Super‐replication in fully incomplete markets

Abstract

In this work, we introduce the notion of fully incomplete markets. We prove that for these markets, the super‐replication price coincides with the model‐free super‐replication price. Namely, the knowledge of the model does not reduce the super‐replication price. We provide two families of fully incomplete models: stochastic volatility models and rough volatility models. Moreover, we give several computational examples. Our approach is purely probabilistic.

Continuity of Utility Maximization under Weak Convergence

“…On the other hand, the model given by S is a fully incomplete market (see Definition 2.1 and Example 2.5 in [12]). In [12,36] it was proved that in fully incomplete markets the super-replication price is prohibitively high and lead to buy-and-hold strategies. Namely, the super-hedging price of a call option is equal to the initial stock price S 0 = 1.…”

Pathwise superhedging on prediction sets

“…In fact, for stochastic volatility or rough volatility models, it turns out that the classical superhedging price coincides with the model-independent one and is so high that for Markovian payoffs of the form Γ(S T ), like e.g. the European Call and Put option, the optimal superhedging strategy can be chosen to be of buy-and-hold type, see [9,23]. To reduce the model-independent superhedging price, inspired by the work of [22], [15] introduced the concept of prediction sets, where agents may allow to exclude paths which they consider to be impossible to model future price paths.…”

Buy-and-Hold Property for Fully Incomplete Markets When Super-Replicating Markovian Claims

“…When a financial agent is allowed to invest in the risky asset S and statically in up to finite many liquid options, the classical super-replication price (defined with respect to the given initial law P of the price process) of a (possibly pathdependent) European option G(S) with G : C[0, T ] → R + being uniformly continuous and bounded coincides with the robust super-replication price, where the super-hedging property must hold for any path. This follows from the result proven in Dolinsky & Neufeld (2018) that for fully incomplete markets, the set of all equivalent local martingale measures are weakly dense in the set of all local martingale measures defined on the continuous path space. For more papers related to robust pricing, in particular to duality results, we refer to Acciaio et al (2016); Bartl et al (2018Bartl et al ( , 2017; Burzoni et al (2017); Dolinsky & Soner (2014, 2015; Guo et al (2017); Hobson (1998) ;Hou & Obłój (2018) to name but a few.…”

“…Fully incomplete markets were introduced in Dolinsky & Neufeld (2018). Roughly speaking, a financial market is fully incomplete if for any volatility process α one can find an equivalent local martingale measure Q under which α is close to the volatility process ν of the price process S. It turns out that it is a natural appearance for incomplete markets to be fully incomplete.…”