Almost sure optimal hedging strategy Almost sure optimal hedging strategy emmanuel.gobet@polytechnique.edu Centre de Mathématiques Appliquées, Ecole Polytechnique and CNRS With the support of Joint work with N. Landon. E. Gobet Conférence Finance Quantitative et Statistique - Université Paris 7 - 1er mars 2012 p. 1/15
Almost sure optimal hedging strategy Problem: Hedging errors due to discrete rebalancing • Payoff: g ( S T ) with d -dimensional Itô diffusion ( S t ) t ≥ 0 • Price function: u ( t, x ) = E ( g ( S T ) | S t = x ) (zero interest-rate) � t • Price process: V t = u ( t , S t ) = u ( 0 , S 0 ) + 0 D x u ( θ, S θ ) · d S θ . • Rebalancing strategy along time grid: π = { 0 = t 0 < ... < t i < ... < t N = T } • Hedging portfolio based on π : � t V N t = u ( 0 , S 0 ) + D x u ϕ ( θ ) · d S θ , 0 where ϕ ( t ) = max { t j ∈ π : t j ≤ t } . � t • Hedging error: Z N t = V t − V N � � t = D x u θ − D x u ϕ ( θ ) · d S θ . 0 Purpose: compute the optimal grids π minimizing a.s. N � Z N . � T as N → ∞ , over the set of deterministic and stopping times strategies π . E. Gobet Conférence Finance Quantitative et Statistique - Université Paris 7 - 1er mars 2012 p. 2/15
Almost sure optimal hedging strategy 1. Literature background 1. Literature background � t Z N � � π = { 0 = t 0 < ... < t i < ... < t N = T } , = D x u θ − D x u ϕ ( θ ) · d S θ . t 0 √ NZ N • Weak convergence: T weakly converges to a Gaussian mixture – when π is deterministic [Bertsimas, Kogan, Lo ’01; Hayashi, Mykland ’05] (under rather weak assumptions) – when π consists of stopping times [Fukasawa ’11] (under conditions easy to check in dim 1, and hardly tractable in higher dimension) T ) 2 = E � Z N � T (under the RN measure) E ( Z N • L 2 norm: – for uniform grids: E � Z N � T ∼ CN − α where α ∈ (0 , 1] is the fractional regularity index of g ( S T ) ; [Zhang’ 99, G’-Temam ’ 01, Geiss-Geiss ’ 04, Geiss-Hujo’ 07, G’-Makhlouf ’10] – appropriate deterministic non uniform grids give E � Z N � T ∼ CN − 1 ; – best n -stopping times [Martini-Patry ’99] (optimal multi-stopping pb); – Asymptotic minimization over stopping times: lim inf E ( N ) E � Z N � T . Convex payoff in dimension 1, mainly within BS model [Fukasawa 10] . E. Gobet Conférence Finance Quantitative et Statistique - Université Paris 7 - 1er mars 2012 p. 3/15
Almost sure optimal hedging strategy 2. Lower bounds and minimizing stopping times 2. Lower bounds and minimizing stopping times Purpose: 1. Compute the a.s. n →∞ N n T � Z n � T lim inf over the set of admissible sequence of strategies. The meaning of n → ∞ is given later. 2. Provide a minimizing sequence. Assumptions • Model of d risky assets: � t � t S t = S 0 + b s d s + σ s d B s . 0 0 W.l.o.g. b ≡ 0 . To simplify σ t = σ ( t, S t ) with σ ( . ) Lipschitz. • Pathwise ellipticity: 0 < λ min ( σ t σ ∗ t ) , ∀ 0 ≤ t ≤ T . E. Gobet Conférence Finance Quantitative et Statistique - Université Paris 7 - 1er mars 2012 p. 4/15
Almost sure optimal hedging strategy 2. Lower bounds and minimizing stopping times About the assumption on Greeks First Greeks are a.s. finite in a small tube around the ( t, S t ) : ! ˛ ˛ ˛ ˛ ˛ ˛ ˛ + ˛ + ˛ D 2 ˛ D 2 ˛ D 3 ( ⋆ ) δ → 0 sup lim sup ( xx u ( t , x ) tx u ( t , x ) xxx u ( t , x ) ˛ ) < + ∞ = 1 . P 0 ≤ t < T | x − S t |≤ δ Much weaker than L p integrability assumption. • In the BS model in dimension 1, for Gamma of the Call 0.18 Call option g ( S ) = ( S − K ) + : OK 0.16 0.14 0.12 0.1 0.08 because the strike K is negligible for 0.06 0.04 0.02 0 the law of S T . 0.3 115 0.25 110 0.2 105 100 0.15 95 0.1 • Digital payoff g ( S ) = 1 S ≥ K : OK 90 0.05 85 80 0 • General diffusion with smooth coefficients: – g ( S ) = Φ ( S ) where Φ is smooth: OK – g ( S ) = 1 S ∈ F Φ ( S ) and F is a closed set which boundary has zero Lebesgue-measure: OK under ellipticity or hypo-ellipticity assumption. � includes all the "usual" continuous and discontinuous payoffs. • Open issue : find a payoff g violating the ( ⋆ ) -condition. E. Gobet Conférence Finance Quantitative et Statistique - Université Paris 7 - 1er mars 2012 p. 5/15
