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

Abstract

We show that when the price process S represents a fully incomplete market, the optimal super-replication of any Markovian claim g(S T ) with g(·) being nonnegative and lower semicontinuous is of buy-and-hold type. Since both (unbounded) stochastic volatility models and rough volatility models are examples of fully incomplete markets, one can interpret the buy-and-hold property when super-replicating Markovian claims as a natural phenomenon in incomplete markets.

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.…”

“…For the shortfall risk measure we show that the truncation error can be controlled, see Lemma 7.1, so our result applies to the non-truncated Heston model. It is well known (see [9,16,12,36]) that in the Heston model the superreplication price is prohibitively high and lead to buy-and-hold strategies. Namely, the cheapest way to super-hedge a European call option is to buy one stock at the initial time and keep that position till maturity.…”

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.…”

Model-free price bounds under dynamic option trading

“…In practice it is a well-known and often faced problem that given a specific market model, superhedging strategies for financial derivatives are very expensive to implement (see for example [14], [21], [29,Example 7.21], [36], or [39]). If an investor is interested in considering hedging strategies that super-replicate the payoff of a derivative under parameter-uncertainty of a specific model class or even completely model-independent, then prices for super-hedges become even higher than model-specific hedges, compare for this the model-independent approaches from [11], [24] as well as the robust approach from [15].…”