Case 0 < c min < c max < ∞

In this subsection, we study some types of arbitrage opportunities for a non convex market model. Inspired from Lemma 6.3.2, in the definition below we introduce two notionsNAVRandNWAVRwhich naturally are asymptotic analogs ofNA and NWA. We make use of LV^∞ instead of A^∞_T since the former has a naturally economical interpretation. More suprisingly is the theorem 6.4.8 below which states that NAVR and NWAVR are in fact not really a strict generalization ofNA and NWA in the case 0 < c_min <

c_max<∞.

Definition 6.4.7. We say that the NAVR condition holds if, for all se- quence ξⁿ ∈ R⁰_T such that L_T(ξⁿ) ≥ −β_n for all n where β_n → 0 and ξⁿ→ξ ∈G_T a.s., we have L_T(ξ) = 0.

The market is said to be satisfied the NWAVR condition if, for all t = 1, . . . , T and all sequenceξⁿ∈R⁰_t,T such that L_T(ξⁿ)≥ −βn for all nwhere β_n→0 andξⁿ→ξ a.s.,L_T(ξ)≥ψ_t∈L⁰₊(Ft) we have ψ_t= 0.

Theorem 6.4.8. If 0 < cmin < cmax < ∞ then four conditions NA, NAVR, NWA andNWAVR are equivalent.

Proof.By elementary arguments we can show thatNAVR⇒NA⇒NWA andNAVR⇒NWAVR⇒NWA. It then suffices to verify thatNWA⇒ NAVR. Suppose on the contrary that, if NAVRdoes not satisfy, then we can find a sequence V_Tⁿ ∈ R⁰_T such that L_T(V_Tⁿ) ≥ −β_n for all n where β_n→0and V_Tⁿ→X_T ∈G_T a.s. such thatL_T(X_T)≥0,L_T(X_T)6= 0.

We now aim at constructing a new sequence of porfolioVˆ_Tⁿ:=PT

t=1ξˆⁿ_t such thatVˆ_TⁿV_Tⁿ and ξˆ_tⁿ= ˆξ_tⁿ1_−ξˆn

t∈Kt+Ct. Let us define, for each1≤t≤T : Vˆ_tⁿ:=V_tⁿ+

Xt

u=1

[L_u(V_u−1ⁿ −V_uⁿ)e₁+r_u⁺(V_u−1ⁿ −V_uⁿ)], wherer⁺_t is defined byr⁺_t (x) :=r_t(x)1_rt(x)∈R^d

+, r_t(x) :=x−L_t(x)e₁denoting the remaining part of x after liquidating. The above equality shows that Vˆ_tⁿV_tⁿ. We also get that

Vˆ_t−1ⁿ −Vˆ_tⁿ=V_t−1ⁿ −V_tⁿ−L_t(V_t−1ⁿ −V_tⁿ)e₁−r⁺_t (V_t−1ⁿ −V_tⁿ)].

We deduce from this equality that−ξˆ_tⁿ∈K_t+C_t⊆G_twhereξˆ_tⁿ= ˆV_tⁿ−Vˆ_t−1ⁿ . Therefore,Vˆ_tⁿ is a portfolio satisfying all required properties.

We now choosen₀ large enough such that β_n<−c_min∀n≥n₀.SinceVˆ_Tⁿ V_Tⁿ we haveL_T( ˆV_Tⁿ)≥L_T(V_Tⁿ). We then get that

lim infL_T( ˆV_Tⁿ)≥L_T(X_T)≥0,L_T(X_T)6= 0.

Therefore, we can deduce that there exists n₁ ≥n₀ such thatVˆ_Tⁿ¹ 6= 0.We rewrite Vˆ_Tⁿ¹ = PT

t=0ξˆ_tⁿ¹ where ξˆ_tⁿ¹ ∈ −G_t. For ease of notations we will denoteV_T^∗ = ˆV_Tⁿ¹.We also suppose thatV_T^∗ =L_T(V_T^∗)e₁.Let us define

π^∗ := min{t≥0 : ξ^∗_t 6= 0}, and

t^∗:= min{t≥0 : P(π^∗ =t)>0}.

This means that we start making transactions from the instant t^∗ and do nothing before t^∗. DenoteB_t^∗ :={ξ_t^∗∗ 6= 0}, we haveP(B_t^∗)>0. From the construction of V_T^∗ we have −ξ_t^∗∗ ∈ Kt+Ct on Bt^∗ By using Lemma 6.4.1 i), we can construct a new porfolioV˜_T in the anlarged market K associated to V_T^∗ such that V˜_T = V_T^∗ +C_T^∗e₁ where C_T^∗ is the cumulated fixed costs incuring in the portfolioV_T^∗.It is now clear thatC_T^∗ ≥cminon the eventBt^∗. AsV_T^∗=L_T(V_T^∗)e₁ we get that

L^K_T ( ˜VT)≥cmin−β_n¹ =:ε >0 onBt^∗ :.

