Buy-and-Hold Property for Fully Incomplete Markets when Super-replicating Markovian Claims

TL;DR

In fully incomplete markets, the optimal super-replication strategy for Markovian claims simplifies to a buy-and-hold approach, highlighting a natural property of such markets.

Contribution

This paper proves that super-replication of Markovian claims in fully incomplete markets is achieved through buy-and-hold strategies, extending understanding to models like stochastic and rough volatility.

Findings

Super-replication in fully incomplete markets is of buy-and-hold type.

Applicable to stochastic and rough volatility models.

Highlights natural property of incomplete markets.

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 $g(\cdot)$ 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.