On the structure of general mean-variance hedging strategies

Abstract

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},

Mean-variance hedging with oil futures

“…On the proof of Theorem 3.3. a) This can be deduced as a special case of Theorem 4.10 inČerný and Kallsen [5], who establish the structure of the solution to the mean-variance hedging problem for a general semimartingale. This generality is not necessary for our setup, and simpler proofs can be found in the literature when asset prices are continuous.…”

“…Part a) is Theorem 2.8 inČerný and Kallsen [6], where the inclusion ⊇ follows from Theorems 1.2 and 2.2 in Delbaen and Schachermayer [8], and part b) is Corollary 2.9 part 1 inČerný and Kallsen [5]. In particular, a) says that the set of admissible strategies A coincides with the set of strategies used in Gourieroux, Laurent and Pham [11].…”

“…In the following we solve (3.3) -(3.6) in the context of the Gaussian convenience yield model of section 2.4. For more background and general results on problem (3.6), we refer to Schweizer [21] andČerný and Kallsen [5].…”

“…Several versions of Theorem 3.3 a) can be found in a number of papers including [18], [11] and [5]. A brief discussion of relations between these results and the proof can be found in appendix A.2.…”

“…The Markowitz problems (1.1) -(1.3) are closely related to the quadratic hedging problem of minimizing the expected quadratic hedging error (see (3.6) below), which has been extensively studied in various setups and levels of generality by many articles such as [18], [11], [16] and [5], to name but a few. Although the general structure of the solution to this problem is well-understood, obtaining closed form expressions for the solution in a concrete model can be quite tricky in applications.…”