Almost-sure hedging under permanent price impact Y.Zou Universit´ e Paris Dauphine April 20, 2016 Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 1 / 16
Outline 1 Introduction 2 Process dynamics and hedging problems Impact rule and continuous trading dynamics Hedging problem: covered options 3 Super-replication under gamma constraint Statement of pricing PDE Viscosity solution of the pricing PDE Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 2 / 16
Introduction Objective: Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 3 / 16
Introduction Objective: 1 Consider a pricing model with impact and liquidity cost. Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 3 / 16
Introduction Objective: 1 Consider a pricing model with impact and liquidity cost. 2 Not high frequency (no bid-ask spread), but still impact on prices. Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 3 / 16
Introduction Objective: 1 Consider a pricing model with impact and liquidity cost. 2 Not high frequency (no bid-ask spread), but still impact on prices. 3 Permanent impact. Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 3 / 16
Introduction Objective: 1 Consider a pricing model with impact and liquidity cost. 2 Not high frequency (no bid-ask spread), but still impact on prices. 3 Permanent impact. Approach: Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 3 / 16
Introduction Objective: 1 Consider a pricing model with impact and liquidity cost. 2 Not high frequency (no bid-ask spread), but still impact on prices. 3 Permanent impact. Approach: 1 Define a continuous time trading dynamics from a discrete time rule. Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 3 / 16
Introduction Objective: 1 Consider a pricing model with impact and liquidity cost. 2 Not high frequency (no bid-ask spread), but still impact on prices. 3 Permanent impact. Approach: 1 Define a continuous time trading dynamics from a discrete time rule. 2 Provide the PDE characterization: stochastic target tool. Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 3 / 16
Impact rule & Trading signal Basic rule: an order δ moves the price by X t − − → X t = X t − + δf ( X t − ) , and costs δX t − + 1 2 δ 2 f ( X t − ) = δ ( X t − + X t ) . 2 Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 4 / 16
Impact rule & Trading signal Basic rule: an order δ moves the price by X t − − → X t = X t − + δf ( X t − ) , and costs δX t − + 1 2 δ 2 f ( X t − ) = δ ( X t − + X t ) . 2 A trading signal is an Itˆ o process of the form � · � · Y = Y 0 + b s ds + a s dW s . 0 0 Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 4 / 16
Discrete trading dynamics Trade at times t n i = iT/n the quantity δ n i = Y t n i − Y t n i − 1 . t n Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 5 / 16
Discrete trading dynamics Trade at times t n i = iT/n the quantity δ n i = Y t n i − Y t n i − 1 . t n The stock price evolves according to � · X = X t n i + σ ( X s ) dW s t n i between two trades.(can add a drift or be multivariate.) Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 5 / 16
Discrete trading dynamics Trade at times t n i = iT/n the quantity δ n i = Y t n i − Y t n i − 1 . t n The stock price evolves according to � · X = X t n i + σ ( X s ) dW s t n i between two trades.(can add a drift or be multivariate.) Dynamics of the wealth and of the asset: discrete trading and pass to the limit. Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 5 / 16
Continuous trading dynamics Passing to the limit n → ∞ , we have the following convergence in S 2 � · � · Y = Y 0 + b s ds + a s dW s 0 0 � · � · � · ( µ + a s σf ′ )( X s ) ds X = X 0 + σ ( X s ) dW s + f ( X s ) dY s + 0 0 0 � · � · Y s dX s + 1 a 2 V = V 0 + s f ( X s ) ds, 2 0 0 at a speed √ n , where V = cash part + Y X = “portfolio value” . Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 6 / 16
Super-replication under gamma constraint Covered options: No initial/final market impact. Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 7 / 16
