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 paper suggests Lévy copulas in order to characterize the dependence among components of multidimensional Lévy processes. This concept parallels the notion of a copula on the level of Lévy measures. As for random vectors, a version of Sklar's theorem states that the law of a general multivariate Lévy process is obtained by combining arbitrary univariate Lévy processes with an arbitrary Lévy copula. We construct parametric families of Lévy copulas and prove a limit theorem, which indicates how to obtain the Lévy copula of a multivariate Lévy process X from the ordinary copula of the random vector X t for small t.

We determine the variance-optimal hedge when the logarithm of the underlying price follows a process with stationary independent increments in discrete or continuous time. Although the general solution to this problem is known as backward recursion or backward stochastic differential equation, we show that for this class of processes the optimal endowment and strategy can be expressed more explicitly. The corresponding formulas involve the moment, respectively, cumulant generating function of the underlying process and a Laplace-or Fourier-type representation of the contingent claim. An example illustrates that our formulas are fast and easy to evaluate numerically.

Abstract. We propose a mean-reverting model for the spot price dynamics of electricity which includes seasonality of the prices and spikes. The dynamics is a sum of non-Gaussian Ornstein-Uhlenbeck processes with jump processes giving the normal variations and spike behaviour of the prices. The amplitude and frequency of jumps may be seasonally dependent. The proposed dynamics ensures that spot prices are positive, and that the dynamics is simple enough to allow for analytical pricing of electricity forward and futures contracts. Electricity forward and futures contracts have the distinctive feature of delivery over a period rather than at a fixed point in time, which leads to quite complicated expressions when using the more traditional multiplicative models for spot price dynamics. We demonstrate in a simulation example that the model seems to be sufficiently flexible to capture the observed dynamics of electricity spot prices. We also discuss the pricing of European call and put options written on electricity forward contracts.

Abstract. In this paper two kinds of cumulant processes are studied in a general setting. These processes generalize the cumulant of an infinitely divisible random variable and they appear as the exponential compensator of a semimartingale. In a financial context cumulant processes lead to a generalized Esscher transform. We also provide some new criteria for uniform integrability of exponential martingales.