A Backward dual representation for the quantile hedging of Bermudan - PowerPoint PPT Presentation

Problem definition Dynamic programming and dual backward algorithm Proof of the dual backward algorithm References A Backward dual representation for the quantile hedging of Bermudan options G´ eraldine Bouveret Imperial College London joint work with B. Bouchard (Paris Dauphine) and J.F. Chassagneux (Imperial College London) Workshop on Stochastic and Quantitative Finance , Imperial College , Friday 28th November 2014 G´ eraldine Bouveret Quantile hedging of Bermudan options

Problem definition Problem formulation Dynamic programming and dual backward algorithm Literature review Proof of the dual backward algorithm Problem reduction References Problem formulation · (Ω , F , F , P ), F := {F t , 0 ≤ t ≤ T } and W a d -dim. BM, · ∀ ( t , x ) ∈ [0 , T ] × (0 , ∞ ) d , T > 0 and for s ≥ t : � s � s X t , x µ ( r , X t , x σ ( r , X t , x = x + ) d r + ) d W r , s r r t t with µ : [0 , T ] × (0 , ∞ ) d → R d and σ : [0 , T ] × (0 , ∞ ) d → M d Lipschitz continuous , · σ is invertible and λ := σ − 1 µ is bounded, d P · Q t , x ∼ P is unique and is s.t. d Q t , x = Q t , x , 1 where for s ≥ t : d Q t , x , 1 ( s ) = λ ( s , X t , x ( s )) Q t , x , 1 ( s ) d W Q t , x ∈ (0 , ∞ ) , s Q t , x , 1 ( t ) = 1 . G´ eraldine Bouveret Quantile hedging of Bermudan options

Problem definition Problem formulation Dynamic programming and dual backward algorithm Literature review Proof of the dual backward algorithm Problem reduction References Problem formulation (cont.) An admissible financial strategy is a d -dimensional predictable process ν s.t. �� T � | ν ⊤ ) | 2 dr E Q t , x r σ ( r , X t , x < ∞ , r t and the corresponding wealth process � · Y t , x , y ,ν := y + ν ⊤ r d X t , x ≥ 0 , on [ t , T ] , r t given ( t , x ) and y ≥ 0. U t , x , y is the collection of admissible financial strategies. G´ eraldine Bouveret Quantile hedging of Bermudan options

Problem definition Problem formulation Dynamic programming and dual backward algorithm Literature review Proof of the dual backward algorithm Problem reduction References Problem formulation (cont.) Fix a finite collection of times T t := { t 0 = 0 ≤ · · · ≤ t i ≤ · · · ≤ t n = T } ∩ ( t , T ] , together with non-negative payoff functions x ∈ (0 , ∞ ) d �→ g ( t i , x ) , Lipschitz continuous for all i ≤ n . The quantile hedging problem is v ( t , x , p ) := inf Γ( t , x , p ) , where Γ( t , x , p ) � � � � � { Y t , x , y ,ν ≥ g ( s , X t , x := y ≥ 0 : ∃ ν ∈ U t , x , y s.t. P ) } ≥ p . s s s ∈ T t G´ eraldine Bouveret Quantile hedging of Bermudan options

Problem definition Problem formulation Dynamic programming and dual backward algorithm Literature review Proof of the dual backward algorithm Problem reduction References Problem formulation (cont.) Remark (Preliminary remarks) Meaning of v ( t , · , 1)... E Q t , x [( v ∨ g )( t i +1 , X t , x v ( t , x , 1) = t i +1 , 1)] , for t ∈ [ t i , t i +1 ) , with i < n and g ( t , x , p ) := g ( t , x ) ✶ { 0 < p ≤ 1 } + ∞ ✶ { p > 1 } , for p ∈ R . p �→ v ( · , p ) is non-decreasing. v ( · , p ) = 0 if p ≤ p min ( t , x ) where p min ( t , x ) := P [ g ( s , X t , x ) ✶ { s < T } = 0 for all s ∈ T t ] . s Hyp: p min ( t , · ) < 1, for t < T ⇒ v ( t , x , 1) > 0 , for t < T . G´ eraldine Bouveret Quantile hedging of Bermudan options

Problem definition Problem formulation Dynamic programming and dual backward algorithm Literature review Proof of the dual backward algorithm Problem reduction References What can we find in the literature? (A) Markovian Framework: (1) Incomplete market case: (a) European Case: Soner and Touzi in [7] and [8], Bouchard, Elie and Touzi in [2], (b) American Case: Bouchard and Vu in [3], (2) Complete market case : (a) European Case: Bouchard, Elie and Touzi in [2] and F¨ ollmer and Leukert in [5]. (B) Non-Markovian Framework: Bouchard, Elie and Reveillac in [1] and Jiao, Klopfenstein and Tankov in [6]. G´ eraldine Bouveret Quantile hedging of Bermudan options

Problem definition Problem formulation Dynamic programming and dual backward algorithm Literature review Proof of the dual backward algorithm Problem reduction References Problem reduction Before all reduce the initial problem to a standard stochastic target one (see [2])...... To this aim introduce the set A t , p of square integrable predictable processes such that � · P t , p ,α := p + α ⊤ r d W r ∈ [0 , 1] , on [ t , T ] . t We denote ˆ U t , x , y , p := U t , x , y × A t , p . Proposition Fix ( t , x , p ) ∈ [0 , T ] × (0 , ∞ ) d × [0 , 1] , then y ≥ 0 : ∃ ( ν, α ) ∈ ˆ � � U t , x , y , p s.t. Γ( t , x , p ) = (1) . Y t , x , y ,ν ≥ g ( · , X t , x ) ✶ { P t , p ,α > 0 } on T t G´ eraldine Bouveret Quantile hedging of Bermudan options

