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Register a new userArbitrage-free modeling under Knightian Uncertainty
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We study the Fundamental Theorem of Asset Pricing for a general financial market under Knightian Uncertainty. We adopt a functional analytic approach which require neither specific assumptions on the class of priors $mathcal{P}$ nor on the structure of the state space. Several aspects of modeling under Knightian Uncertainty are considered and analyzed. We show the need for a suitable adaptation of the notion of No Free Lunch with Vanishing Risk and discuss its relation to the choice of an appropriate filtration. In an abstract setup, we show that absence of arbitrage is equivalent to the existence of emph{approximate} martingale measures sharing the same polar set of $mathcal{P}$. We then specialize the results to a discrete-time framework in order to obtain true martingale measures.
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We reconsider the microeconomic foundations of financial economics. Motivated by the importance of Knightian Uncertainty in markets, we present a model that does not carry any probabilistic structure ex ante, yet is based on a common order. We derive the fundamental equivalence of economic viability of asset prices and absence of arbitrage. We also obtain a modified version of the Fundamental Theorem of Asset Pricing using the notion of sublinear pricing measures. Differe
In a model independent discrete time financial market, we discuss the richness of the family of martingale measures in relation to different notions of Arbitrage, generated by a class $mathcal{S}$ of significant sets, which we call Arbitrage de la classe $mathcal{S}$. The choice of $mathcal{S}$ reflects into the intrinsic properties of the class of polar sets of martingale measures. In particular: for S=${Omega}$ absence of Model Independent Arbitrage is equivalent to the existence of a martingale measure; for $mathcal{S}$ being the open sets, absence of Open Arbitrage is equivalent to the existence of full support martingale measures. These results are obtained by adopting a technical filtration enlargement and by constructing a universal aggregator of all arbitrage opportunities. We further introduce the notion of market feasibility and provide its characterization via arbitrage conditions. We conclude providing a dual representation of Open Arbitrage in terms of weakly open sets of probability measures, which highlights the robust nature of this concept.
We propose a hybrid method for generating arbitrage-free implied volatility (IV) surfaces consistent with historical data by combining model-free Variational Autoencoders (VAEs) with continuous time stochastic differential equation (SDE) driven models. We focus on two classes of SDE models: regime switching models and Levy additive processes. By projecting historical surfaces onto the space of SDE model parameters, we obtain a distribution on the parameter subspace faithful to the data on which we then train a VAE. Arbitrage-free IV surfaces are then generated by sampling from the posterior distribution on the latent space, decoding to obtain SDE model parameters, and finally mapping those parameters to IV surfaces.
The no-arbitrage property is widely accepted to be a centerpiece of modern financial mathematics and could be considered to be a financial law applicable to a large class of (idealized) markets. The paper addresses the following basic question: can one characterize the class of transformations that leave the law of no-arbitrage invariant? We provide a geometric formalization of this question in a non probabilistic setting of discrete time, the so-called trajectorial models. The paper then characterizes, in a local sense, the no-arbitrage symmetries and illustrates their meaning in a detailed example. Our context makes the result available to the stochastic setting as a special case
We study an intertemporal consumption and portfolio choice problem under Knightian uncertainty in which agents preferences exhibit local intertemporal substitution. We also allow for market frictions in the sense that the pricing functional is nonlinear. We prove existence and uniqueness of the optimal consumption plan, and we derive a set of sufficient first-order conditions for optimality. With the help of a backward equation, we are able to determine the structure of optimal consumption plans. We obtain explicit solutions in a stationary setting in which the financial market has different risk premia for short and long positions.
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