Super-replication under gamma constraint Covered options: No initial/final market impact. Super-replication: The minimum initial total wealth to cover the final payoff in almost sure sense. Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 7 / 16
Super-replication under gamma constraint Covered options: No initial/final market impact. Super-replication: The minimum initial total wealth to cover the final payoff in almost sure sense. Gamma constraint: Re-write the dynamics of Y as below: dY = a dW + b dt = γ a ( X ) dX + µ a,b Y ( X ) dt with γ a = a/ ( σ + af ). Then γ a ( X ) is bounded by some function. Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 7 / 16
Mathematical definition Super-hedging price under gamma constraint: v( t, x ) := inf { v = c + yx : ( c, y ) ∈ R 2 s.t. G γ ( t, x, v , y ) � = ∅} where G γ ( t, x, v , y ) is the set of ( a, b ) s.t. φ := ( y, a, b ) satisfies: V t,x, v ,φ ≥ g ( X t,x,φ ) , and γ a ( X ) ≤ ¯ γ ( X ) T T γ < 1 − ǫ , ǫ > 0. with f ¯ Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 8 / 16
Informal derivation If we follow the delta-hedging rule: V = v( · , X ) , and Y = ∂ x v( · , X ) Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 9 / 16
Informal derivation If we follow the delta-hedging rule: V = v( · , X ) , and Y = ∂ x v( · , X ) Equating the dt terms on V = v( · , X ) & dX terms on Y = ∂ x v( · , X ): 1 ∂ t v( · , X ) + 1 2 a 2 f ( X ) 2( σ + af ) 2 ( X ) ∂ 2 xx v( · , X ) = ∂ 2 γ a = xx v( · , X ) Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 9 / 16
Informal derivation If we follow the delta-hedging rule: V = v( · , X ) , and Y = ∂ x v( · , X ) Equating the dt terms on V = v( · , X ) & dX terms on Y = ∂ x v( · , X ): 1 ∂ t v( · , X ) + 1 2 a 2 f ( X ) 2( σ + af ) 2 ( X ) ∂ 2 xx v( · , X ) = ∂ 2 γ a = xx v( · , X ) By definition of γ a and some calculate: � � σ 2 − ∂ t v − 1 xx v) ∂ 2 ( · , X ) = 0 xx v (1 − f∂ 2 2 Gamma constraint insures the non-singularity of the PDE. Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 9 / 16
Pricing PDE and Terminal condition Pricing PDE: on [0 , T ) × R � � σ 2 γ − 1 γ ) ∂ 2 γ − ∂ 2 − ∂ t v ¯ F [v ¯ γ ]( t, x ) := min xx v ¯ γ , ¯ xx v ¯ γ (1 − f∂ 2 2 xx v ¯ = 0 Propagation of the gamma constraint to the boundary: γ ( T − , · ) = ˆ v ¯ g on R where ˆ g is the smallest (viscosity) super-solution of γ − ∂ 2 min { ϕ − g, ¯ xx ϕ } = 0 Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 10 / 16
Super-solution property Theorem (Geometric Dynamic Programming Principle) If V 0 > v(0 , X 0 ) , then there exists ( a, b, Y 0 ) such that V θ ≥ v( θ, X θ ) for any stopping time θ ∈ [0 , T ] . Result: A super-solution v ¯ γ ≤ v ¯ γ . cf. N.Touzi, M.Soner. Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 11 / 16
Sub-solution property: shaken operator γ ≥ v ¯ Objective: Construct a solution ¯ v ¯ γ . Theorem Define the shaken operator � � σ 2 ( x ′ ) F ǫ [ ϕ ]( t, x ) := γ ( x ′ ) − ∂ 2 − ∂ t ϕ + x ′ ∈ B ǫ ( x ) min min xx ϕ ) , ¯ xx ϕ ( t, x ) 2(1 − f ( x ′ ) ∂ 2 v ǫ then ∀ ǫ > 0 , ∃ ¯ γ the unique continuous viscosity solution of ¯ F ǫ [ ϕ ]( t, x ) ✶ [0 ,T ) + [ ϕ − (ˆ g + ǫ )] ✶ { T } = 0 v ǫ Moreover, ¯ γ ≥ ˆ g + ǫ/ 2 on [ T − c ǫ , T ] × R . ¯ Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 12 / 16
✶ ✶ Sub-solution property: shaken operator Objective: Construct a solution ¯ v ¯ γ ≥ v ¯ γ . Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 13 / 16
Sub-solution property: shaken operator Objective: Construct a solution ¯ v ¯ γ ≥ v ¯ γ . By the stability of viscosity solution: v ǫ γ → ¯ as ǫ → 0 ¯ v ¯ γ , ¯ where ¯ v ¯ γ is the unique viscosity solution of F [ ϕ ]( t, x ) ✶ [0 ,T ) + ( ϕ − ˆ g ) ✶ { T } = 0 Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 13 / 16
Sub-solution property: shaken operator Objective: Construct a solution ¯ v ¯ γ ≥ v ¯ γ . By the stability of viscosity solution: v ǫ γ → ¯ as ǫ → 0 ¯ v ¯ γ , ¯ where ¯ v ¯ γ is the unique viscosity solution of F [ ϕ ]( t, x ) ✶ [0 ,T ) + ( ϕ − ˆ g ) ✶ { T } = 0 γ ≥ v ¯ Remain to prove: ¯ v ¯ γ . Y.Zou (Universit´ e Paris Dauphine) JPS les Houches April 20, 2016 13 / 16
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