In this section, we provide a brief review of the theory of risk measures and of the pricing-hedging duality.
2.1 Risk measures and the pricing-hedging duality: the classical setup
The notion of subhedging price is one of the most analyzed concepts in financial mathematics. Although specular considerations can be done for the superhedging price, in this introduction we focus on the subhedging price. We are assuming a discrete-time market model with zero interest rate. It may be convenient for the reader to have at hand the summary described in Table
1. In the classical setup of stochastic securities market models, one considers an adapted stochastic process
\(X=(X_{t})_{t}\),
\(t=0,...,T,\) defined on a filtered probability space
\((\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t},P),\) representing the price of some underlying asset. Let
\({\mathcal {P}}(P)\) be the set of all probability measures on
\(\Omega \) that are absolutely continuous with respect to
P,
\(\textrm{Mart}(\Omega )\) be the set of all probability measures on
\(\Omega \) under which
X is a martingale and
\({\mathcal {M}}(P)=\) \( {\mathcal {P}}(P)\cap \textrm{Mart}(\Omega ).\) We also let
\({\mathcal {H}}\) be the class of admissible integrands and
\(I^{\Delta }:=I^{\Delta }(X)\) be the stochastic integral of
X with respect to
\(\Delta \in {\mathcal {H}}\). Under reasonable assumptions on
\({\mathcal {H}}\), the equality
$$\begin{aligned} E_{Q}\left[ I^{\Delta }(X)\right] =0 \end{aligned}$$
(1)
holds for all
\(Q\in {\mathcal {M}}(P)\) and, as well known, all linear pricing functionals compatible with no arbitrage are expectations
\(E_{Q}[\cdot ]\) under some probability
\(Q\in {\mathcal {M}}(P)\) such that
\(Q\sim P\).
We denote with
p the
subhedging price of a contingent claim
\(Z:=c(X_{T})\) written on the payoff
\(X_{T}\) of the underlying asset. If we let
\({\mathcal {L}}(P)\subseteq L^{0}(\Omega ,{\mathcal {F}}_{T},P)\) be the space of random payoffs, then
\( p:{\mathcal {L}}(P)\rightarrow {\mathbb {R}}\) is defined by
$$\begin{aligned} p(Z):=\sup \left\{ m\in {\mathbb {R}}\mid \exists \Delta \in {\mathcal {H}}\text { s.t. }m+I^{\Delta }(X)\le Z\text {, }P-\text {a.s.}\right\} . \end{aligned}$$
(2)
The subhedging price is independent from the preferences of the agents, but it depends on the reference probability measure via the class of
P-null events. It satisfies the following two key properties:
(CA)
Cash Additivity on \({\mathcal {L}}(P)\): \(p(Z+k)=p(Z)+k,\) for all \( k\in {\mathbb {R}}\), \(Z\in {\mathcal {L}}(P).\)
(IA)
Integral Additivity on \({\mathcal {L}}(P)\): \(p(Z+I^{\Delta })=p(Z),\) for all \(\Delta \in {\mathcal {H}}\), \(Z\in {\mathcal {L}}(P).\)
When a functional
p satisfies (CA), then
Z,
k and
p(
Z) must be expressed in the same monetary unit and this allows for the
monetary interpretation of
p, as the price of the contingent claim. This will be one of the key features that we will require also in the novel definition of the nonlinear subhedging value. The (IA) property and
\(p(0)=0\) imply that the
p price of any stochastic integral
\(I^{\Delta }(X)\) is equal to zero, as in (
1).
Since the seminal works of El Karoui and Quenez (
1995), Karatzas (
1997), Delbaen and Schachermayer (
1994), it was discovered that, under the no arbitrage assumption, the dual representation of the subhedging price
p is
$$\begin{aligned} p(Z)=\inf _{Q\in {\mathcal {M}}(P)}E_{Q}\left[ Z\right] . \end{aligned}$$
(3)
More or less in the same period, the concept of a
coherent risk measure was introduced in the pioneering work by Artzner et al. (
1999). A coherent risk measure
\(\rho :{\mathcal {L}}(P)\rightarrow {\mathbb {R}}\) determines the minimal capital required to make acceptable a financial position and its dual formulation is assigned by
$$\begin{aligned} -\rho (Y)=\inf _{Q\in {\mathcal {Q}}\subseteq {\mathcal {P}}(P)}E_{Q}\left[ Y\right] , \end{aligned}$$
(4)
where
Y is a random variable representing future profit and loss and
\( {\mathcal {Q}}\subseteq {\mathcal {P}}(P)\). Coherent risk measures
\(\rho \) are convex, cash additive, monotone and positively homogeneous. We take the liberty to label both the representations in (
3) and (
4) as the “
sublinear case ”.