Problem definition Problem formulation Dynamic programming and dual backward algorithm Literature review Proof of the dual backward algorithm Problem reduction References Problem reduction (cont.) Proof. Obvious at T . Fix t < T . Let y ∈ ¯ Γ( t , x , p ) with ¯ Γ the RHS in (1) and fix ( ν, α ) ∈ ˆ U t , x , y , p s.t. Y t , x , y ,ν ≥ g ( · , X t , x , P t , p ,α ) on T t . Then, { Y t , x , y ,ν ≥ g ( · , X t , x ) } ⊃ { P t , p ,α > 0 } on T t . Since P t , p ,α ∈ [0 , 1] and ✶ { P t , p ,α > 0 } ≥ P t , p ,α , we have � � � � � � { Y t , x , y ,ν ≥ g ( s , X t , x { P t , p ,α ) } ≥ P > 0 } P s s s s ∈ T t s ∈ T t    P t , p ,α �  . ≥ E ✶ { P t , p ,α T > 0 } s s ∈ T t \{ T } Noticing that the process P t , p ,α is a martingale, for s ∈ T t , { P t , p ,α = 0 } ⊂ { P t , p ,α = 0 } we obtain y ∈ Γ( t , x , p ). s T G´ eraldine Bouveret Quantile hedging of Bermudan options

Problem definition Problem formulation Dynamic programming and dual backward algorithm Literature review Proof of the dual backward algorithm Problem reduction References Problem reduction (cont.) Proof. (cont.) Fix y ∈ Γ( t , x , p ) and choose ν ∈ U t , x , p s.t. p ′ := P s ∈ T t { Y t , x , y ,ν ≥ g ( s , X t , x �� � ) } ≥ p . By the martingale s s representation theorem, we can find α ∈ A t , p ′ such that ) } = P t , p ′ ,α ≥ P t , p ,α . ✶ � s ∈ T t { Y t , x , y ,ν ≥ g ( s , X t , x T T s s Modifying appropriately α we have α ∈ A t , p . Moreover ) } ≥ P t , p ,α , s ∈ T t . ✶ { Y t , x , y ,ν ≥ g ( s , X t , x T s s Now take the conditional expectation and use the fact that P t , p ,α is a martingale to get ✶ { Y t , x , y ,ν ≥ g ( · , X t , x ) } ≥ P t , p ,α ⇔ Y t , x , y ,ν ≥ g ( · , X t , x ) ✶ { P t , p ,α > 0 } on T t . Hence, y ∈ ¯ Γ( t , x , p ). G´ eraldine Bouveret Quantile hedging of Bermudan options

Problem definition Dynamic programming and dual backward algorithm Dynamic programming Proof of the dual backward algorithm Dual backward algorithm References Dynamic programming A first way to compute the value function v ... Theorem (Dynamic Programming) Fix 0 ≤ i ≤ n − 1 and ( t , x , p ) ∈ [ t i , t i +1 ) × (0 , ∞ ) d × [0 , 1] , α ∈A t , p E Q t , x � t i +1 , X t , x t i +1 , P t , p ,α � �� v ( t , x , p ) = inf ( v ∨ g ) . t i +1 Standard arguments should lead to a characterization of v as a viscosity solution on each interval [ t i , t i +1 ), i < n of − ∂ t ϕ + α ⊤ λ D p ϕ � � sup = 0 , − 1 Tr[ σσ ⊤ D 2 xx ϕ ] + 2 Tr[ α ⊤ σ ⊤ D 2 xp ϕ ] + | α | 2 D 2 � � pp ϕ α ∈ R d 2 with the boundary condition v ( t i +1 − , · ) = ( v ∨ g )( t i +1 , · ) . G´ eraldine Bouveret Quantile hedging of Bermudan options

Problem definition Dynamic programming and dual backward algorithm Dynamic programming Proof of the dual backward algorithm Dual backward algorithm References Dual backward algorithm: intuition of the main result As with Bouchard, Elie and Touzi in [2] for n = 1, take the Fenchel transform v ♯ of v , i.e. v ♯ ( t , x , q ) := sup ( pq − v ( t , x , p )) , p ∈ R to deduce that v ♯ should be a viscosity solution of the linear PDE on each interval [ t i , t i +1 ), i < n of − ∂ t ϕ − 1 � � Tr [ σσ ⊤ D 2 xx ϕ ] + 2 q Tr[ λ ⊤ σ ⊤ D 2 xq ϕ ] + | λ | 2 q 2 D 2 = 0 , qq ϕ 2 with the boundary condition v ♯ ( t i +1 − , · ) = ( v ∨ g ) ♯ ( t i +1 , · ) . G´ eraldine Bouveret Quantile hedging of Bermudan options

Problem definition Dynamic programming and dual backward algorithm Dynamic programming Proof of the dual backward algorithm Dual backward algorithm References Dual backward algorithm: intuition of the main result (cont.) and main result By the Feynman-Kac representation this corresponds to the following backward algorithm for i < n � w ( T , x , q ) := q + ∞ ✶ { q < 0 } , ( w ♯ ∨ g ) ♯ ( t i +1 , X t , x t i +1 , Q t , x , q E Q t , x � � w ( t , x , q ) := t i +1 ) t ∈ [ t i , t i +1 ) , Main result... Theorem (Main result) v = w ♯ on [0 , T ] × (0 , ∞ ) d × [0 , 1] . Aim: Prove the result by relying on dual arguments only! G´ eraldine Bouveret Quantile hedging of Bermudan options