Introduction : the Cetin-Jarrow-Protter liquidity model A Reference Dominating Measure Second Order Stochastic Target Problems Dual Formulation of Second order Target Problems Nizar TOUZI Ecole Polytechnique Paris Joint with Mete SONER and Jianfeng ZHANG Special Semester in Financial Mathematics Stochastic Control Linz, October 19-24, 2008 Nizar TOUZI Dual Formulation of Second order Target Problems
Introduction : the Cetin-Jarrow-Protter liquidity model A Reference Dominating Measure Second Order Stochastic Target Problems Outline 1 Introduction : the Cetin-Jarrow-Protter liquidity model 2 A Reference Dominating Measure 3 Second Order Stochastic Target Problems Nizar TOUZI Dual Formulation of Second order Target Problems
Introduction : the Cetin-Jarrow-Protter liquidity model A Reference Dominating Measure Second Order Stochastic Target Problems Supply Function Models Price of an order depends on volumes S t ( ω, ν ) . S may be estimated from orders book : Quantity 10 35 20 100 Price 110 112 117 125 Note that the price by share is non-decreasing. But there is no influence of a large trade on the next moment orders book... (Çetin-Jarrow-Protter ’06, Rogers-Singh ’05) This includes Proportional Transaction Costs models S t ( ν ) = ( 1 + λ ) S t ( 0 ) 1 I R + ( ν ) + ( 1 − µ ) S t ( 0 ) 1 I R − ( ν ) Nizar TOUZI Dual Formulation of Second order Target Problems
Introduction : the Cetin-Jarrow-Protter liquidity model A Reference Dominating Measure Second Order Stochastic Target Problems The discrete-time model (Çetin, Jarrow and Protter 2004, 2006) Risky asset price is defined by the marginal price S t , t ≥ 0 the supply curve ν �− → S ( ., ν ) : S ( S t , ν ) price per share of ν risky assets with S ( s , 0 ) = s Z 0 t : holdings in cash, Z t : holdings in risky asset Z 0 t + dt − Z 0 t + ( Z t + dt − Z t ) S ( S t , Z t + dt − Z t ) = 0 � Z 0 Z 0 = ⇒ = 0 − ( Z t + dt − Z t ) S ( S t , Z t + dt − Z t ) T Z 0 � = 0 + Z t ( S t + dt − S t ) + .... Nizar TOUZI Dual Formulation of Second order Target Problems
Introduction : the Cetin-Jarrow-Protter liquidity model A Reference Dominating Measure Second Order Stochastic Target Problems Continuous-time formulation of Model Set Y t := Z 0 t + Z t S t , then : � Y T = Y 0 + Z t ( S t + dt − S t ) � − ( Z t + dt − Z t ) [ S ( S t , Z t + dt − Z t ) − S ( S t , Z t + dt − Z t )] Assume ν �− → S ( ., ν ) is smooth (unlike proportional transaction costs models), then : � T � T ∂ S ∂ν ( S t , 0 ) d � Z c � t Y T = Y 0 + Z t dS t − 0 0 � − ∆ Z t [ S ( S t , ∆ Z t ) − S t ] t ≤ T • d � Z c � t = Γ 2 t d � Z c � t : the so-called Gamma... Nizar TOUZI Dual Formulation of Second order Target Problems
Introduction : the Cetin-Jarrow-Protter liquidity model A Reference Dominating Measure Second Order Stochastic Target Problems The Hedging Problem Option / contingent claim : g ( S T ) , where g : R + − → R has linear growth Super-hedging problem � � y : Y y , Z V := inf ≥ g ( S T ) P − a.s. for some "admissible" Z T • For this formulation to be consistent with the financial problem, we assume there is no liquidity cost at maturity T • Here, admissibility is the crucial issue • Non-Markov case : with new results, should be possible... Nizar TOUZI Dual Formulation of Second order Target Problems
Introduction : the Cetin-Jarrow-Protter liquidity model A Reference Dominating Measure Second Order Stochastic Target Problems The Çetin-Jarrow-Protter Negative Result Without further restrictions on trading strategies, the problem reduces to Black-Scholes ! Reason for this result is the following result of Bank-Baum 04 Lemma For predictable W − integ. càdlàg process φ , and ε > 0 � t � t � � � � φ ε sup φ r dW r − r dW r ≤ ε � � 0 ≤ t ≤ 1 � � 0 0 � t for some a.c. predictable process φ ε t = φ ε 0 + α r dr 0 = ⇒ If the "admissibility" set allows for arbitrary a.c. portfolio � t 0 α u du , then V = V BS (with Γ = 0 !) Z t = Z 0 + Nizar TOUZI Dual Formulation of Second order Target Problems
Introduction : the Cetin-Jarrow-Protter liquidity model A Reference Dominating Measure Second Order Stochastic Target Problems A Convenient Set of Admissible Strategies We show that liquidity cost does affect V , perfect replication is possible, and hedging strategy can be described (formally) Definition Z ∈ A if it is of the form � t � t N − 1 � Z t = z n 1 I { t <τ n + 1 } + α u du + Γ u dS u 0 0 n = 0 • ( τ n ) is an ր seq. of stop. times, z n are F τ n − measurable, � N � ∞ < ∞ • Z and Γ are L ∞ − bounded up to some polynomial of S � t � t • Γ t = Γ 0 + 0 a u du + 0 ξ u dW u , 0 ≤ t ≤ T , and � � | φ r | � � � α � B , b + � a � B , b + � ξ � B , 2 < ∞ , � φ � B , b := � sup � � 1 + S B � � 0 ≤ t ≤ T t � L b Nizar TOUZI Dual Formulation of Second order Target Problems
