Mathematical Finance最新文献

{"title":"A general approximation method for optimal stopping and random delay","authors":"Pengzhan Chen,&nbsp;Yingda Song","doi":"10.1111/mafi.12380","DOIUrl":"10.1111/mafi.12380","url":null,"abstract":"<p>This study examines the continuous-time optimal stopping problem with an infinite horizon under Markov processes. Existing research focuses on finding explicit solutions under certain assumptions of the reward function or underlying process; however, these assumptions may either not be fulfilled or be difficult to validate in practice. We developed a continuous-time Markov chain (CTMC) approximation method to find the optimal solution, which applies to general reward functions and underlying Markov processes. We demonstrated that our method can be used to solve the optimal stopping problem with a random delay, in which the delay could be either an independent random variable or a function of the underlying process. We established a theoretical upper bound for the approximation error to facilitate error control. Furthermore, we designed a two-stage scheme to implement our method efficiently. The numerical results show that the proposed method is accurate and rapid under various model specifications.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"34 1","pages":"5-35"},"PeriodicalIF":1.6,"publicationDate":"2023-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47079969","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Markov decision processes under model uncertainty","authors":"Ariel Neufeld,&nbsp;Julian Sester,&nbsp;Mario Šikić","doi":"10.1111/mafi.12381","DOIUrl":"10.1111/mafi.12381","url":null,"abstract":"<p>We introduce a general framework for Markov decision problems under model uncertainty in a discrete-time infinite horizon setting. By providing a dynamic programming principle, we obtain a local-to-global paradigm, namely solving a local, that is, a one time-step robust optimization problem leads to an optimizer of the global (i.e., infinite time-steps) robust stochastic optimal control problem, as well as to a corresponding worst-case measure. Moreover, we apply this framework to portfolio optimization involving data of the <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>S</mi>\u0000 <mo>&amp;</mo>\u0000 <mi>P</mi>\u0000 <mspace></mspace>\u0000 <mn>500</mn>\u0000 </mrow>\u0000 <annotation>$S&amp;Pnobreakspace 500$</annotation>\u0000 </semantics></math>. We present two different types of ambiguity sets; one is fully data-driven given by a Wasserstein-ball around the empirical measure, the second one is described by a parametric set of multivariate normal distributions, where the corresponding uncertainty sets of the parameters are estimated from the data. It turns out that in scenarios where the market is volatile or bearish, the optimal portfolio strategies from the corresponding robust optimization problem outperforms the ones without model uncertainty, showcasing the importance of taking model uncertainty into account.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"33 3","pages":"618-665"},"PeriodicalIF":1.6,"publicationDate":"2023-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43218515","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Preference robust distortion risk measure and its application","authors":"Wei Wang,&nbsp;Huifu Xu","doi":"10.1111/mafi.12379","DOIUrl":"10.1111/mafi.12379","url":null,"abstract":"Distortion risk measure (DRM) plays a crucial role in management science and finance particularly actuarial science. Various DRMs have been introduced but little is discussed on which DRM at hand should be chosen to address a decision maker's (DM's) risk preference. This paper aims to fill out the gap. Specifically, we consider a situation where the true distortion function is unknown either because it is difficult to identify/elicit and/or because the DM's risk preference is ambiguous. We introduce a preference robust distortion risk measure (PRDRM), which is based on the worst‐case distortion function from an ambiguity set of distortion functions to mitigate the impact arising from the ambiguity. The ambiguity set is constructed under well‐known general principles such as concavity and inverse S‐shapedness of distortion functions (overweighting on events from impossible to possible or possible to certainty and underweighting on those from possible to more possible) as well as new user‐specific information such as sensitivity to tail losses, confidence intervals to some lotteries, and preferences to certain lotteries over others. To calculate the proposed PRDRM, we use the convex and/or concave envelope of a set of points to characterize the curvature of the distortion function and derive a tractable reformulation of the PRDRM when the underlying random loss is discretely distributed. Moreover, we show that the worst‐case distortion function is a nondecreasing piecewise linear function and can be determined by solving a linear programming problem. Finally, we apply the proposed PRDRM to a risk capital allocation problem and carry out some numerical tests to examine the efficiency of the PRDRM model.","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"33 2","pages":"389-434"},"PeriodicalIF":1.6,"publicationDate":"2023-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/mafi.12379","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45665490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Improving reinforcement learning algorithms: Towards optimal learning rate policies","authors":"Othmane Mounjid,&nbsp;Charles-Albert Lehalle","doi":"10.1111/mafi.12378","DOIUrl":"10.1111/mafi.12378","url":null,"abstract":"<p>This paper shows how to use results of statistical learning