In the study of incomplete markets, the concept of the (buyer)
indifference price \(p^{b}\), originally introduced by Hodges and Neuberger (
1989), received, in the early 2000, increasing consideration (see Frittelli (
2000), Rouge and El Karoui (
2000), Delbaen et al. (
2002), Bellini and Frittelli (
2002)) as a tool to assess,
consistently with the no arbitrage principle, the value of nonreplicable contingent claims, and not just to determine an upper bound (the superhedging price) or a lower bound (the subhedging price) for the price of the claim. Differently from the notion of subhedging,
\(p^{b}\) is based on some concave increasing utility function
\(u:{\mathbb {R}}\rightarrow [-\infty ,+\infty )\) of the agent. By defining the indirect utility function
$$\begin{aligned} U(w_{0}):=\sup _{\Delta \in {\mathcal {H}}}E_{P}[u(w_{0}+I^{\Delta }(X))], \end{aligned}$$
where
\(w_{0}\in {\mathbb {R}}\) is the initial wealth, the indifference price
\( p^{b}\) is defined as
$$\begin{aligned} p^{b}(Z):=\sup \left\{ m\in {\mathbb {R}}\mid U(Z-m)\ge U(0)\right\} . \end{aligned}$$
Under suitable assumptions, the dual formulation of
\(p^{b}\) is
$$\begin{aligned} p^{b}(Z)=\inf _{Q\in {\mathcal {M}}(P)}\left\{ E_{Q}\left[ Z\right] +\alpha _{u}(Q)\right\} , \end{aligned}$$
(5)
and the penalty term
\(\alpha _{u}:{\mathcal {M}}(P)\rightarrow [0,+\infty ]\) is associated with the particular utility function
u appearing in the definition of
\(p^{b}\) via the Fenchel conjugate of
u. We observe that in case of the exponential utility function
\(u(x)=1-\exp (-x),\) the penalty is
\(\alpha _{\exp }(Q):=H(Q,P)-\min _{Q\in {\mathcal {M}}(P)}H(Q,P),\) where
$$\begin{aligned} H(Q,P):=\int F\left( \frac{\textrm{d}Q}{\textrm{d}P}\right) \,\textrm{d}P \text {,\quad if }Q\ll P\text { and }F(y)=y\ln (y), \end{aligned}$$
is the relative entropy. In this case, the penalty
\(\alpha _{\exp }\) is a divergence functional, similarly, e.g., to those considered in (
11). Observe that the functional
\(p^{b}\) is concave, monotone increasing and satisfies both properties (CA) and (IA), but it is not necessarily linear on the space of all contingent claims. As recalled in the conclusion of Frittelli (
2000), “there is no reason why a price functional defined on the whole space of bundles and consistent with no arbitrage should be linear also outside the space of marketed bundles”.
It was exactly the particular form (
5) of the indifference price that suggested to Frittelli and Rosazza Gianin (
2002) to introduce the concept of
convex risk measure (also independently introduced by Föllmer and Schied (
2002)), as a map
\(\rho :{\mathcal {L}}(P)\rightarrow {\mathbb {R}}\) that is convex, cash additive and monotone decreasing. Under good continuity properties, the Fenchel–Moreau theorem shows that any convex risk measure admits the following representation
$$\begin{aligned} -\rho (Y)=\inf _{Q\in {\mathcal {P}}(P)}\left\{ E_{Q}\left[ Y\right] +\alpha (Q)\right\} \end{aligned}$$
(6)
for some penalty
\(\alpha :{\mathcal {P}}(P)\rightarrow [0,+\infty ]\). We will then label functional in the form (
5) or (
6) as the “
convex case”. As a consequence of the cash additivity property, in the dual representations (
5) or (
6) the infimum is taken with respect to
probability measures, namely with respect to normalized nonnegative elements in the dual space, which in this case can be taken as
\( L^{1}(P)\). Differently from the indifference price
\(p^{b}\), convex risk measures do not necessarily take into account the presence of the stochastic security market, as reflected by the absence of any reference to martingale measures in the dual formulation (
6) and (
4), in contrast to (
5) and (
3). The discussion and comparison regarding convex/coherent risk measures are summarized in rows 1–2 of Table
1, while rows 3–4 compare the subheding price with the indifference price.