By Lemma 6.4.1 ii) we can construct a new porfolio V_T^k in the market G such that

V_T^k=kV˜_T −C_Te₁, k >0,

6.4 Market with one risky asset 153 whereC_T ≤c_maxT. By choosing klarge enough we have

L_T(V_T^k)≥kε−cmaxT >0 onBt^∗:, i.e.V_T^k is a Weak Arbitrage Opportunity in the market G.

Corollary 6.4.9. Suppose that 0 < cmin < cmax < ∞. Then NAVR is equivalent to LV^∞_0,T ∩L⁰(R₊,FT) = {0} and NWAVR is equivalent to LV^∞_t,T ∩L⁰(R₊,Ft) ={0}.

Chapitre 7

Utility Maximization under Target Risk Constraints

Abstract

This chapter studies the classical utility maximization problem in a general continuous-time financial markets model under target risk constraints, i.e.

given in terms of some expected loss constraints imposed on the final wealth.

The standard duality technics are used to solve the problem. In a complete market, we relate the utility maximization problem to some kind of hedging problem with multiple targets. The latter is solved by using both backward stochastic differential equation (BSDE) and convex duality technics.

Keywords : Utility maximization, Duality method, Shortfall risk, Convex risk measures, Optimal portfolio choice, Quantile hedging.

Note. This chapter is based on the article Utility Maximization under Target Risk Constraints, T. Tran. Preprint. It was done under the supervision of Bruno Bouchard and I want to thank him for his advice and helpful comments.

7.1 Introduction

In this chapter, we consider the utility maximization problem under expected loss constraints. First introduced by Merton (1971) in the non constraint case, the problem has attracted a lot of attentions from academicians and practitioners. Especially in the case of complete markets, it is solved by the martingale and duality methods, see for example Karatzas, Lehoczky and Shreve (1987) or Cox and Huang (1989). These technics appear to be very powerful even for incomplete markets. In the non-Markovian case without constraints, the existence of solutions and the characterizations of optimal strategies are given by He and Pearson (1991), Karatzas, Lehoczky, Shreve and Xu (1991). Similar results have been derived by Cvitanic and Karatzas (1992) or Cuoco (1997) in the case of portfolio constraints.

155

For financial institutions the measurement and management of downside risk is a key issue. Regulators, for example, might impose a risk constraint to cer- tain companies, a manager of a firm might require his traders to stay within some risk limit, or an investor might wish to bound his own risk exposure.

It is then natural to consider the utility maximization problem under risk constraints. Optimal investment policies under downside risk constraints in terms of value at risk and and a second risk functional have been studied in a Brownian setting by Basak & Shapiro (2001) and Gabih et al. (2005).

A complete solution in a general semimartingale with utility-based shortfall risk constraints is given by Gundel and Weber (2005). In the latter, the au- thors reduce the problem to a static optimization one under constraints and then solve it by means of classical Lagrange multipliers. These mutipliers corespond to the expected loss and the budget constraints.

In this chapter, we investigate the utility maximization problem under expec- ted loss constraints in a general setting, where the price process is described by a general semimartingale and the risk constraint is represented by a non decreasing concave function. This framework is more or less similar to that of Gundel and Weber (2005). The difference is that we chose the convex duality theory in the spirit of Kramkov and Schachermayer (1999) or Schachermayer (2001) to tackle the problem. Therefore, rather than providing a closed-form solution to the optimization problem, we only characterize the optimal solu- tion through the duality relation between the primal and the dual problem by apealing to the set of equivalent local martingale measures. The duality technic is reviewed in different situations and with different ways. We consi- der both complete and incomplete markets ; the wealth might be negative or non-negative. We also consider both unconstrained and constrained Fenchel dual functions and provide the link between utility maximization problem and hedging-type problem in the complete market case.

The remainder of the chapter is structured as follows. In Section 6.2, we consider the problem with positive wealth constraint in an incomplete mar- ket framework. We show that the duality technic used in Kramkov and Schachermayer (1999) can be easily adapted to our new setting. Section 6.3 relaxes the positivity constraints by approximating the given utility function by a sequence of new utility functions bounded from below as being done in Schachermayer (2001). The case of complete markets is studied in Section 6.4 and Section 6.5 by two approaches which are still based on duality tech- nics. By introducing new random shortfall thresholds which is inspired from the paper of Bouchard-Touzi-Elie (2009), in Section 6.4 we boil down the problem from one expected loss constraint to an infinite number of almost sure constraints. The latter is simpler to deal with by means of constrained Fenchel dual function, which leads to a new kind of duality in the case of complete markets. In Section 6.5, we relate the utility maximization problem to the hedging-type problem with multiple targets. This is a special kind of minimal solutions to a BSDE with weak terminal conditions as considered in Bouchard-Elie-Réveillac (2013).