theory and stochastic algorithms to have a better understanding of the convergence of Reinforcement Learning (RL) once it is formulated as a fixed point problem. This can be used to propose improvement of RL learning rates. First, our analysis shows that the classical asymptotic convergence rate <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>O</mi>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <msqrt>\u0000 <mi>N</mi>\u0000 </msqrt>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$O(1/sqrt {N})$</annotation>\u0000 </semantics></math> is pessimistic and can be replaced by <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>O</mi>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>log</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>N</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>/</mo>\u0000 <mi>N</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mi>β</mi>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$O((log (N)/N)^{beta })$</annotation>\u0000 </semantics></math> with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mfrac>\u0000 <mn>1</mn>\u0000 <mn>2</mn>\u0000 </mfrac>\u0000 <mo>≤</mo>\u0000 <mi>β</mi>\u0000 <mo>≤</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$frac{1}{2}le beta le 1$</annotation>\u0000 </semantics></math>, and <math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math> the number of iterations. Second, we propose a dynamic optimal policy for the choice of the learning rate used in RL. We decompose our policy into two interacting levels: the inner and outer levels. In the inner level, we present the PASS algorithm (for “PAst Sign Search”) which, based on a predefined sequence of learning rates, constructs a new sequence for which the error decreases faster. The convergence of PASS is proved and error bounds are established. In the outer level, we propose an optimal methodology for the selection of the predefined sequence. Third, we show empirically that our selection methodology of the learning rate outperforms significantly standard algorithms used in RL for the three following applications: the estim","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"34 2","pages":"588-621"},"PeriodicalIF":1.6,"publicationDate":"2023-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/mafi.12378","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74600791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal measure preserving derivatives revisited","authors":"Brendan K. Beare","doi":"10.1111/mafi.12377","DOIUrl":"10.1111/mafi.12377","url":null,"abstract":"<p>This article clarifies the relationship between pricing kernel monotonicity and the existence of opportunities for stochastic arbitrage in a complete and frictionless market of derivative securities written on a market portfolio. The relationship depends on whether the payoff distribution of the market portfolio satisfies a technical condition called adequacy, meaning that it is atomless or is comprised of finitely many equally probable atoms. Under adequacy, pricing kernel nonmonotonicity is equivalent to the existence of a strong form of stochastic arbitrage involving distributional replication of the market portfolio at a lower price. If the adequacy condition is dropped then this equivalence no longer holds, but pricing kernel nonmonotonicity remains equivalent to the existence of a weaker form of stochastic arbitrage involving second-order stochastic dominance of the market portfolio at a lower price. A generalization of the optimal measure preserving derivative is obtained, which achieves distributional replication at the minimum cost of all second-order stochastically dominant securities under adequacy.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"33 2","pages":"370-388"},"PeriodicalIF":1.6,"publicationDate":"2023-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/mafi.12377","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45039531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Reverse stress testing: Scenario design for macroprudential stress tests","authors":"Michel Baes,&nbsp;Eric Schaanning","doi":"10.1111/mafi.12373","DOIUrl":"10.1111/mafi.12373","url":null,"abstract":"<p>We propose a systematic algorithmic reverse-stress testing methodology to create “worst case” scenarios for regulatory stress tests by accounting for losses that arise from distressed portfolio liquidations. First, we derive the optimal bank response for any given shock. Then, we introduce an algorithm which systematically generates scenarios that exploit the key vulnerabilities in banks' portfolio holdings and thus maximize contagion despite banks' optimal response to the shock. We apply our methodology to data of the 2016 European Banking Authority (EBA) stress test, and design worst case scenarios for the portfolio holdings of European banks at the time. Using spectral clustering techniques, we group 10,000 worst-case scenarios into twelve geographically concentrated families. Our results show that even though there is a wide range of different scenarios within these 12 families, each cluster tends to affect the same banks. An “Anna Karenina” principle of stress testing emerges: <i>Not all stressful scenarios are alike, but every stressful scenario stresses the same banks</i>. These findings suggest that the precise specification of a scenario is not of primal importance as long as the most vulnerable banks are targeted and sufficiently stressed. Finally, our methodology can be used to uncover the weakest links in the financial system and thereby focus supervisory attention on these, thus building a bridge between macroprudential and microprudential stress tests.