2.2 Pathwise finance
In the classical setting of nonlinear pricing recalled before, it was implicitly assumed that a reference probability
P was fixed and known a priori. The financial crises in 2008, however, somehow inspired and motivated an increasing interest in the case where uncertainty in the selection of a reference probability occurs. The classical notions of arbitrage and pricing-hedging duality have been therefore investigated in this new framework. Two main approaches have been adopted to deal with uncertainty in
P. One approach consisted in replacing the single reference probability
P with a family of—a priori nondominated—probability measures, leading to the theory of quasi-sure stochastic analysis (see Bayraktar and Zhang (
2016), Bayraktar and Zhou (
2017), Bouchard and Nutz (
2015), Cohen (
2012), Denis and Martini (
2006), Peng (
2019), Soner et al. (
2011)). An alternative approach, even more radical, developed a probability-free, pathwise, theory of financial markets, see Acciaio et al. (
2016), Burzoni et al. (
2016), Burzoni et al. (
2017), Burzoni et al. (
2019), Riedel (
2015). In such framework, optimal transport theory became a very powerful tool to prove pathwise pricing-hedging duality results with relevant contributions by many authors (Beiglböck et al. (
2013), Davis et al. (
2014), Dolinsky and Soner (
2014) and Dolinsky and Soner (
2015), Galichon et al. (
2014), Henry-Labordère (
2013), Henry-Labordère et al. (
2016); Hou and Obłój (
2018), Tan and Touzi (
2013), Bartl et al. (
2019), Cheridito et al. (
2020) and Cheridito et al. (
2017), Guo and Obłój (
2019), Backhoff-Veraguas and Pammer (
2020), Neufeld and Sester (
2021), Sester (
2023) and Sester (
2023)).
These contributions mainly deal with what we labeled above as the sublinear case, while our main interest in this paper is to develop the convex case theory, as explained below.
From now on, we will abandon the classical setup described above and work without a reference probability measure. We consider a finite horizon
\(T\in {\mathbb {N}}\),
\( T\ge 1\), and
$$\begin{aligned} \Omega :=K_{0}\times \dots \times K_{T} \end{aligned}$$
for
\(K_{0},\dots ,K_{T}\) subsets of
\({\mathbb {R}}\) and assume that
\(K_{0}\) is a singleton, that is,
\(K_{0}=\{x_0\}\),
\(x_0 \in {\mathbb {R}} \). We let
\( X_{0},\dots ,X_{T}\) be the canonical projections
\(X_{t}:\Omega \rightarrow K_{t} \), for
\(t=0,1,...,T\). We denote
$$\begin{aligned} \textrm{Mart}(\Omega ):=\{\text {Martingale probability measures for the canonical process of }\Omega \}\,, \end{aligned}$$
and when
\(\mu \) is a measure defined on the Borel
\(\sigma \)-algebra of
\( (K_{0}\times \dots \times K_{T})\), its marginals will be denoted with
\(\mu _{0},\dots ,\mu _{T}\). We consider a contingent claim
\(c:\Omega \rightarrow (-\infty ,+\infty ]\) which is now allowed to depend on the whole path, and for hedging purposes, we will adopt semistatic trading strategies. In other words, in addition to dynamic trading in
X via the admissible integrands
\(\Delta \in {\mathcal {H}}\), we may invest in “vanilla” options
\(\varphi _{t}:K_{t}\rightarrow {\mathbb {R}}\). For modeling purposes, we take vector subspaces
\({\mathcal {E}}_{t}\subseteq {\mathcal {C}}_{b}(K_{t})\) for
\( t=0,\dots ,T\), where
\({\mathcal {C}}_{b}(K_{t})\) is the space of real-valued, continuous, bounded functions on
\(K_{t}\). For each
t,
\({\mathcal {E}}_{t}\) represents the set of options to be used for hedging, say affine combinations of options with same maturity
t but different strikes. The key assumption in the robust, optimal transport-based formulation is that the marginals
\(\mathcal {(}\widehat{Q}_{0},\widehat{Q}_{1},...,\widehat{Q} _{T})\) of the underlying price process
X are known. This assumption can be justified (see the seminal papers by Breeden and Litzenberger (
1978) and Hobson (
1998), as well as the many contributions by Hobson (
2011), Cox and Obłój (
2011a), Cox and Obłój (
2011b), Cox and Wang (
2013), Henry-Labordère et al. (
2016), Brown et al. (
2001), Hobson and Klimmek (
2013)) by assuming the knowledge of a sufficiently large number of plain vanilla options maturing at each intermediate date, implying then the possibility of calibration.
