2 ∈ C 0 ([ t , T ]) . • Define the spaces C 1 , 2 (Λ) and C 1 , 2 b (Λ) in obvious sense Logo Jianfeng ZHANG (USC) Martingale Approach for fBM and PPDE