Almost sure optimal hedging strategy 2. Lower bounds and minimizing stopping times Asymptotic framework • Positive deterministic real numbers ( ε n ) n ≥ 0 such that P n ≥ 0 ε 2 n < + ∞ • Strategy (indexed by n = 0 , 1 , . . . ) = sequence of stopping times T n := { τ n 0 = 0 < τ n 1 < ... < τ n i < ... ≤ τ n N n T = T } ( T may be random). N n 3 ) . A sequence of strategies ( T n ) n ≥ 0 is admissible if a.s. • Let ρ N ∈ [ 1 , 4 ` ´ ` ´ ε − 1 ε 2 ρ N N n sup sup sup | S t − S τ n i − 1 | < + ∞ , sup < + ∞ . n n T 1 ≤ i ≤ N n t ∈ ( τ n i − 1 ,τ n n ≥ 0 n ≥ 0 i ] T T ∼ Cε − 2 ρ N • Deterministic times: if ρ N > 1 , a sequence of strategy with N n n i ≤ Cε 2 ρ N T ∆ τ n deterministic times and mesh size sup 1 ≤ i ≤ N n is admissible. n • Hitting times of random "ellipsoids": the strategy given by n o τ n τ n t ≥ τ n i − 1 ) ∗ H τ n i − 1 ) > ε 2 0 := 0 , i := inf i − 1 : ( S t − S τ n i − 1 ( S t − S τ n ∧ T, n defines an admissible sequence if H is a continuous adapted positive-definite d × d -matrix process. E. Gobet Conférence Finance Quantitative et Statistique - Université Paris 7 - 1er mars 2012 p. 6/15
Almost sure optimal hedging strategy 2. Lower bounds and minimizing stopping times n →∞ N n T � Z n � T ≥ . . . Sketch of proof of lim inf Lemma (matrix equation) Let c ∈ S d ( R ) . There is a unique solution x ( c ) ∈ S d + ( R ) to the equation 2 Tr( x ) x + 4x 2 = c 2 and c �→ x ( c ) is continuous. Heuristic proof: Z s Z s ` ´ Z n ( D 2 s = D x u t − D x u ϕ ( t ) · d S t = xx u ϕ ( t ) ∆ S t ) · d S t + Errors 0 0 Z T � Z n � T = ∆ B ∗ t σ ∗ ϕ ( t ) D 2 xx u ϕ ( t ) σ ϕ ( t ) σ ∗ ϕ ( t ) D 2 xx u ϕ ( t ) σ ϕ ( t ) ∆ B t d t + . . . 0 | {z } := c 2 ϕ ( t ) “ ” 2 X ∆ B ∗ ≈ i X τ n i − 1 ∆ B τ n + stochastic integrals +. . . (Matrix equation) τ n i τ n i − 1 <T „ « 2 X N T � Z n � T ≥ ∆ B ∗ i X τ n i − 1 ∆ B τ n + . . . (Cauchy-Schwarz ineq. and X ≥ 0 ) τ n i τ n i − 1 <T “ Z T ” 2 ≥ Tr ( X t ) d t + . . . (Convergence of quadratic variation) . 0 Most difficult part: error estimates without using L p estimates and in a.s. sense. E. Gobet Conférence Finance Quantitative et Statistique - Université Paris 7 - 1er mars 2012 p. 7/15
Almost sure optimal hedging strategy 2. Lower bounds and minimizing stopping times Main results Theorem (lower bound) Let X be the solution of the matrix equation with c := σ ∗ D 2 xx uσ . Then, for any admissible sequence of strategies, “ Z T ” 2 n → + ∞ N n T � Z n � T ≥ lim inf Tr ( X t ) d t , a.s. . 0 Theorem (minimizing sequence) For any µ > 0 , we can exhibit an admissible sequence of strategies such that ˛ Z T ˛ ˛ ˛ ˛ N n T � Z n � T − ( Tr( X t )d t ) 2 lim sup ˛ ≤ µ C µ a.s. , ˛ ˛ n → + ∞ 0 where the random variable C µ is a.s. finite (locally uniformly in µ ). • If the Gamma matrix is positive-definite (unif. in ( t, ω ) ), we can take µ = 0 . • If λ min ( D 2 xx u t )( ω ) > 0 for any t < T , one may have C µ ( ω ) = 0 ( optimal strategy ). • The µ -optimal strategies are of the hitting time form � deterministic times are suboptimal. E. Gobet Conférence Finance Quantitative et Statistique - Université Paris 7 - 1er mars 2012 p. 8/15
Almost sure optimal hedging strategy 2. Lower bounds and minimizing stopping times Explicit representation of the optimal strategies Let χ ( . ) ∈ C ∞ ( R ) with 1 ] −∞ , 1 / 2] ≤ χ ( . ) ≤ 1 ] −∞ , 1] . Set χ µ ( x ) = χ ( x/µ ) . • In the one dimensional case, the µ -optimal stopping times read τ n 0 := 0 and 8 9 < = ε n τ n : t ≥ τ n i = inf i − 1 : | S t − S τ n i − 1 | > ; ∧ T . √ √ q | D 2 6 + µχ µ ( | D 2 i − 1 | / i − 1 | / 6 ) xx u τ n xx u τ n Rebalancing frequency depends on the Gamma of the option. • In the general case, we have to set Λ t := ( σ − 1 ) ∗ X t σ − 1 Λ µ and t := Λ t + µχ µ ( λ min (Λ t )) I d . t t Then the µ -optimal strategy is defined by n o i − 1 ) ∗ Λ µ τ n t ≥ τ n i − 1 ) > ε 2 i = inf i − 1 : ( S t − S τ n i − 1 ( S t − S τ n ∧ T . τ n n � Hitting times of ellipsoids. E. Gobet Conférence Finance Quantitative et Statistique - Université Paris 7 - 1er mars 2012 p. 9/15
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