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"33 2","pages":"209-256"},"PeriodicalIF":1.6,"publicationDate":"2023-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/mafi.12373","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49236648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Model-free portfolio theory: A rough path approach","authors":"Andrew L. Allan,&nbsp;Christa Cuchiero,&nbsp;Chong Liu,&nbsp;David J. Prömel","doi":"10.1111/mafi.12376","DOIUrl":"10.1111/mafi.12376","url":null,"abstract":"<p>Based on a rough path foundation, we develop a model-free approach to stochastic portfolio theory (SPT). Our approach allows to handle significantly more general portfolios compared to previous model-free approaches based on Föllmer integration. Without the assumption of any underlying probabilistic model, we prove a pathwise formula for the relative wealth process, which reduces in the special case of functionally generated portfolios to a pathwise version of the so-called master formula of classical SPT. We show that the appropriately scaled asymptotic growth rate of a far reaching generalization of Cover's universal portfolio based on controlled paths coincides with that of the best retrospectively chosen portfolio within this class. We provide several novel results concerning rough integration, and highlight the advantages of the rough path approach by showing that (nonfunctionally generated) log-optimal portfolios in an ergodic Itô diffusion setting have the same asymptotic growth rate as Cover's universal portfolio and the best retrospectively chosen one.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"33 3","pages":"709-765"},"PeriodicalIF":1.6,"publicationDate":"2023-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/mafi.12376","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43072598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A model-free approach to continuous-time finance","authors":"Henry Chiu,&nbsp;Rama Cont","doi":"10.1111/mafi.12370","DOIUrl":"10.1111/mafi.12370","url":null,"abstract":"<p>We present a pathwise approach to continuous-time finance based on causal functional calculus. Our framework does not rely on any probabilistic concept. We introduce a definition of continuous-time self-financing portfolios, which does not rely on any integration concept and show that the value of a self-financing portfolio belongs to a class of nonanticipative functionals, which are pathwise analogs of martingales. We show that if the set of market scenarios is <i>generic</i> in the sense of being stable under certain operations, such self-financing strategies do not give rise to arbitrage. We then consider the problem of hedging a path-dependent payoff across a generic set of scenarios. Applying the transition principle of Rufus Isaacs in differential games, we obtain a pathwise dynamic programming principle for the superhedging cost. We show that the superhedging cost is characterized as the solution of a path-dependent equation. For the Asian option, we obtain an explicit solution.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"33 2","pages":"257-273"},"PeriodicalIF":1.6,"publicationDate":"2023-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/mafi.12370","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46352051","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Reconstructing volatility: Pricing of index options under rough volatility","authors":"Peter K. Friz,&nbsp;Thomas Wagenhofer","doi":"10.1111/mafi.12374","DOIUrl":"10.1111/mafi.12374","url":null,"abstract":"<p>Avellaneda et al. (2002, 2003) pioneered the pricing and hedging of index options – products highly sensitive to implied volatility and correlation assumptions – with large deviations methods, assuming local volatility dynamics for all components of the index. We present an extension applicable to non-Markovian dynamics and in particular the case of rough volatility dynamics.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"33 1","pages":"19-40"},"PeriodicalIF":1.6,"publicationDate":"2023-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/mafi.12374","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46365160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal investment with correlated stochastic volatility factors","authors":"Maxim Bichuch,&nbsp;Jean-Pierre Fouque","doi":"10.1111/mafi.12371","DOIUrl":"10.1111/mafi.12371","url":null,"abstract":"<p>The problem of portfolio allocation in the context of stocks evolving in random environments, that is with volatility and returns depending on random factors, has attracted a lot of attention. The problem of maximizing a power utility at a terminal time with only one random factor can be linearized thanks to a classical distortion transformation. In the present paper, we address the situation with several factors using a perturbation technique around the case where these factors are perfectly correlated reducing the problem to the case with a single factor. Our proposed approximation requires to solve numerically two linear equations in lower dimension instead of a fully nonlinear HJB equation. A rigorous accuracy result is derived by constructing sub- and super-solutions so that their difference is at the desired order of accuracy. We illustrate our result with a particular model for which we have explicit formulas for the approximation. In order to keep the notations as explicit as possible, we treat the case with one stock and two factors and we describe an extension to the case with two stocks and two factors.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"33 2","pages":"342-369"},"PeriodicalIF":1.6,"publicationDate":"2023-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44746595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}