Introduction Quadratic hedging GKW- versus FS-decomposition (Counter)examples References F¨ ollmer-Schweizer or Galtchouck-Kunita-Watanabe Decomposition? A Comparison and Description Tahir Choulli, Nele Vandaele, Mich` ele Vanmaele Workshop on Actuarial and Financial Statistics August 29 - 30 , 2011 Mich` ele Vanmaele — F¨ ollmer-Schweizer or Galtchouck-Kunita-Watanabe Decomposition? A Comparison and Description 1/44
Introduction Quadratic hedging GKW- versus FS-decomposition (Counter)examples References Outline 1 Introduction 2 Quadratic hedging 3 GKW- versus FS-decomposition 4 (Counter)examples 5 References Mich` ele Vanmaele — F¨ ollmer-Schweizer or Galtchouck-Kunita-Watanabe Decomposition? A Comparison and Description 2/44
Introduction Quadratic hedging GKW- versus FS-decomposition (Counter)examples References T. Choulli, N. Vandaele, and M. Vanmaele. The F¨ ollmer-Schweizer decomposition: Comparison and description. Stochastic Processes and their Applications , 120(6):853–872, 2010. Mich` ele Vanmaele — F¨ ollmer-Schweizer or Galtchouck-Kunita-Watanabe Decomposition? A Comparison and Description 3/44
Introduction Quadratic hedging GKW- versus FS-decomposition (Counter)examples References Outline 1 Introduction Hedging Complete market Incomplete market 2 Quadratic hedging 3 GKW- versus FS-decomposition 4 (Counter)examples 5 References Mich` ele Vanmaele — F¨ ollmer-Schweizer or Galtchouck-Kunita-Watanabe Decomposition? A Comparison and Description 4/44
Introduction Quadratic hedging GKW- versus FS-decomposition (Counter)examples References Hedging problem Financial product P ( t , S t ) Depends on a risky asset S A bank sells an option and wants to replicate its payoff P ( T , S T ) by trading in stocks (liquid assets). Hedging strategy ϕ = ( ξ, η ) Investment in risky asset and cash in order to reduce the risk related to a financial product. Hedging portfolio V t = ξ t S t + η t Cost process � C = V − ξ dS = V − ξ · S Mich` ele Vanmaele — F¨ ollmer-Schweizer or Galtchouck-Kunita-Watanabe Decomposition? A Comparison and Description 5/44
Introduction Quadratic hedging GKW- versus FS-decomposition (Counter)examples References Hedging problem Financial product P ( t , S t ) Depends on a risky asset S A bank sells an option and wants to replicate its payoff P ( T , S T ) by trading in stocks (liquid assets). Hedging strategy ϕ = ( ξ, η ) Investment in risky asset and cash in order to reduce the risk related to a financial product. Hedging portfolio V t = ξ t S t + η t Cost process � C = V − ξ dS = V − ξ · S Mich` ele Vanmaele — F¨ ollmer-Schweizer or Galtchouck-Kunita-Watanabe Decomposition? A Comparison and Description 5/44
Introduction Quadratic hedging GKW- versus FS-decomposition (Counter)examples References Hedging problem Financial product P ( t , S t ) Depends on a risky asset S A bank sells an option and wants to replicate its payoff P ( T , S T ) by trading in stocks (liquid assets). Hedging strategy ϕ = ( ξ, η ) Investment in risky asset and cash in order to reduce the risk related to a financial product. Hedging portfolio V t = ξ t S t + η t Cost process � C = V − ξ dS = V − ξ · S Mich` ele Vanmaele — F¨ ollmer-Schweizer or Galtchouck-Kunita-Watanabe Decomposition? A Comparison and Description 5/44
Introduction Quadratic hedging GKW- versus FS-decomposition (Counter)examples References Complete market Hedging in Black-Scholes model dS t = σ S t dW t (martingale measure Q , no interest rate) perfect replication by self-financing strategies martingale representation � T � T P ( T , S T ) = E Q [ P ( T , S T )]+ Z t dW t = E Q [ P ( T , S T )]+ ξ t dS t 0 0 where in case of a European option ξ t = ∂ P ( t , S t ) with P ( t , s ) = E Q [ P ( T , S T ) | S t = s ] ∂ s delta-hedge Mich` ele Vanmaele — F¨ ollmer-Schweizer or Galtchouck-Kunita-Watanabe Decomposition? A Comparison and Description 6/44
Introduction Quadratic hedging GKW- versus FS-decomposition (Counter)examples References Complete market Hedging in Black-Scholes model dS t = σ S t dW t (martingale measure Q , no interest rate) perfect replication by self-financing strategies martingale representation � T � T P ( T , S T ) = E Q [ P ( T , S T )]+ Z t dW t = E Q [ P ( T , S T )]+ ξ t dS t 0 0 where in case of a European option ξ t = ∂ P ( t , S t ) with P ( t , s ) = E Q [ P ( T , S T ) | S t = s ] ∂ s delta-hedge Mich` ele Vanmaele — F¨ ollmer-Schweizer or Galtchouck-Kunita-Watanabe Decomposition? A Comparison and Description 6/44
