Discrete hedging in models with jumps Peter Tankov CMAP, Ecole - PowerPoint PPT Presentation

Introduction Model setup Weak convergence L 2 convergence Discrete hedging in models with jumps Peter Tankov CMAP, Ecole Polytechnique Partly joint work with E. Voltchkova (Universit´ e Toulouse 1) Workshop on Optimization and Optimal Control, Linz, October 20–24, 2008 Peter Tankov Discrete hedging in models with jumps

Introduction Model setup Hedging in incomplete markets Weak convergence Discrete hedging L 2 convergence Hedging in incomplete markets Incomplete market: exact replication impossible. Hedging is now an approximation problem. Industry practice: sensitivities to risk factors Delta = ∂ C ( t , S t ) : infinitesimal moves, hedge with stock ∂ S Gamma = ∂ 2 C ( t , S t ) : bigger moves; hedge with liquid options ∂ S 2 Quadratic hedging: control the residual error � T � 2 � min φ E c + φ t dS t − Y 0 All these strategies require a continuously rebalanced portfolio. Peter Tankov Discrete hedging in models with jumps

Introduction Model setup Hedging in incomplete markets Weak convergence Discrete hedging L 2 convergence Discrete hedging Continuous rebalancing is unfeasible: in practice, the strategy φ t is replaced with a discrete strategy, leading to the hedging error of the “second type”: error of approximating the continuous portfolio with a discrete one. The simplest choice is φ n t := φ h [ t / h ] , h = T / n . This discretization error has only been studied in the case of continuous processes. Two main approaches: weak convergence (CLT for hedging error) and L 2 convergence Peter Tankov Discrete hedging in models with jumps

Introduction Model setup Hedging in incomplete markets Weak convergence Discrete hedging L 2 convergence Discrete hedging: the complete market case Bertsimas, Kogan and Lo ’98 introduced an asymptotic approach allowing to study discrete hedging in continuous time. Suppose dS t = µ ( t , S t ) dt + σ ( t , S t ) dW t S t and we want to hedge a European option with payoff h ( S T ) using delta-hedging φ t = ∂ C ∂ S . Peter Tankov Discrete hedging in models with jumps

Introduction Model setup Hedging in incomplete markets Weak convergence Discrete hedging L 2 convergence CLT for hedging error The discrete hedging error is defined by � T ε n φ n T = h ( S T ) − t dS t 0 T → 0 but the renormalized error √ n ε n Then ε n T converges to � T � ∂ 2 C T ∂ S 2 S 2 t σ 2 t dW ∗ t , 2 0 where W ∗ is a Brownian motion independent of W . √ Hedging error decays as h . It is orthogonal to the stock price. The amplitude is determined by the gamma ∂ 2 C ∂ S 2 Peter Tankov Discrete hedging in models with jumps

Introduction Model setup Hedging in incomplete markets Weak convergence Discrete hedging L 2 convergence Approximating hedging portfolios Hayashi and Mykland ’05 interpreted the discrete hedging error as � T the error of approximating the “ideal” hedging portfolio 0 φ t dS t � T 0 φ n with a feasible hedging portfolio t dS t • This makes sense in incomplete markets Suppose φ and S are Itˆ o process: d φ t = ˜ µ t dt + ˜ σ t dW t and dS t = µ t dt + σ t dW t . Then � t � σ t = ∂ 2 C √ n ε n T � � σ s σ s dW ∗ t ⇒ ˜ s , ˜ ∂ S 2 S t σ t 2 0 � t ε n ( φ t − φ n where t := t ) dS t . 0 • Weak convergence of processes in the Skorokhod topology on the space D of c` adl` ag functions Peter Tankov Discrete hedging in models with jumps

Introduction Model setup Hedging in incomplete markets Weak convergence Discrete hedging L 2 convergence L 2 hedging error for continuous processes Result by Zhang (1999): for call/put options, the L 2 hedging error converges to the expected square of the weak limit. �� T � 2 � � ∂ 2 C T ) 2 ] = T n →∞ nE [( ε n S 4 t σ 4 lim 2 E s ds . ∂ S 2 0 The constant may be improved by an intelligent choice of rebalancing dates (Brod´ en and Wiktorsson ’08) but the convergence rate cannot be improved. See also related results by Gobet and Temam (01) and Geiss (02), (06), (07). Peter Tankov Discrete hedging in models with jumps

Introduction Model setup Hedging in incomplete markets Weak convergence Discrete hedging L 2 convergence Discretization error in presence of jumps Our idea: study the discretization error � t ε n ( φ t − − φ n t := t − ) dS t 0 in presence of jumps in the underlying and the hedging strategy. Approximation error of the L´ evy-driven Euler scheme: Jacod and Protter (98), Jacod (04) Related results in the approximation of quadratic variation by realized volatility � T X 2 T = X 2 0 + 2 X t − dX t + [ X , X ] T 0 Limit theorems for the approximation error of quadratic variation: Jacod (08). Peter Tankov Discrete hedging in models with jumps

