In a numéraire-independent framework, we study a financial market with [Formula: see text] assets which are all treated in a symmetric way. We define the fundamental value ∗S of an asset [Formula: see text] as its super-replication price and say that the market has a strong bubble if ∗S and [Formula: see text] deviate from each other. None of these concepts needs any mention of martingales. Our main result then shows that under a weak absence-of-arbitrage assumption (basically NUPBR), a market has a strong bubble if and only if in all numéraire s for which there is an equivalent local martingale measure (ELMM), asset prices are strict local martingales under all possible ELMMs. We show by an example that our bubble concept lies strictly between the existing notions from the literature. We also give an example where asset prices are strict local martingales under one ELMM, but true martingales under another, and we show how our approach can lead naturally to endogenous bubble birth.

We study how trading costs are reflected in equilibrium returns. To this end, we develop a tractable continuous-time risk-sharing model, where heterogeneous mean-variance investors trade subject to a quadratic transaction cost. The corresponding equilibrium is characterized as the unique solution of a system of coupled but linear forward-backward stochastic differential equations. Explicit solutions are obtained in a number of concrete settings. The sluggishness of the frictional portfolios makes the corresponding equilibrium returns mean-reverting. Compared to the frictionless case, expected returns are higher if the more risk-averse agents are net sellers or if the asset supply expands over time.Mathematics Subject Classification (2010): 91G10, 91G80.JEL Classification: C68, D52, G11, G12. We are grateful to Michalis Anthropelos, Peter Bank, Paolo Guasoni, and Felix Kübler for stimulating discussions and detailed comments. Moreover, we thank an anonymous referee for his or her careful reading and pertinent remarks.

We study risk-sharing economies where heterogenous agents trade subject to quadratic transaction costs. The corresponding equilibrium asset prices and trading strategies are characterised by a system of nonlinear, fully-coupled forward-backward stochastic differential equations. We show that a unique solution generally exists provided that the agents' preferences are sufficiently similar. In a benchmark specification with linear state dynamics, the illiquidity discounts and liquidity premia observed empirically correspond to a positive relationship between transaction costs and volatility.Mathematics Subject Classification: (2010) 91G10, 91G80, 60H10.JEL Classification: C68, D52, G11, G12.between the effects of transaction costs on asset prices, expected returns, and volatilities. To wit, the "liquidity discount" of asset prices compared to their frictionless counterparts, the "liquidity premia" that distinguish their expected returns, and the adjustment of the corresponding volatilities all have the same sign in our model, determined by the difference of the agents' risk aversion parameters. In the empirically relevant case of positive illiquidity discounts and liquidity premia [3,9,44], our model predicts a positive relation between transaction costs and volatility, corroborating empirical evidence of [49,30,27], numerical results of [1,12], and findings in a risk-neutral model with asymmetric information [15]. In addition to these systematic shifts, transaction costs also endogenously lead to mean-reverting expected returns as in the reduced-form models of [33,16,40,23]: for illiquid assets, supply-demand imbalances do not offset immediately but only gradually, thereby leading to partially predictable returns.Without transaction costs, the equilibrium dynamics of the risky asset are determined by a scalar purely quadratic BSDE in our model, which leads to explicit formulas in concrete examples. With quadratic transaction costs on the agents' trading rates, we show that the corresponding equilibria are characterised by fully-coupled systems of FBSDEs. To wit, the optimal risky positions evolve forward from the given initial allocations. In contrast, the corresponding trading rates controlling these positions need to be determined from their zero terminal values -near the terminal time, trading stops since additional trades can no longer earn back the costs that would need to be paid to implement them. If a constant volatility is given exogenously as in [8], then these forward-backward dynamics suffice to pin down the equilibrium returns. In this case, the FBSDEs are linear, and therefore can be solved explicitly in terms of Riccati equations and conditional expectations of the endowment processes [24,6]. In the present context, where the volatility is determined endogenously from the terminal condition for the risky asset, the corresponding FBSDEs are coupled to an additional backward equation arising from this extra constraint. Due to the quadratic preferences and trading costs, the resulting forward-backward ...

The classic approach to modeling financial markets consists of four steps. First, one fixes a currency unit. Second, one describes in that unit the evolution of financial assets by a stochastic process. Third, one chooses in that unit a numéraire, usually the price process of a positive asset. Fourth, one divides the original price process by the numéraire and considers the class of admissible strategies for trading. This approach has one fundamental drawback: Almost all concepts, definitions, and results, including no‐arbitrage conditions like NA, NFLVR, and NUPBR depend by their very definition, at least formally, on initial choices of a currency unit and a numéraire. In this paper, we develop a new framework for modeling financial markets, which is not based on ex‐ante choices of a currency unit and a numéraire. In particular, we introduce a “numéraire‐independent” notion of no‐arbitrage and derive its dual characterization. This yields a numéraire‐independent version of the fundamental theorem of asset pricing (FTAP). We also explain how the classic approach and other recent approaches to modeling financial markets and studying no‐arbitrage can be embedded in our framework.