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Markovian projections for Itô semimartingales with jumps 有跳跃的伊托半马勒的马尔可夫投影
arXiv - QuantFin - Mathematical Finance Pub Date : 2024-03-24 DOI: arxiv-2403.15980
Martin Larsson, Shukun Long
{"title":"Markovian projections for Itô semimartingales with jumps","authors":"Martin Larsson, Shukun Long","doi":"arxiv-2403.15980","DOIUrl":"https://doi.org/arxiv-2403.15980","url":null,"abstract":"Given a general It^o semimartingale, its Markovian projection is an It^o\u0000process, with Markovian differential characteristics, that matches the\u0000one-dimensional marginal laws of the original process. We construct Markovian\u0000projections for It^o semimartingales with jumps, whose flows of\u0000one-dimensional marginal laws are solutions to non-local\u0000Fokker--Planck--Kolmogorov equations (FPKEs). As an application, we show how\u0000Markovian projections appear in building calibrated diffusion/jump models with\u0000both local and stochastic features.","PeriodicalId":501084,"journal":{"name":"arXiv - QuantFin - Mathematical Finance","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140301122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On Merton's Optimal Portfolio Problem under Sporadic Bankruptcy 论零星破产下的默顿最优投资组合问题
arXiv - QuantFin - Mathematical Finance Pub Date : 2024-03-23 DOI: arxiv-2403.15923
Yaacov Kopeliovich, Michael Pokojovy
{"title":"On Merton's Optimal Portfolio Problem under Sporadic Bankruptcy","authors":"Yaacov Kopeliovich, Michael Pokojovy","doi":"arxiv-2403.15923","DOIUrl":"https://doi.org/arxiv-2403.15923","url":null,"abstract":"Consider a stock market following a geometric Brownian motion and a riskless\u0000asset continuously compounded at a constant rate. Assuming the stock can go\u0000bankrupt, i.e., lose all of its value, at some exogenous random time\u0000(independent of the stock price) modeled as the first arrival time of a\u0000homogeneous Poisson process, we study the Merton's optimal portfolio problem\u0000consisting of maximizing the expected logarithmic utility of the total wealth\u0000at a preselected finite maturity time. First, we present a heuristic derivation\u0000based on a new type of Hamilton-Jacobi-Bellman equation. Then, we formally\u0000reduce the problem to a classical controlled Markovian diffusion with a new\u0000type of terminal and running costs. A new version of Merton's ratio is\u0000rigorously derived using Bellman's dynamic programming principle and validated\u0000with a suitable type of verification theorem. A real-world example comparing\u0000the latter ratio to the classical Merton's ratio is given.","PeriodicalId":501084,"journal":{"name":"arXiv - QuantFin - Mathematical Finance","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140301123","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Hull-White model with volatility smile for Valuation Adjustments 关于估值调整的带波动微笑的赫尔-怀特模型
arXiv - QuantFin - Mathematical Finance Pub Date : 2024-03-21 DOI: arxiv-2403.14841
T. van der Zwaard, L. A. Grzelak, C. W. Oosterlee
{"title":"On the Hull-White model with volatility smile for Valuation Adjustments","authors":"T. van der Zwaard, L. A. Grzelak, C. W. Oosterlee","doi":"arxiv-2403.14841","DOIUrl":"https://doi.org/arxiv-2403.14841","url":null,"abstract":"Affine Diffusion dynamics are frequently used for Valuation Adjustments (xVA)\u0000calculations due to their analytic tractability. However, these models cannot\u0000capture the market-implied skew and smile, which are relevant when computing\u0000xVA metrics. Hence, additional degrees of freedom are required to capture these\u0000market features. In this paper, we address this through an SDE with\u0000state-dependent coefficients. The SDE is consistent with the convex combination\u0000of a finite number of different AD dynamics. We combine Hull-White one-factor\u0000models where one model parameter is varied. We use the Randomized AD (RAnD)\u0000technique to parameterize the combination of dynamics. We refer to our SDE with\u0000state-dependent coefficients and the RAnD parametrization of the original\u0000models as the rHW model. The rHW model allows for efficient semi-analytic\u0000calibration to European swaptions through the analytic tractability of the\u0000Hull-White dynamics. We use a regression-based Monte-Carlo simulation to\u0000calculate exposures. In this setting, we demonstrate the significant effect of\u0000skew and smile on exposures and xVAs of linear and early-exercise interest rate\u0000derivatives.","PeriodicalId":501084,"journal":{"name":"arXiv - QuantFin - Mathematical Finance","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140301119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
New Stochastic Fubini Theorems 新随机富比尼定理
arXiv - QuantFin - Mathematical Finance Pub Date : 2024-03-20 DOI: arxiv-2403.13791
Tahir Choulli, Martin Schweizer