Introduction Quadratic hedging GKW- versus FS-decomposition (Counter)examples References Complete market Hedging in Black-Scholes model dS t = σ S t dW t (martingale measure Q , no interest rate) perfect replication by self-financing strategies martingale representation � T � T P ( T , S T ) = E Q [ P ( T , S T )]+ Z t dW t = E Q [ P ( T , S T )]+ ξ t dS t 0 0 where in case of a European option ξ t = ∂ P ( t , S t ) with P ( t , s ) = E Q [ P ( T , S T ) | S t = s ] ∂ s delta-hedge Mich` ele Vanmaele — F¨ ollmer-Schweizer or Galtchouck-Kunita-Watanabe Decomposition? A Comparison and Description 6/44
Introduction Quadratic hedging GKW- versus FS-decomposition (Counter)examples References Complete market Hedging in Black-Scholes model dS t = σ S t dW t (martingale measure Q , no interest rate) perfect replication by self-financing strategies martingale representation � T � T P ( T , S T ) = E Q [ P ( T , S T )]+ Z t dW t = E Q [ P ( T , S T )]+ ξ t dS t 0 0 where in case of a European option ξ t = ∂ P ( t , S t ) with P ( t , s ) = E Q [ P ( T , S T ) | S t = s ] ∂ s delta-hedge Mich` ele Vanmaele — F¨ ollmer-Schweizer or Galtchouck-Kunita-Watanabe Decomposition? A Comparison and Description 6/44
Introduction Quadratic hedging GKW- versus FS-decomposition (Counter)examples References Incomplete market jumps, stochastic volatility or trading constraints martingale representation above does not hold ‘every claim attainable and replicated by self-financing strategy’ is not valid relax one of these two conditions hedging is an approximation problem Mich` ele Vanmaele — F¨ ollmer-Schweizer or Galtchouck-Kunita-Watanabe Decomposition? A Comparison and Description 7/44
Introduction Quadratic hedging GKW- versus FS-decomposition (Counter)examples References Incomplete market utility maximization: non-linear pricing/hedging rule � � T � max U ( c + ξ t dS t − H ) E ξ 0 quadratic hedging: linear pricing/hedging rule � � � T ξ t dS t − H ) 2 min ξ E ( c + (mean-variance) 0 � ( C T − C t ) 2 � � F t ] min ((local) risk minimization) ξ E optimal hedging portfolio (if exists) is L 2 -projection of H onto the (linear) subspace of hedgeable claims Mich` ele Vanmaele — F¨ ollmer-Schweizer or Galtchouck-Kunita-Watanabe Decomposition? A Comparison and Description 8/44
Introduction Quadratic hedging GKW- versus FS-decomposition (Counter)examples References Outline 1 Introduction 2 Quadratic hedging Risk-minimization GKW-decomposition Local risk-minimization F¨ ollmer-Schweizer decomposition 3 GKW- versus FS-decomposition 4 (Counter)examples 5 References Mich` ele Vanmaele — F¨ ollmer-Schweizer or Galtchouck-Kunita-Watanabe Decomposition? A Comparison and Description 9/44
Introduction Quadratic hedging GKW- versus FS-decomposition (Counter)examples References Quadratic hedging finding optimal hedging portfolio ⇔ finding GKW-decomposition or finding FS-decomposition Martingale case: easy to determine ξ which is same for RM and MVH ( η differs) Semimartingale case = martingale + drift LRM: general solution MVH: no general solution due to self-financing condition Mich` ele Vanmaele — F¨ ollmer-Schweizer or Galtchouck-Kunita-Watanabe Decomposition? A Comparison and Description 10/44
Introduction Quadratic hedging GKW- versus FS-decomposition (Counter)examples References Quadratic hedging finding optimal hedging portfolio ⇔ finding GKW-decomposition or finding FS-decomposition Martingale case: easy to determine ξ which is same for RM and MVH ( η differs) Semimartingale case = martingale + drift LRM: general solution MVH: no general solution due to self-financing condition Mich` ele Vanmaele — F¨ ollmer-Schweizer or Galtchouck-Kunita-Watanabe Decomposition? A Comparison and Description 10/44
Introduction Quadratic hedging GKW- versus FS-decomposition (Counter)examples References Risk-minimization S : local martingale under measure P T -contingent claim H ∈ L 2 ( P ) not self-financing strategy but mean self-financing strategy, i.e. cost process is martingale H -admissible strategy: value process has terminal value H value process V of discounted portfolio: V t = E [ H |F t ] Mich` ele Vanmaele — F¨ ollmer-Schweizer or Galtchouck-Kunita-Watanabe Decomposition? A Comparison and Description 11/44
Introduction Quadratic hedging GKW- versus FS-decomposition (Counter)examples References GKW-decomposition F¨ ollmer and Sondermann (1986): solution to risk-minimization problem can be found by Galtchouk-Kunita-Watanabe decomposition � T H = E [ H ] + ξ u dS u + L T 0 with L local martingale orthogonal to S by martingale property � t V t = E [ H |F t ] = E [ H ] + ξ u dS u + L t 0 Hedging strategy: ϕ = ( ξ t , V t − ξ t S t ) Mich` ele Vanmaele — F¨ ollmer-Schweizer or Galtchouck-Kunita-Watanabe Decomposition? A Comparison and Description 12/44
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