Introduction Model setup Weak convergence L 2 convergence Model setup: L´ evy-Itˆ o processes � t � t � t � γ s ( z )˜ X t = X 0 + µ s ds + σ s dW s + J ( ds × dz ) 0 0 0 | z |≤ 1 � t � + γ s ( z ) J ( ds × dz ) . 0 | z | > 1 • J : Poisson random measure with intensity dt × ν • µ and σ are c` adl` ag ( F t )-adapted • γ : Ω × [0 , T ] × R → R is such that ( ω, z ) �→ γ t ( z ) is F t × B ( R )-measurable ∀ t and t → γ t ( z ) is c` agl` ad ∀ ω, z ; � γ t ( z ) 2 ≤ A t ρ ( z ) , ρ ( z ) ν ( dz ) < ∞ | z |≤ 1 with ρ positive deterministic and A c` agl` ad ( F t )-adapted. Peter Tankov Discrete hedging in models with jumps

Introduction Model setup Weak convergence L 2 convergence Model setup The stock price S is a L´ evy-Itˆ o process with coefficients µ, σ, γ ; The continuous-time strategy φ is a L´ evy-Itˆ o process with coefficients ˜ µ, ˜ σ, ˜ γ . The agent uses the discrete strategy φ n t := φ h [ t / h ] instead of the continuous strategy φ t . Peter Tankov Discrete hedging in models with jumps

Introduction The asymptotic error process Model setup Proof of the weak convergence Weak convergence Delta-hedging in a L´ evy market L 2 convergence Weak convergence: the normalizing sequence The normalizing factor need not be equal to √ n . Suppose φ and S move only by finite-intensity jumps. If there is only one jump between t i and t i +1 , � t i +1 � t i +1 φ n φ t − dS t = t − dS t t i t i Therefore P [ ε n t � = 0] = O (1 / n ) and n α ε n t → 0 in probability ∀ α . More generally, if S and φ are L´ evy-Itˆ o processes without diffusion parts, √ n ε n t → 0 in probability uniformly on t . Peter Tankov Discrete hedging in models with jumps

Introduction The asymptotic error process Model setup Proof of the weak convergence Weak convergence Delta-hedging in a L´ evy market L 2 convergence Weak convergence The discretization error satisfies � t � √ √ n ε n T � σ s dW ∗ � t → σ s ˜ s + T ∆ φ T i ζ i ξ i σ T i 2 0 i : T i ≤ t √ � � 1 − ζ i ξ ′ + ∆ S T i i ˜ σ T i − . T i : T i ≤ t W ∗ is a standard BM independent from W and J , ( ξ k ) k ≥ 1 and ( ξ ′ k ) k ≥ 1 are two sequences of independent N (0 , 1), ( ζ k ) k ≥ 1 is sequence of independent U ([0 , 1]) ( T i ) i ≥ 1 are the jump times of J enumerated in any order. Peter Tankov Discrete hedging in models with jumps

Introduction The asymptotic error process Model setup Proof of the weak convergence Weak convergence Delta-hedging in a L´ evy market L 2 convergence Remarks on convergence The hedging error √ n ε n t converges weakly in finite-dimensional laws but not in Skorohod topology. The discretized error process √ n ε n h [ t / h ] converges in Skorohod topology to the same limit. Peter Tankov Discrete hedging in models with jumps

Introduction The asymptotic error process Model setup Proof of the weak convergence Weak convergence Delta-hedging in a L´ evy market L 2 convergence Idea of the proof Main tool: if ( X n ) and ( Y n ) are two sequences of processes such that t | X n t − Y n sup t | → 0 in probability and X n → X weakly then Y n → X weakly. Peter Tankov Discrete hedging in models with jumps

Introduction The asymptotic error process Model setup Proof of the weak convergence Weak convergence Delta-hedging in a L´ evy market L 2 convergence Idea of the proof Step 1 Remove the big jumps Step 2 Remove the small jumps Step 3 Now we can write t + S j S t = S 0 + S d t + S c t � t � � � S d t = µ s + γ s ( z ) ν ( dz ) ds 0 � t S c t = σ s dW s 0 � t � S j t = γ s ( z ) J ( ds × dz ) 0 t + φ j and φ t = φ 0 + φ d t + φ c t . Peter Tankov Discrete hedging in models with jumps

Introduction The asymptotic error process Model setup Proof of the weak convergence Weak convergence Delta-hedging in a L´ evy market L 2 convergence Idea of the proof The leading terms in the hedging error are � t � √ n � T t − φ c , n σ s dW ∗ ( φ c t ) dS c t → σ s ˜ s 2 0 � r ( T i ) √ n √ n ∆ φ T i � ( φ j t − φ j , n � t ) dS c t = σ s dW s T i i √ � � → T ∆ φ T i ζ i ξ i σ T i i : T i ≤ t � T i √ n √ n ∆ S T i � t − φ c , n t ) dS j � ( φ c t = ˜ σ s dW s l ( T i ) i √ � 1 − ζ i ξ ′ � → T ∆ S T i i ˜ σ T i − . i : T i ≤ t Peter Tankov Discrete hedging in models with jumps