{"title":"New Stochastic Fubini Theorems","authors":"Tahir Choulli, Martin Schweizer","doi":"arxiv-2403.13791","DOIUrl":"https://doi.org/arxiv-2403.13791","url":null,"abstract":"The classic stochastic Fubini theorem says that if one stochastically\u0000integrates with respect to a semimartingale $S$ an $eta(dz)$-mixture of\u0000$z$-parametrized integrands $psi^z$, the result is just the $eta(dz)$-mixture\u0000of the individual $z$-parametrized stochastic integrals $intpsi^z{d}S.$ But\u0000if one wants to use such a result for the study of Volterra semimartingales of\u0000the form $ X_t =int_0^t Psi_{t,s}dS_s, t geq0,$ the classic assumption that\u0000one has a fixed measure $eta$ is too restrictive; the mixture over the\u0000integrands needs to be taken instead with respect to a stochastic kernel on the\u0000parameter space. To handle that situation and prove a corresponding new\u0000stochastic Fubini theorem, we introduce a new notion of measure-valued\u0000stochastic integration with respect to a general multidimensional\u0000semimartingale. As an application, we show how this allows to handle a class of\u0000quite general stochastic Volterra semimartingales.","PeriodicalId":501084,"journal":{"name":"arXiv - QuantFin - Mathematical Finance","volume":"293 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140203760","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Max-stability under first-order stochastic dominance 一阶随机优势下的最大稳定性
arXiv - QuantFin - Mathematical Finance Pub Date : 2024-03-19 DOI: arxiv-2403.13138
Christopher Chambers, Alan Miller, Ruodu Wang, Qinyu Wu
{"title":"Max-stability under first-order stochastic dominance","authors":"Christopher Chambers, Alan Miller, Ruodu Wang, Qinyu Wu","doi":"arxiv-2403.13138","DOIUrl":"https://doi.org/arxiv-2403.13138","url":null,"abstract":"Max-stability is the property that taking a maximum between two inputs\u0000results in a maximum between two outputs. We investigate max-stability with\u0000respect to first-order stochastic dominance, the most fundamental notion of\u0000stochastic dominance in decision theory. Under two additional standard axioms\u0000of monotonicity and lower semicontinuity, we establish a representation theorem\u0000for functionals satisfying max-stability, which turns out to be represented by\u0000the supremum of a bivariate function. Our characterized functionals encompass\u0000special classes of functionals in the literature of risk measures, such as\u0000benchmark-loss Value at Risk (VaR) and $Lambda$-quantile.","PeriodicalId":501084,"journal":{"name":"arXiv - QuantFin - Mathematical Finance","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140203694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Risk premium and rough volatility 风险溢价和粗略波动
arXiv - QuantFin - Mathematical Finance Pub Date : 2024-03-18 DOI: arxiv-2403.11897
Ofelia Bonesini, Antoine Jacquier, Aitor Muguruza
{"title":"Risk premium and rough volatility","authors":"Ofelia Bonesini, Antoine Jacquier, Aitor Muguruza","doi":"arxiv-2403.11897","DOIUrl":"https://doi.org/arxiv-2403.11897","url":null,"abstract":"One the one hand, rough volatility has been shown to provide a consistent\u0000framework to capture the properties of stock price dynamics both under the\u0000historical measure and for pricing purposes. On the other hand, market price of\u0000volatility risk is a well-studied object in Financial Economics, and empirical\u0000estimates show it to be stochastic rather than deterministic. Starting from a\u0000rough volatility model under the historical measure, we take up this challenge\u0000and provide an analysis of the impact of such a non-deterministic risk for\u0000pricing purposes.","PeriodicalId":501084,"journal":{"name":"arXiv - QuantFin - Mathematical Finance","volume":"199 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140168793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Nonconcave Robust Utility Maximization under Projective Determinacy 投影确定性下的非凹鲁棒性效用最大化
arXiv - QuantFin - Mathematical Finance Pub Date : 2024-03-18 DOI: arxiv-2403.11824
Laurence Carassus, Massinissa Ferhoune
{"title":"Nonconcave Robust Utility Maximization under Projective Determinacy","authors":"Laurence Carassus, Massinissa Ferhoune","doi":"arxiv-2403.11824","DOIUrl":"https://doi.org/arxiv-2403.11824","url":null,"abstract":"We study a robust utility maximization problem in a general discrete-time\u0000frictionless market. The investor is assumed to have a random, nonconcave and\u0000nondecreasing utility function, which may or may not be finite on the whole\u0000real-line. She also faces model ambiguity on her beliefs about the market,\u0000which is modeled through a set of priors. We prove, using only primal methods,\u0000the existence of an optimal investment strategy when the utility function is\u0000also upper-semicontinuous. For that, we introduce the new notion of\u0000projectively measurable functions. We show basic properties of these functions\u0000as stability under sums, differences, products, suprema, infima and\u0000compositions but also assuming the set-theoretical axiom of Projective\u0000Determinacy (PD) stability under integration and existence of\u0000$epsilon$-optimal selectors. We consider projectively measurable random\u0000utility function and price process and assume that the graphs of the sets of\u0000local priors are projective sets. Our other assumptions are stated on a\u0000prior-by-prior basis and correspond to generally accepted assumptions in the\u0000literature on markets without ambiguity.","PeriodicalId":501084,"journal":{"name":"arXiv - QuantFin - Mathematical Finance","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140168546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Mean-Field Game of Market Entry: Portfolio Liquidation with Trading Constraints 市场进入的平均场博弈：有交易限制的投资组合清算
