1,711 research outputs found
Arbitrage and hedging in model-independent markets with frictions
No full textWe provide a fundamental theorem of asset pricing and a superhedging theorem for a model indepen- dent discrete time financial market with proportional transaction costs. We consider a probability- free version of the robust no arbitrage condition introduced by Schachermayer in [Math. Finance, 14 (2004), pp. 1948] and show that this is equivalent to the existence of consistent price systems. More- over, we prove that the superhedging price for a claim g coincides with the frictionless superhedging price of g for a suitable process in the bid-ask spread
Risk Measures Based on Benchmark Loss Distributions
No full textWe introduce a class of quantile-based risk measures that generalize Value at Risk (VaR) and, likewise Expected Shortfall (ES), take into account both the frequency and the severity of losses. Under VaR a single confidence level is assigned regardless of the size of potential losses. We allow for a range of confidence levels that depend on the loss magnitude. The key ingredient is a benchmark loss distribution (BLD), that is, a function that associates to each potential loss a maximal acceptable probability of occurrence. The corresponding risk measure, called Loss VaR (LVaR), determines the minimal capital injection that is required to align the loss distribution of a risky position to the target BLD. By design, one has full flexibility in the choice of the BLD profile and, therefore, in the range of relevant quantiles. Special attention is given to piecewise constant functions and to tail distributions of benchmark random losses, in which case the acceptability condition imposed by the BLD boils down to first-order stochastic dominance. We investigate the main theoretical properties of LVaR with a focus on their comparison with VaR and ES and discuss applications to capital adequacy, portfolio risk management, and catastrophic risk
A Tikhonov Theorem for McKean–Vlasov Two-Scale Systems and a New Application to Mean Field Optimal Control Problems
No full textAdjusted Expected Shortfall
No full textWe introduce and study the main properties of a class of convex risk measures that refine Expected Shortfall by simultaneously controlling the expected losses associated with different portions of the tail distribution. The corresponding adjusted Expected Shortfalls quantify risk as the minimum amount of capital that has to be raised and injected into a financial position X to ensure that Expected Shortfall ESp (X) does not exceed a pre-specified threshold g(p) for every probability level p is an element of [0, 1]. Through the choice of the benchmark risk profile gone can tailor the risk assessment to the specific application of interest. We devote special attention to the study of risk profiles defined by the Expected Shortfall of a benchmark random loss, in which case our risk measures are intimately linked to second-order stochastic dominance
Mean field games with absorption and common noise with a model of bank run
No full textWe consider a mean field game describing the limit of a stochastic differential game of -players whose state dynamics are subject to idiosyncratic and common noise and that can be absorbed when they hit a prescribed region of the state space. We provide a general result for the existence of weak mean field equilibria which, due to the absorption and the common noise, are given by random flow of sub-probabilities. We first use a fixed point argument to find solutions to the mean field problem in a reduced setting resulting from a discretization procedure and then we prove convergence of such equilibria to the desired solution. We exploit these ideas also to construct ɛ-Nash equilibria for the -player game. Since the approximation is two-fold, one given by the mean field limit and one given by the discretization, some suitable convergence results are needed. We also introduce and discuss a novel model of bank run that can be studied within this framework
On the quasi-sure superhedging duality with frictions
No full textWe prove the superhedging duality for a discrete-time financial market with proportional transaction costs under model uncertainty. Frictions are modelled through solvency cones as in the original model of Kabanov (Finance Stoch. 3:237–248, 1999) adapted to the quasi-sure setup of Bouchard and Nutz (Ann. Appl. Probab. 25:823–859, 2015). Our approach allows removing the restrictive assumption of no arbitrage of the second kind considered in Bouchard et al. (Math. Finance 29:837–860, 2019) and showing the duality under the more natural condition of strict no arbitrage. In addition, we extend the results to models with portfolio constraints
A MODEL-FREE ANALYSIS OF DISCRETE TIME FINANCIAL MARKETS
No full textWe discuss fundamental questions of Mathematical Finance such as arbitrage and hedging in the context of a discrete time market with no reference probability. We show how different notions of arbitrage can be studied under the same general framework by specifying a class S of significant sets, and we investigate the richness of the family of martingale measures in relation to the choice of S. We also provide a superhedging duality theorem. We show that the initial cost of the cheapest portfolio that dominates a contingent claim on every possible path, might be strictly greater than the upper bound of the no-arbitrage prices. We therefore characterize the subset of trajectories on which this duality gap disappears and observe how this is related to no-arbitrage considerations. We finally consider the extension of the previous results to markets with frictions
Model-free superhedging duality
No full textIn a model free discrete time financial market, we prove the superhedging duality theorem, where trading is allowed with dynamic and semi-static strategies. We also show that the initial cost of the cheapest portfolio that dominates a contingent claim on every possible path , might be strictly greater than the upper bound of the no-arbitrage prices. We therefore characterize the subset of trajectories on which this duality gap disappears and prove that it is an analytic set
On the properties of the Lambda value at risk: robustness, elicitability and consistency
No full textRecently, the financial industry and regulators have enhanced the debate on the good properties of a risk measure. A fundamental issue is the evaluation of the quality of a risk estimation. On the one hand, a backtesting procedure is desirable for assessing the accuracy of such an estimation and this can be naturally achieved by elicitable risk measures. For the same objective, an alternative approach has been introduced by Davis [Stat. Risk Model. Appl. Finance Insurance, 2016, 33, 67–93] through the so-called consistency property. On the other hand, a risk estimation should be less sensitive with respect to small changes in the available data-set and exhibit qualitative robustness. A new risk measure, the Lambda value at risk (), has been recently proposed by Frittelli et al. [Math. Finance, 2014, 24, 442–463], as a generalization of VaR with the ability to discriminate the risk among P&L distributions with different tail behaviour. In this article, we show that also satisfies the properties of robustness, elicitability and consistency under some conditions
Short Communication: Robust Market-Adjusted Systemic Risk Measures
No full textIn this note we consider a system of financial institutions and study systemic risk measures in the presence of a financial market and in a robust setting, namely, where no reference probability is assigned. We obtain a dual representation for convex robust systemic risk measures adjusted to the financial market and show its relation to some appropriate no-arbitrage conditions
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