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Title
Abstract
We consider the problem of optimally sharing a financial position among agents with potentially different reference risk measures. The problem is equivalent to computing the infimal convolution of the risk metrics and finding the so-called optimal allocations. We propose a neural network-based framework to solve the problem and we prove the convergence of the approximated inf-convolution, as well as the approximated optimal allocations, to the corresponding theoretical values. We support our findings with several numerical experiments.
Bibtex
@misc{BDMC22, url = {https://arxiv.org/abs/2212.11752}, author = {Burzoni, Matteo and Doldi, Alessandro and Compagnoni, Enea Monzio}, keywords = {Risk Management (q-fin.RM), Probability (math.PR), FOS: Economics and business, FOS: Economics and business, FOS: Mathematics, FOS: Mathematics}, title = {Risk sharing with deep neural networks}, publisher = {arXiv}, year = {2022}, copyright = {arXiv.org perpetual, non-exclusive license}}
Title
Abstract
We provide a new version of the Tikhonov theorem for two-scale systems of stochastic differential equations which also covers the McKean-Vlasov case. Differently from what is usually done in the literature, we prove a type of convergence for the "fast" variable, which allows the limiting process to be discontinuous. This is relevant for the second part of the paper, where we present a new application of this theory to the approximation of the solution of mean field control problems. Towards this aim, we construct a two-scale system whose "fast" component converges to the optimal control process, while the "slow" component converges to the optimal state process. The interest in such a procedure is that it allows to approximate the solution of the control problem avoiding the usual step of the minimization of the Hamiltonian.
Bibtex
@misc{BC22, url = {https://arxiv.org/abs/2212.12293}, author = {Burzoni, Matteo and Cosso, Andrea}, keywords = {Probability (math.PR), Optimization and Control (math.OC), FOS: Mathematics, FOS: Mathematics, 60H10, 49N80, 49M99}, title = {A Tikhonov theorem for McKean-Vlasov two-scale systems and a new application to mean field optimal control problems}, publisher = {arXiv}, year = {2022}, copyright = {arXiv.org perpetual, non-exclusive license}}
Title
Abstract
We consider a mean field game describing the limit of a stochastic differential game of N-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 ε-Nash equilibria for the N-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.
Title
Abstract
In 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.
Abstract
We 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 ES_p(X) does not exceed a pre-specified threshold g(p) for every probability level p in [0,1]. Through the choice of the benchmark risk profile g one 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.
Bibtex
@article{BMW20adjusted, title = {Adjusted Expected Shortfall}, journal = {Journal of Banking & Finance}, volume = {134}, pages = {106297}, year = {2022}, issn = {0378-4266}, doi = {https://doi.org/10.1016/j.jbankfin.2021.106297}, url = {https://www.sciencedirect.com/science/article/pii/S0378426621002491}, author = {Matteo Burzoni and Cosimo Munari and Ruodu Wang}, }
Title
Abstract
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. Different versions of the Efficient Market Hypothesis are related to the assumptions one is willing to impose on the common order.
Bibtex
@article{BRS21, author = {Burzoni, Matteo and Riedel, Frank and Soner, H. Mete}, title = {Viability and Arbitrage Under Knightian Uncertainty}, journal = {Econometrica}, volume = {89}, number = {3}, pages = {1207-1234}, keywords = {Viability, Knightian uncertainty, no arbitrage, robust finance}, doi = {https://doi.org/10.3982/ECTA16535}, url = {https://onlinelibrary.wiley.com/doi/abs/10.3982/ECTA16535}, eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.3982/ECTA16535}, year = {2021} }
Title
Abstract
We study a class of non linear integro-differential equations on the Wasserstein space related to the optimal control of McKean--Vlasov jump--diffusions. We develop an intrinsic notion of viscosity solutions that does not rely on the lifting to an Hilbert space and prove a comparison theorem for these solutions. We also show that the value function is the unique viscosity solution.
Title
Abstract
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 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 approximate martingale measures sharing the same polar sets of P. We then specialize the results to a discrete-time framework in order to obtain true martingale measures.
Title
Abstract
We 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.
Title
Abstract
We analyze the martingale selection problem of Rokhlin (2006) in a pointwise (robust) setting. We derive conditions for solvability of this problem and show how it is related to the classical no-arbitrage deliberations. We obtain versions of the Fundamental Theorem of Asset Pricing in examples spanning frictionless markets, models with proportional transaction costs and also models for illiquid markets. In all these examples, we also incorporate trading constraints.
Title
Abstract
We 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), i.e.~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 provide a comprehensive study of the main finance theoretical and statistical properties of LVaR with a focus on their comparison with VaR and ES. Merits and drawbacks are discussed and applications to capital adequacy, portfolio risk management and catastrophic risk are presented.
Title
Abstract
We develop a robust framework for pricing and hedging of derivative securities in discrete-time financial markets. We consider markets with both dynamically and statically traded assets and make minimal measurability assumptions. We obtain abstract (pointwise) fundamental theorem of asset pricing and pricing–hedging duality. Our results are general and, in particular, cover both the so-called model independent case as well as the classical probabilistic case of Dalang--Morton--Willinger. Our analysis is scenario-based: a model specification is equivalent to a choice of scenarios to be considered. The choice can vary between all scenarios and the set of scenarios charged by a given probability measure. In this way, our framework interpolates between a model with universally acceptable broad assumptions and a model based on a specific probabilistic view of future asset dynamics.
Bibtex
@article{BFHMO19pointwise, author = {Burzoni, Matteo and Frittelli, Marco and Hou, Zhaoxu and Maggis, Marco and Ob\l \'oj, Jan}, title = {Pointwise Arbitrage Pricing Theory in Discrete Time}, journal = {Mathematics of Operations Research}, volume = {44}, number = {3}, pages = {1034-1057}, year = {2019}}
Title
Abstract
Recently, 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 Lambda vale at risk also satisfies the properties of robustness, elicitability and consistency under some conditions.
Title
Abstract
We provide a Fundamental Theorem of Asset Pricing and a Superhedging Theorem for a model independent 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. Fin. (2004): 14(1), 19-48) and show that this is equivalent to the existence of Consistent Price Systems. Moreover, 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.
Title
Abstract
In a model-free discrete time financial market, we prove the superhedging duality theorem, where trading is allowed with dynamic and semistatic 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.
Title
Abstract
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 S of significant sets, which we call arbitrage de la classe S. The choice of S reflects the intrinsic properties of the class of polar sets of martingale measures. In particular, for S absence of model-independent arbitrage is equivalent to the existence of a martingale measure; for S=Ω 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.
Bibtex
@article{BFM16universal, Author = {Burzoni, Matteo and Frittelli, Marco and Maggis, Marco}, Fjournal = {Finance and Stochastics}, Journal = {Finance Stoch.}, Number = {1}, Pages = {1--50}, Title = {Universal arbitrage aggregator in discrete-time markets under uncertainty}, Volume = {20}, Year = {2016}}
Title
Abstract
We 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.