arXiv - QuantFin - Mathematical Finance Pub Date : 2024-03-15 DOI: arxiv-2403.10441
Guanxing Fu, Paul P. Hager, Ulrich Horst
{"title":"A Mean-Field Game of Market Entry: Portfolio Liquidation with Trading Constraints","authors":"Guanxing Fu, Paul P. Hager, Ulrich Horst","doi":"arxiv-2403.10441","DOIUrl":"https://doi.org/arxiv-2403.10441","url":null,"abstract":"We consider both $N$-player and mean-field games of optimal portfolio\u0000liquidation in which the players are not allowed to change the direction of\u0000trading. Players with an initially short position of stocks are only allowed to\u0000buy while players with an initially long position are only allowed to sell the\u0000stock. Under suitable conditions on the model parameters we show that the games\u0000are equivalent to games of timing where the players need to determine the\u0000optimal times of market entry and exit. We identify the equilibrium entry and\u0000exit times and prove that equilibrium mean-trading rates can be characterized\u0000in terms of the solutions to a highly non-linear higher-order integral equation\u0000with endogenous terminal condition. We prove the existence of a unique solution\u0000to the integral equation from which we obtain the existence of a unique\u0000equilibrium both in the mean-field and the $N$-player game.","PeriodicalId":501084,"journal":{"name":"arXiv - QuantFin - Mathematical Finance","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140147905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Robust SGLD algorithm for solving non-convex distributionally robust optimisation problems 解决非凸分布鲁棒优化问题的鲁棒 SGLD 算法
arXiv - QuantFin - Mathematical Finance Pub Date : 2024-03-14 DOI: arxiv-2403.09532
Ariel Neufeld, Matthew Ng Cheng En, Ying Zhang
{"title":"Robust SGLD algorithm for solving non-convex distributionally robust optimisation problems","authors":"Ariel Neufeld, Matthew Ng Cheng En, Ying Zhang","doi":"arxiv-2403.09532","DOIUrl":"https://doi.org/arxiv-2403.09532","url":null,"abstract":"In this paper we develop a Stochastic Gradient Langevin Dynamics (SGLD)\u0000algorithm tailored for solving a certain class of non-convex distributionally\u0000robust optimisation problems. By deriving non-asymptotic convergence bounds, we\u0000build an algorithm which for any prescribed accuracy $varepsilon>0$ outputs an\u0000estimator whose expected excess risk is at most $varepsilon$. As a concrete\u0000application, we employ our robust SGLD algorithm to solve the (regularised)\u0000distributionally robust Mean-CVaR portfolio optimisation problem using real\u0000financial data. We empirically demonstrate that the trading strategy obtained\u0000by our robust SGLD algorithm outperforms the trading strategy obtained when\u0000solving the corresponding non-robust Mean-CVaR portfolio optimisation problem\u0000using, e.g., a classical SGLD algorithm. This highlights the practical\u0000relevance of incorporating model uncertainty when optimising portfolios in real\u0000financial markets.","PeriodicalId":501084,"journal":{"name":"arXiv - QuantFin - Mathematical Finance","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140148263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Hydrodynamics of Markets:Hidden Links Between Physics and Finance 市场的流体力学：物理学与金融学之间的隐秘联系
arXiv - QuantFin - Mathematical Finance Pub Date : 2024-03-14 DOI: arxiv-2403.09761
Alexander Lipton
{"title":"Hydrodynamics of Markets:Hidden Links Between Physics and Finance","authors":"Alexander Lipton","doi":"arxiv-2403.09761","DOIUrl":"https://doi.org/arxiv-2403.09761","url":null,"abstract":"An intriguing link between a wide range of problems occurring in physics and\u0000financial engineering is presented. These problems include the evolution of\u0000small perturbations of linear flows in hydrodynamics, the movements of\u0000particles in random fields described by the Kolmogorov and Klein-Kramers\u0000equations, the Ornstein-Uhlenbeck and Feller processes, and their\u0000generalizations. They are reduced to affine differential and\u0000pseudo-differential equations and solved in a unified way by using Kelvin waves\u0000and developing a comprehensive math framework for calculating transition\u0000probabilities and expectations. Kelvin waves are instrumental for studying the\u0000well-known Black-Scholes, Heston, and Stein-Stein models and more complex\u0000path-dependent volatility models, as well as the pricing of Asian options,\u0000volatility and variance swaps, bonds, and bond options. Kelvin waves help to\u0000solve several cutting-edge problems, including hedging the impermanent loss of\u0000Automated Market Makers for cryptocurrency trading. This title is also\u0000available as Open Access on Cambridge Core.","PeriodicalId":501084,"journal":{"name":"arXiv - QuantFin - Mathematical Finance","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140148267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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