Introduction : the Cetin-Jarrow-Protter liquidity model A Reference Dominating Measure Second Order Stochastic Target Problems PDE characterization Let � − 1 � 4 ∂ S ℓ ( s ) := ∂ν ( s , 0 ) Theorem Let − C ≤ g ( . ) ≤ C ( 1 + . ) for some C > 0. Then V ( t , s ) is the unique continuous viscosity solution of the dynamic programming equation � + 2 − V t ( t , s ) + 1 � V ss ( t , s ) 4 s 2 σ ( t , s ) 2 ℓ ( s ) 1 − + 1 = 0 ℓ ( s ) with V ( T , s ) = g ( s ) and − C ≤ V ( t , s ) ≤ C ( 1 + s ) for every ( t , s ) . • Notice that there is no boundary layer = ⇒ perfect hedge Nizar TOUZI Dual Formulation of Second order Target Problems
Introduction : the Cetin-Jarrow-Protter liquidity model A Reference Dominating Measure Second Order Stochastic Target Problems Hedging a Convex Payoff in the Frictionless BS Model For a convex payoff : only possibility to super-hedge is the Black-Scholes perfect replication strategy Nizar TOUZI Dual Formulation of Second order Target Problems
Introduction : the Cetin-Jarrow-Protter liquidity model A Reference Dominating Measure Second Order Stochastic Target Problems Hedging a Concave Payoff in the Frictionless BS Model For a concave payoff : two possibilities to super-hedge Black-Scholes perfect replication = ⇒ Γ � = 0 so pay liquidity cost Buy-and-hold = ⇒ Γ = 0 no liquidity cost, but hedge might be too expensive Nizar TOUZI Dual Formulation of Second order Target Problems
Introduction : the Cetin-Jarrow-Protter liquidity model A Reference Dominating Measure Second Order Stochastic Target Problems Hedging a Concave Payoff in the Frictionless BS Model Nizar TOUZI Dual Formulation of Second order Target Problems
Introduction : the Cetin-Jarrow-Protter liquidity model A Reference Dominating Measure Second Order Stochastic Target Problems Formal Description of a Hedging Strategy • v ss < − ℓ ( s ) : Then the PDE satisfied by V reduces to − V t ( t , s ) + 1 4 s 2 σ ( t , s ) 2 ℓ ( s ) = 0 (degenerate !) buy-and-hold strategy is more interesting because liquidity cost is too expensive v ss ≥ − ℓ ( s ) : Then the PDE satisfied by V reduces to 2 s 2 σ ( t , s ) 2 V ss − s 2 σ ( t , s ) 2 − V t ( t , s ) − 1 V 2 = 0 ss 4 ℓ ( s ) perfect replication Nizar TOUZI Dual Formulation of Second order Target Problems
Introduction : the Cetin-Jarrow-Protter liquidity model A Reference Dominating Measure Second Order Stochastic Target Problems The Technical Difficulty Nizar TOUZI Dual Formulation of Second order Target Problems
Introduction : the Cetin-Jarrow-Protter liquidity model A Reference Dominating Measure Second Order Stochastic Target Problems A New Formulation : Intuition Recall the state dynamics in Stratonovitch form : � 1 � σ 2 t S 2 dY t = Z t ◦ dS t − 2 Γ t + S ν ( S t , 0 ) t dt and the corresponding "natural" PDE : � 1 � ∂ V σ 2 s 2 = − 2 V ss + S ν ( s , 0 ) ∂ t Main observation : We would obtain the same PDE if the volatility of S is modified : � 1 � 2 dt Z t ◦ dS ′ 2 Γ t + S ν ( S ′ σ 2 t S ′ dY t = t − t , 0 ) t dS ′ σ ′ t S ′ = t dW t t Nizar TOUZI Dual Formulation of Second order Target Problems
Introduction : the Cetin-Jarrow-Protter liquidity model A Reference Dominating Measure Second Order Stochastic Target Problems A New Formulation : Relax Controls and Change Volatility (Intuition from L. Denis and C. Martini) Consider the super-hedging problem : � 2 � ˆ y : Y T ≥ g ( S T ) ˆ ˆ V := inf P − a.s. for some Z ∈ SM where � 1 � 2 Γ t + S ν ( S t , 0 )Γ 2 σ 2 t S t 2 dt dY t = Z t ◦ dS t − t Compare with V := inf { y : Y T ≥ g ( S T ) P − a.s. for some Z ∈ A} Then, ˆ V = V Nizar TOUZI Dual Formulation of Second order Target Problems
Introduction : the Cetin-Jarrow-Protter liquidity model A Reference Dominating Measure Second Order Stochastic Target Problems Outline 1 Introduction : the Cetin-Jarrow-Protter liquidity model 2 A Reference Dominating Measure 3 Second Order Stochastic Target Problems Nizar TOUZI Dual Formulation of Second order Target Problems
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