Robust efficient hedging

Let us explain how the efficient hedging problem solved by Föllmer and Leukert[24] and the problem of definition 2.2 are related. For this pur- pose, let HT be a European option with superhedging cost πsup(HT) =

sup_{P ∈M}EP[HT], and let c be an initial wealth with 0 ≤ c < πsup(HT). For a

loss function l, the efficient hedging asks for a c-admissible strategy ξ∗ ∈ Adc

with ER[l((HT − Vc,ξ ∗ T ) +_{)] = inf} ξ∈Adc ER[l((HT − VTc,ξ) +_)].

Loosely speaking, the value process of any c-admissible strategy ξ ∈ Adc

yields a nontrivial shortfall (HT−VTc,ξ)+, and the strategy is selected through

a loss-based criterion specified by the loss function l.

But in the utility or loss representation of a preference order, it has been assumed that the probabilistic structure specified by the probability measure

R is well determined. A more realistic formulation should allow for model

uncertainty where some probabilistic aspects are unclear. This is captured by the robust formulation of preferences due to Gilboa and Schmeidler[30]. Accordingly, we assume that the agent has a convex set Q of probability measures or priors Q, and valuates a payoff-profile W through the utility functional

inf

Q∈QEQ[u(W )] (2.3)

where u is a utility function. Alternatively, a loss-profile S is valuated ac- cording to the loss functional

sup

Q∈Q

EQ[l(S)]

where l is a loss function. Thus, we are led to quantify the robust shortfall risk by

ξ ∈ Adc → sup Q∈Q

EQ[l((HT − VTc,ξ)

This can be seen as a robust version of efficient hedging for European op- tions, a problem which was introduced and discussed by Kirch[39].

Let us now move on to the American case.

We are taking the point of view of the seller, and so the liquidation date is uncertain. If the option is exercised in a stopping time θ ∈ T , then the correspondence

ξ ∈ Adc → sup Q∈Q

EQ[l((Hθ− Vθc,ξ)

+_)]

gives a robust quantification of the shortfall risk at time θ. But the seller has no control over the time of exercise. If he takes a worst-case attitude regarding stopping times, then this is quantified by the functional

ξ ∈ Adc → sup θ∈T sup Q∈Q EQ[l((Hθ− Vθc,ξ) +_)].

In this robust framework, efficient hedging for American options asks for a

c-admissible strategy ξ∗ ∈ Adc with

sup θ∈T sup Q∈Q EQ[l((Hθ− Vc,ξ ∗ θ ) +_{)] = inf} ξ∈Adc sup θ∈T sup Q∈Q EQ[l((Hθ− Vθc,ξ) +_)].

This is the robust partial hedging problem 2.2 in the special case f (h, v) =

l((h − v)+_).

Stochastic optimization of utility with discretionary stopping. In the previous paragraph we explained that the robust partial hedging prob- lem P H(·) is motivated by a robust version of efficient hedging of American options, where model uncertainty is explicitly taken into account. Our for- mulation combined two lines of ideas. In the first, preferences are represented by robust utility or loss functionals. In the second, in order to incorporate the dynamic nature of American options, we assumed a worst-case attitude whereby the seller is pessimistic regarding the buyer’s selection of a stopping time. In this way, we obtained a robust stochastic optimization problem of expected shortfall with discretionary stopping. The class of problems where expected utility optimization is combined with discretionary stopping is quite recent, and it has been previously studied in the financial literature with pur- poses other than partial hedging. Let us cite a few papers.

Davis and Zariphopoulou[6] and Oberman and Zariphopoulou[46] studied two stochastic problems of maximizing expected utility with discretionary stopping. They adopted an indifference-price approach in order to valuate

early exercise contingent claims. Their analysis was based on variational in- equalities.

Karatzas and Wang[37] studied an optimal portfolio management problem combined with discretionary stopping. Their analysis was based on the martingale-method and they established a criterion to apply convex-duality which in the cases of logarithmic and moment utilities led to explicit results. Letting aside the different motivations, a common feature in the afore men- tioned papers is that stopping and portfolio selection are decision variables under our control. This is the main conceptual difference with our problem here.

In the indifference-pricing approach studied in [6, 46], a price is given to an early exercise contingent claim from the perspective of an investor having a long position on the claim. The investor simultaneously searches an optimal exercise and an optimal portfolio allocation, and hence a utility functional is maximized over portfolio strategies and over stopping times.

In [37] the problem is of utility maximization from consumption and termi- nal wealth, stopping times are introduced to search for the best liquidation date. Here again, a utility functional is maximized over portfolio strategies and over stopping times.

In the robust partial hedging problem 2.2 we have taken the point of view of the seller of an American option, portfolios are not investment opportunities but hedging strategies, and stopping times are adverse variables. Loosely speaking, the criteria in the afore mentioned papers are of “maxmax” type, while here we are considering a “minimax” criterion.

Robust utility maximization. We conclude this section with some re- marks about numerical representations of preference orders and about robust utility maximization. The axiomatic treatment on preference orders and its numerical representations began with Von Neumann and Morgenstern[53] and Savage[50]. They formulated a set of axioms to be satisfied by a prefer- ence order, and constructed a numerical representations of the form

EQ[u(·)].

The interpretation is that, given two payoffs X1 and X2, the first is “more

preferred” than the second if and only if EQ[u(X1)] > EQ[u(X2)], see e.g.,

section 2.5 in Föllmer and Schied[27]. However, Ellsberg’s paradox (see ex- ample 2.75 in [27]) illustrates that this numerical representation does not account for model-uncertainty aversion. An uncertainty aversion axiom led Gilboa and Schmeidler[30] to obtain a robust numerical representation of the

form (2.3):

inf

Q∈QEQ[u(·)].

Maximization of robust utility in the context of financial markets is re- cent. Let us give a partial list of the related literature.

Schied[51] studies the problem of robust utility maximization in a com- plete market. For the special case of priors

Qλ := ( Q  P | dQ dP ≤ 1 λ ) ,

corresponding to the risk measure AVaR, explicit solutions are obtained, using the robust version of the Neyman-Pearson lemma due to Huber and Strassen[32].

Kirch[39] studies a robust version of efficient hedging for European op- tions. His solution reduces the problem into a Neyman-Pearson type problem for composite hypotheses against composite alternatives and for non linear power functions.

Föllmer and Gundel[21] consider the robust utility maximization problem in incomplete markets. They extend the method of Goll and Rüschendorf[31] and obtain a least favorable pair of probability measures (Q∗, P∗) where Q∗ is an element of the set of priors and P∗ is an extended martingale measure. This pair reduces the robust problem to a classical problem of utility max- imization with respect to Q∗ and for P∗-affordable claims. Their approach allows to obtain the least favorable pair as the solution of a dual optimization problem.

Schied and Wu[52] consider the problem of robust utility in an incom- plete market. Their approach extends the duality results of Kramkov and Schachermayer[42] to the robust setting.