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

Abstract

We pursue the robust approach to pricing and hedging in which no probability measure is fixed, but call or put options with different maturities and strikes can be traded initially at their market prices. We allow the inclusion of robust modelling assumptions by specifying a set of feasible paths on which (super)hedging arguments are required to work. In a discrete-time setup with no short selling, we characterise absence of arbitrage and show that if call options are traded, then the usual pricinghedging duality is preserved. In contrast, if only put options are traded, a duality gap may appear. Embedding the results into a continuous-time framework, we show that the duality gap may be interpreted as a financial bubble and link it to strict local martingales. This provides an intrinsic justification of strict local martingales as models for financial bubbles arising from a combination of trading restrictions and current market prices.Keywords Robust pricing and hedging · Financial bubble · Local martingale models · Pricing-hedging duality · Trading restrictions · Short selling constraint · Martingale optimal transport Mathematics Subject Classification (2010) 91G20 · 91B24 · 60G42 · 60G44 JEL Classification C61 · G13 · E32

Financial asset price bubbles under model uncertainty

“…This second situation is precisely the one considered in Cox et al (2016) in order to detect a bubble. This means that S * can always be viewed at least as the worst model price, among the models considered by the investor.…”

“…Hence, in the general case, under RND any bubble would be the result of a duality gap in (35), which is the case considered in Cox et al (2016).…”

“…The same submartingale behavior is in fact described for some cases also in Biagini et al (2014), but it is the result of a smooth shift from an equivalent pricing measure to another. To the best of our knowledge this description of bubbles is new, as it distinguishes itself also from the robust setting outlined in Cox et al (2016), where bubbles arise as a consequence of constraints on possible trading strategies in a different setup.…”

Quantile Hedging in a Semi-Static Market with Model Uncertainty

“…Arbitrage and superhedging duality with model uncertainty or semi-static trading are studied by many researchers, see e.g. [5,11,7,28,6,9,1,3,13,14]. [11] is particularly relevant, which proves the Fundamental Theorem of Asset Pricing and superhedging duality in our setting.…”

A Risk-Neutral Equilibrium Leading to Uncertain Volatility Pricing

“…The non-Markovian version of this pricing operator is known as the G-expectation [50,51]. More recently, a rich literature considering a variety of hedging instruments and underlying models has emerged; see, among many others, [1,3,9,10,23,45] for models in discrete time and [8,13,15,16,18,19,22,25,26,27,29,31,32,43,46] for continuous-time models. We refer to [30,48] for surveys.…”