We provide a new characterization of mean-variance hedging strategies in a general semimartingale market. The key point is the introduction of a new probability measure P ⋆ which turns the dynamic asset allocation problem into a myopic one. The minimal martingale measure relative to P ⋆ coincides with the variance-optimal martingale measure relative to the original probability measure P .A.ČERNÝ AND J. KALLSEN paper are actually special semimartingales, we use from now on the (otherwise forbidden) "truncation" functionwhich simplifies a number of expressions considerably.By X, Y we denote the P -compensator of [X, Y ] provided that X, Y are semimartingales such that [X, Y ] is P -special (cf. [27], page 37). If X and Y are vector-valued, then [X, Y ] and X, Y are to be understood as matrixvalued processes with components [X i , Y j ] and X i , Y j , respectively. Moreover, if both Y and a predictable process ϑ are R d -valued, then the notation ϑ • [X, Y ] (and accordingly ϑ • X, Y ) refers to the vector-valued process whose components ϑIn the whole paper, we write M X for the local martingale part and A X for the predictable part of finite variation in the canonical decomposition X = X 0 + M X + A X of a special semimartingale X. If P ⋆ denotes another probability measure, we write accordinglyIf (b, c, F, A) denote differential characteristics of an R d -valued special semimartingale X, we use the notationc,ĉ for modified second characteristics in the following sense (provided that the integrals exist):Observe that x ⊤ĉ x ≤ x ⊤c x for any x ∈ R d . The notion of modified second characteristics is motivated by the following: Proposition 1.2. Let X be an R d -valued special semimartingale with differential characteristics (b, c, F, A) and modified second characteristics as in (1.2) and (1.3). If the corresponding integrals exist, thenMEAN-VARIANCE HEDGING 5 2. Admissible strategies and quadratic hedging. We work on a filtered probability space (Ω, F , (F t ) t∈[0,T ] , P ), where T ∈ R + denotes a fixed terminal time. The R d -valued process S = (S 1 t , . . . , S d t ) t∈[0,T ] represents the discounted prices of d securities. We assume that sup{E((S i τ ) 2 ) : τ stopping time , i = 1, . . . , d} < ∞, (2.1) that is, S is a L 2 (P )-semimartingale in the sense of [15].Moreover, we make the following standing:Assumption 2.1. There exists some equivalent σ-martingale measure with square-integrable density, that is, some probability measure Q ∼ P with E(( dQ dP ) 2 ) < ∞ and such that S is a Q-σ-martingale.This can be interpreted as a natural no-free-lunch condition in the present quadratic context. More specifically, Théorème 2 in [47] and standard arguments show that Assumption 2.1 is equivalent to the absence of L 2 -free lunches in the sense that K s 2 (0) − L 2 + ∩ L 2 + = {0},

This is the accepted version of the paper.This version of the publication may differ from the final published version. (2008), and, if we restrict our attention only to lognormally distributed returns, also Veµ ceµ r (2002). While the existing convolution algorithms compute the density of the underlying state variable by moving forward on a suitably de…ned state space grid our new algorithm uses backward price convolution, which resembles classical lattice pricing algorithms. For the …rst time in the literature we provide an analytical upper bound for the pricing error caused by the truncation of the state space grid and by the curtailment of the integration range. We highlight the bene…ts of the new scheme and benchmark its performance against existing …nite di¤erence, Monte Carlo, and forward density convolution algorithms. Permanent repository link

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Dedicated to Walter Schachermayer on the occasion of his 60th birthday.Abstract. The choice of admissible trading strategies in mathematical modelling of financial markets is a delicate issue, going back to Harrison and Kreps [HK79]. In the context of optimal portfolio selection with expected utility preferences this question has been the focus of considerable attention over the last twenty years.We propose a novel notion of admissibility that has many pleasant features -admissibility is characterized purely under the objective measure P ; each admissible strategy can be approximated by simple strategies using finite number of trading dates; the wealth of any admissible strategy is a supermartingale under all pricing measures; local boundedness of the price process is not required; neither strict monotonicity, strict concavity nor differentiability of the utility function are necessary; the definition encompasses both the classical mean-variance preferences and the monotone expected utility.For utility functions finite on R, our class represents a minimal set containing simple strategies which also contains the optimizer, under conditions that are milder than the celebrated reasonable asymptotic elasticity condition on the utility function.

This is the accepted version of the paper.This version of the publication may differ from the final published version. Abstract. The paper presents an incomplete market pricing methodology generating asset price bounds conditional on the absence of attractive investment opportunities in equilibrium. The paper extends and generalises the seminal article of Cochrane and Saá-Requejo who pioneered option pricing based on the absence of arbitrage and high Sharpe Ratios. Our contribution is threefold: Permanent repository link:We base the equilibrium restrictions on an arbitrary utility function, obtaining the Cochrane and Saá-Requejo analysis as a special case with truncated quadratic utility. We extend the de…nition of Sharpe Ratio from quadratic utility to the entire family of CRRA utility functions and restate the equilibrium restrictions in terms of Generalised Sharpe Ratios which, unlike the standard Sharpe Ratio, provide a consistent ranking of investment opportunities even when asset returns are highly non-normal. Last but not least, we demonstrate that for Itô processes the Cochrane and Saá-Requejo price bounds are invariant to the choice of the utility function, and that in the limit they tend to a unique price determined by the minimal martingale measure.