The existence of a self-financing trading strategy that replicates the money market account at a fixed future date at a lower cost than the current value of this account constitutes a money market bubble (MMB). Understanding whether a market exhibits an MMB is crucial, in particular, for derivative pricing. An MMB precludes the existence of a risk-neutral probability measure. The benchmark approach allows to study MMBs and is formulated under the real world probability measure. It does not require the existence of a risk neutral probability measure. Using a range of well-known stochastic volatility models, we study the existence of an MMB in the US economy, and find that the US market exhibits an MMB for all models considered that allow it. This suggests that for derivative pricing and hedging care should be taken when making assumptions pertaining to the existence of a risk-neutral probability measure. Less expensive portfolios are likely to exist for a wide range of long-term derivatives, as typical for pensions.

This paper considers the realistic modelling of derivative contracts on exchange rates. We propose a stochastic volatility model that recovers not only the typically observed implied volatility smiles and skews for short dated vanilla foreign exchange options but allows one also to price payoffs in foreign currencies, lower than possible under classical risk neutral pricing, in particular, for long dated derivatives. The main reason for this important feature is the strict supermartingale property of benchmarked savings accounts under the real world probability measure, which the calibrated parameters identify under the proposed model. Using a real dataset on vanilla option quotes, we calibrate our model on a triangle of currencies and find that the risk neutral approach fails for the calibrated model, while the benchmark approach still works.