Robust pricing and hedging under trading restrictions and the emergence of local martingale models

TL;DR

This paper explores derivative pricing under trading restrictions without probabilistic models, revealing duality gaps and linking to local martingale models and financial bubbles in both discrete and continuous time.

Contribution

It demonstrates the existence of duality gaps when only put options are traded and connects discrete models with continuous local martingale models, providing new insights into financial bubbles.

Findings

Classical duality holds with call options and short-selling restrictions.

A duality gap exists when only put options are traded.

Embedding into continuous time links to local martingale models and financial bubbles.

Abstract

We consider the pricing of derivatives in a setting with trading restrictions, but without any probabilistic assumptions on the underlying model, in discrete and continuous time. In particular, we assume that European put or call options are traded at certain maturities, and the forward price implied by these option prices may be strictly decreasing in time. In discrete time, when call options are traded, the short-selling restrictions ensure no arbitrage, and we show that classical duality holds between the smallest super-replication price and the supremum over expectations of the payoff over all supermartingale measures. More surprisingly in the case where the only vanilla options are put options, we show that there is a duality gap. Embedding the discrete time model into a continuous time setup, we make a connection with (strict) local-martingale models, and derive framework and results often seen in the literature on financial bubbles. This connection suggests a certain natural interpretation of many existing results in the literature on financial bubbles.