Thus,
$$\begin{aligned} {\mathcal {M}}(\widehat{Q}_{0},\widehat{Q}_{1},...\widehat{Q}_{T}):=\left\{ Q\in \textrm{Mart}(\Omega )\mid X_{t}\sim _{Q}\mathcal {\widehat{Q}}_{t}\text { for each }t=0,\dots ,T\right\} \end{aligned}$$
represents the set of arbitrage-free pricing measures that are compatible with the observed prices of the options. In this framework, the set of admissible trading strategies and of the corresponding stochastic integrals are, respectively, given by
$$\begin{aligned} {\mathcal {H}}&:=\left\{ \Delta =[\Delta _{0},\dots ,\Delta _{T-1}]\mid \Delta _{t}\in {\mathcal {C}}_{b}(K_{0}\times \dots \times K_{t}\text {;}{\mathbb {R}} )\right\} \end{aligned}$$
(7)
$$\begin{aligned} {\mathcal {I}}&:=\left\{ I^{\Delta }(x)=\sum _{t=0}^{T-1}\Delta _{t}(x_{0},\dots ,x_{t})(x_{t+1}-x_{t})\mid \Delta \in {\mathcal {H}}\right\} \end{aligned}$$
(8)
and the subhedging duality, obtained in Beiglböck et al. (
2013) Th. 1.1, takes the form:
$$\begin{aligned}{} & {} \inf _{Q\in {\mathcal {M}}(\widehat{Q}_{0},\widehat{Q}_{1},...\widehat{Q} _{T})}E_{Q}\left[ c\right] \nonumber \\{} & {} \quad =\sup \left\{ \sum _{t=0}^{T}E_{\widehat{Q} _{t}}[\varphi _{t}]\mid \exists \Delta \in {\mathcal {H}}\text { s.t. } \sum _{t=0}^{T}\varphi _{t}(x_{t})+I^{\Delta }(x)\le c(x)\text { }\forall x\in \Omega \right\} , \end{aligned}$$
(9)
where the RHS of (
9) is known as the
robust subhedging price of
c. Comparing (
9) with the duality between (
2) and (
3), we observe that: (i) the
\(P-\)a.s. inequality in (
2) has been replaced by an inequality that holds for all
\(x\in \Omega \); (ii) in (
9) the infimum of the price of the contingent claim
c is taken under all martingale measure compatible with the option prices, with no reference to the probability
P; and (iii) in (
9) static hedging with options is allowed.
As can be seen from the LHS of (
9), this case falls into the category labeled above as the
sublinear case, and the purpose of this paper (as well as of Doldi and Frittelli (
2023)) is to investigate the
convex case, in the robust setting, using the tools from entropy optimal transport (EOT) recently developed in Liero et al. (
2018).
Let us first describe the financial interpretation of the problems that we are going to study.
