A utility maximization proof of Strassen s theorem

Transcription

1 Introduction CMAP, Ecole Polytechnique Paris Advances in Financial Mathematics, Paris January, 2014

2 Outline Introduction Notations Strassen s theorem 1 Introduction Notations Strassen s theorem 2

3 General framework Introduction Notations Strassen s theorem The probability space is Ω := R d R d. We denote by (X, Y ) the canonical process on R d R d. For two measures µ and ν on R d, we denotes by P(µ, ν) the set : P(µ, ν) := { P P R 2d : X P µ and Y P ν }. The subset of P(µ, ν) consisting of martingale probability laws is : M(µ, ν) := { P P(µ, ν) : E P [Y X ] = X, µ a.s. }.

4 Introduction Notations Strassen s theorem Existence of a martingale measure with given marginals We observe easily that P(µ, ν) is non-empty since µ ν is one of its elements. The non-emptyness of M(µ, ν) is much more complicated and is given by the following : Theorem (Strassen 1965) Assume that µ and ν are two probability measures on R d with x µ(dx) + y ν(dy) <, then M(µ, ν) if and only if µ ν in convex order, i.e. µ(g) ν(g) for all convex function g, where µ(g) := g(x)µ(dx), ν(g) = g(y)ν(dy). The initial proof provided by Strassen is an application of Hahn-Banach theorem.

5 Introduction Notations Strassen s theorem Short explanation of Strassen s condition The intuition of Strassen s condition for the existence of a martingale measure with given marginal laws is the following : The first condition, x µ(dx) + y ν(dy) <, is indeed obvious since, by definition of a martingale, the canonical process (X, Y ) has to be integrable. The second condition, µ(g) ν(g) for all g convex, comes from the Jensen s inequality. Indeed, if E P [Y X ] = X, then by Jensen s inequality, we have that for g convex that : [ ] ν(g) = E P [g(y )] = E P E P [g(y ) X ] E P [g(x )] = µ(g).

6 General idea Introduction Notations Strassen s theorem Our new proof of Strassen s theorem is based on a proof due to L.C.G. Rogers (1994) of the Fundamental Theorem of Asset Pricing, based on utility maximization technics. We then adapt his framework to the to derive the result. Theorem (FTAP) Consider a probability law P on Ω = R d R d, F 1 := σ(x ) and F 2 := σ(x, Y ). Then the following are equivalent : (i) There exists P equivalent to P such that the canonical process is a martingale under P ; (ii) There is no arbitrage opportunity, i.e. there is no F 1 -measurable r.v. H such that H(Y X ) 0 P-a.s. and P [H(Y X ) > 0] > 0.

7 Outline Introduction 1 Introduction Notations Strassen s theorem 2

8 Introduction Martingale optimal transport setup I We consider a financial situation of a market in discret time 0,1,2, consisting of a family of tradable assets corresponding to a d-dimensional vector S with values S 1 = X and S 2 = Y, and of all European options of maturity 1 and 2 available at time 0. Then under the assumption of linearity of the pricing functionnal, we know that any european position φ of maturity 1 (resp ψ of maturity 2) has the no arbitrage price µ(φ) (resp ν(ψ), where µ (resp ν) is the law of S at time 1 (resp 2) ( identified from the calls and puts prices if we were considering the 1-dimensional case). An agent strategy is then given by a triplet (h, φ, ψ), where h is the classical trading strategy in the underlying, and φ (resp ψ) is the european position taken at time 0 of maturity 1 (resp 2).

9 Introduction Martingale optimal transport setup II For (h, φ, ψ) L 0 (R d ) L 1 (µ) L 1 (ν), we define : h (x, y) := h(x) (y x), φ µ (x) := φ(x) µ(φ), ψ ν (y) := ψ(y) ν(ψ), and φ ψ(x, y) := φ(x) + ψ(y). Then the final wealth of the agent is given by : h (X, Y ) + φ µ ψ ν (X, Y ).

10 Introduction Problem formulation and main result We now describe the utility maximization problem. For a given strategy (h, φ, ψ), the agent s utility is given by : J µ,ν (h, φ, ψ) := E µ ν [ e (h +φ µ ψ ν )(X,Y ) ]. Then the utility maximization problem is : We have the following : V (µ, ν) := sup J µ,ν (h, φ, ψ). h,φ,ψ Proposition ("finiteness" of the utility maximization problem) Assume that µ ν, then : V (µ, ν) < 0.

11 Formal proof I Introduction We deduce from the proposition the main result. Indeed assume the existence of a maximizing triplet (h, φ, ψ ), then the probability law P defined by dp dµ ν := e (h +φ µ ψ ν)(x,y ) V (µ, ν) is an element of M(µ, ν). This is a consequence of the first order condition of the maximization. We describe formally the proof for the first marginal law. The same scheme will lead to the martingale property and the second marginal law.

12 Formal proof II Introduction Formally, we have for any ε and φ 0 that : J µ,ν (h, εφ 0 + φ, ψ) J µ,ν (h, φ, ψ ). Then the first order derivative of this function of ε at 0 is 0, so that : [ 0 = E µ ν e (h +φ µ ψ ν )(X,Y ) ( φ 0 (X ) µ(φ 0 ) )] ( ) = V (µ, ν) E P [φ 0 (X )] µ(φ 0 ). Since this is true for all φ 0, we have that the first marginal law of P is µ.

13 Bibliography Introduction Beiglböck, M., Juillet, N. (2012) : On a problem of optimal transport under marginal martingale constraints, preprint, arxiv : Rogers, L.C.G. (1994). Equivalent martingale measures and no-arbitrage. Stochastics and Stochastics Reports 51, Royer, G. (2014) A utility maximization proof of Strassen s theorem, working paper Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist., 36 :

14 Introduction THANK YOU FOR YOUR ATTENTION