Introduction Heat equation Functional Itô formula Nonlinear extension A Martingale Approach for Fractional Brownian Motions Jianfeng ZHANG University of Southern California Joint work with Frederi Viens 8th WCMF, Seattle, 3/24-3/25, 2017 Logo Jianfeng ZHANG (USC) Martingale Approach for fBM
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
Introduction Heat equation Functional Itô formula Nonlinear extension Forward versus backward problems Logo Jianfeng ZHANG (USC) Martingale Approach for fBM
Introduction Heat equation Functional Itô formula Nonlinear extension Robust hedging ? • Given S on [ 0 , T ] and a terminal payoff ξ (at T ) • Complete market (linear case) : � T E l P [ ξ ] � � Y 0 = I = y : ∃ Z s.t. y + 0 Z t dS t = ξ, l P -a.s. • Incomplete market : l P ∈ P (semimartingale measures) E l P [ ξ ] Y 0 = sup l P ∈P I � T � � = inf y : ∃ Z s.t. y + 0 Z t dS t ≥ ξ, l P -a.s. for all l P ∈ P • Beyond semimartingale framework ? � T � Y 0 = inf y : ∃ Z s.t. y + 0 Z t ( ω ) dS t ( ω ) ≥ ξ ( ω ) , � Logo for "all" rough paths ω Jianfeng ZHANG (USC) Martingale Approach for fBM
Introduction Heat equation Functional Itô formula Nonlinear extension Rough price versus rough volatility • Rough price S : Z t dS t ? • Rough volatility : dS t = S t [ b t dt + σ t dB t ] and σ is rough ⋄ See Huimeng’s talk yesterday ⋄ See the recent work El Euch-Rosenbaum (2017) • A natural model : σ driven by a fractional Brownian motion B H � F B , B H � � � • Goal : characterize Y t := I E t 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 t ξ Logo t Jianfeng ZHANG (USC) Martingale Approach for fBM
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
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 � t • Representation : B H t = 0 K ( t , s ) dW s ⋄ K ( t , s ) ∼ ( t − s ) 2 H − 1 , which blows up at t = s when H < 1 2 F B H = l F W ⋄ l F := l • Two main features : ⋄ B H is not Markovian ( H � = 1 2 ) ⋄ B H is not a semimartingale ( H < 1 2 ) Logo Jianfeng ZHANG (USC) Martingale Approach for fBM
Introduction Heat equation Functional Itô formula Nonlinear extension A Heat equation • Let ξ := g ( B H E t [ g ( B H T ) and V 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 p H ( t , x ) − Ht 2 H − 1 ∂ xx p H ( t , x ) = 0 ∂ t v ( t , x ) + Ht 2 H − 1 ∂ xx v ( t , x ) = 0 , v ( T , x ) = g ( x ) . • V 0 = v ( 0 , 0 ) Logo Jianfeng ZHANG (USC) Martingale Approach for fBM
Introduction Heat equation Functional Itô formula Nonlinear extension A Heat equation • Let ξ := g ( B H E t [ g ( B H T ) and V t := I T )] . • Denote g ( x + B H T − B H � � 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 ) . • However, v ( t , B H t ) is not a martingale : V 0 = v ( 0 , B H 0 ) , V T = v ( T , B H T ) , but V t � = v ( t , B H t ) for 0 < t < T . Logo Jianfeng ZHANG (USC) Martingale Approach for fBM
Introduction Heat equation Functional Itô formula Nonlinear extension An alternative heat equation • Let ξ := g ( B H E t [ g ( B H T ) and V t := I T )] . • Note � � T � �� V t = I E t g 0 K ( T , r ) dW r �� t � T � �� = I E t g 0 K ( T , r ) dW r + t K ( T , r ) dW r � t � � = v t , 0 K ( T , r ) dW r , � T � �� � where v ( t , x ) := I x + t K ( T , r ) dW r E g � t � � • Martingale property : v 0 K ( T , r ) dW r is a martingale t , • 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
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 V t : V t = v 1 ( t , B H t ∧· ) = v 2 ( t , W t ∧· ) = v ( t , Θ t T ) ⋄ v 1 could be smooth but 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
Introduction Heat equation Functional Itô formula Nonlinear extension An extension � T � � g ( B H t f ( s , B H • Denote V t := I E t T ) + s ) ds . • By previous computation : � T E t [ g ( B H E t [ f ( s , B H V 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 V t = u ( t , B t ) is state dependent ⋄ In more general cases, � � t , { B H E t [ B H V t = u s } 0 ≤ s ≤ t ⊗ t { I s ] } t ≤ s ≤ T . Logo Jianfeng ZHANG (USC) Martingale Approach for fBM
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
Introduction Heat equation Functional Itô formula Nonlinear extension The canonical setup • Recall � � t , { B H E t [ B H V 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
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
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
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
Introduction Heat equation Functional Itô formula Nonlinear extension Functional Ito formula : H ≥ 1 2 • Regular case : K ( t , t ) is finite and thus s ∈ [ t , T ] �→ K t s := K ( s , t ) is in C 0 ([ t , T ]) . • Denote : X s := B H Θ t E t [ B H s , 0 ≤ s ≤ t ; s := I s ] , t ≤ s ≤ T • Functional Ito formula : du ( t , X ⊗ t Θ t ) ∂ t u ( · ) dt + � ∂ θ u ( · ) , K t � dW t + 1 2 � ∂ 2 θθ u ( · ) , ( K t , K t ) � dt . = ⋄ If H = 1 2 , K = 1, this is exactly Dupire’s functional Ito formula Logo Jianfeng ZHANG (USC) Martingale Approach for fBM
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