2.3 The dual problem
Differently from the pricing theory in finance where the problem
\(\inf _{Q\in {\mathcal {M}}(\widehat{Q} _{0},\widehat{Q}_{1},...\widehat{Q}_{T})}E_{Q}\left[ c\right] \) in the LHS of (
9) is a dual problem, in
martingale optimal transport (MOT) it represents the primal problem (called henceforth
sublinear case of MOT). In Liero et al. (
2018), the primal
entropy optimal transport (EOT) problem takes the form
$$\begin{aligned} \inf _{\mu \in \textrm{Meas}\mathcal {(}\Omega )}\left( E_{Q}\left[ c\right] +\sum _{t=0}^{T}{\mathcal {D}}_{F_{t},\widehat{Q}_{t}}(\mu _{t})\right) , \end{aligned}$$
(10)
where
\(\textrm{Meas}(\Omega )\) is the set of all positive finite measures
\( \mu \) on
\(\Omega ,\) and
\({\mathcal {D}}_{F_{t},\widehat{Q}_{t}}(\mu _{t})\) is a divergence in the form:
$$\begin{aligned} {\mathcal {D}}_{F_{t},\widehat{Q}_{t}}(\mu _{t}):=\int _{K_{t}}F_{t}\left( \frac{ \textrm{d}\mu _{t}}{\textrm{d}\widehat{Q}_{t}}\right) \,\textrm{d}\widehat{Q} _{t}\text {, if }\mu _{t}\ll \widehat{Q}_{t}\text {;} \end{aligned}$$
(11)
otherwise,
\({\mathcal {D}}_{F_{t},\widehat{Q}_{t}}(\mu _{t}):=+\infty \). We label with
\(F:=(F_{t})_{_{t=0,...,T}}\) the family of divergence functions
\( F_{t}:{\mathbb {R}}\rightarrow \mathbb {R\cup }\left\{ +\infty \right\} \) appearing in (
11). Problem (
10) represents the
convex case of OT theory. Notice that in the EOT primal problem (
10) the typical constraint that
\(\mu \) has prescribed marginals
\((\widehat{Q} _{0},\widehat{Q}_{1},...\widehat{Q}_{T})\) has been relaxed thanks to the introduction of the divergence functional
\({\mathcal {D}}_{F_{t},\widehat{Q}_{t}}(\mu _{t})\), which penalizes those measures
\(\mu \) that are “far” from some reference marginals
\((\widehat{Q}_{0}, \widehat{Q}_{1},...\widehat{Q}_{T}).\) We are then naturally led to the study of the convex case of MOT, i.e., to the entropy martingale optimal transport (EMOT) problem
$$\begin{aligned} \inf _{Q\in \textrm{Mart}(\Omega )}\left( E_{Q}\left[ c\right] +\sum _{t=0}^{T}{\mathcal {D}}_{F_{t},\widehat{Q} _{t}}(Q_{t})\right) \end{aligned}$$
(12)
having also a clear financial interpretation. The marginals are not any more fixed a priori, to capture the fact that the available information might not be enough to detect them with satisfactory precision. So the infimum is taken over
all martingale probability measures, but those that are far from some estimate
\((\widehat{Q}_{0},\widehat{Q}_{1},...\widehat{Q} _{T})\) are appropriately penalized through
\({\mathcal {D}}_{F_{t},\widehat{Q} _{t}}\). Of course, when
\({\mathcal {D}}_{F_{t},\widehat{Q}_{t}}(\cdot )=\delta _{\widehat{Q}_{t}}(\cdot )\), we recover the sublinear MOT problem, where only martingale probability measures with fixed marginals are allowed. Observe that in addition to the martingale property, the elements
\(Q\in \textrm{Mart}(\Omega )\) in (
12) are required to be probability measures, while in the EOT (
10) theory all positive finite measure are allowed. As it was recalled after equation (
6), this normalization feature of the dual elements (
\(\mu (\Omega )=1\)) is not surprising when one deals with dual problems of primal problems with a cash additive objective functional as, for example, in the theory of coherent and convex risk measures.
Potentially, we could push our smoothing argument above even further: In place of the functionals
\({\mathcal {D}}_{F_{t},\widehat{Q}_{t}}(\mu _{t})\),
\( t=0,...,T\), we might as well consider more general marginal penalizations, not necessarily in the divergence form (
11), yielding the problem
$$\begin{aligned} {\mathfrak {D}}(c):=\inf _{Q\in \textrm{Mart}(\Omega )}\left( E_{Q}\left[ c \right] +\sum _{t=0}^{T}{\mathcal {D}}_{t}(Q_{t})\right) \,. \end{aligned}$$
(13)
These penalizations
\({\mathcal {D}}_{0},\dots ,{\mathcal {D}}_{T}\) will be better specified later.
We point out that an additional entropic term has been added to optimal transport problems since the seminal work of Cuturi (
2013) (see also the survey/monograph Peyré and Cuturi (
2019)). On this topic, we also cite Nutz and Wiesel (
2011), Bernton et al. (
2021), Ghosal et al. (
2021), De March and Henry-Labordère (
2020), Henry-Labordère (
2019), Blanchet et al. (
2020). We also point out that in all the works stemming from Cuturi (
2013), the exact matching of the marginals is still required. In this paper such constraint is absent to take into account uncertainty regarding the marginals themselves. In Table
1, rows 5–6, one may compare optimal transport with entropy optimal transport, while in rows 5–7 one may compare optimal transport with martingale optimal transport.