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Robust Lambda-quantiles and extreme probabilities 稳健的 Lambda-quantiles 和极端概率
arXiv - QuantFin - Mathematical Finance Pub Date : 2024-06-19 DOI: arxiv-2406.13539
Xia Han, Peng Liu
{"title":"Robust Lambda-quantiles and extreme probabilities","authors":"Xia Han, Peng Liu","doi":"arxiv-2406.13539","DOIUrl":"https://doi.org/arxiv-2406.13539","url":null,"abstract":"In this paper, we investigate the robust models for $Lambda$-quantiles with\u0000partial information regarding the loss distribution, where $Lambda$-quantiles\u0000extend the classical quantiles by replacing the fixed probability level with a\u0000probability/loss function $Lambda$. We find that, under some assumptions, the\u0000robust $Lambda$-quantiles equal the $Lambda$-quantiles of the extreme\u0000probabilities. This finding allows us to obtain the robust $Lambda$-quantiles\u0000by applying the results of robust quantiles in the literature. Our results are\u0000applied to uncertainty sets characterized by three different constraints\u0000respectively: moment constraints, probability distance constraints via\u0000Wasserstein metric, and marginal constraints in risk aggregation. We obtain\u0000some explicit expressions for robust $Lambda$-quantiles by deriving the\u0000extreme probabilities for each uncertainty set. Those results are applied to\u0000optimal portfolio selection under model uncertainty.","PeriodicalId":501084,"journal":{"name":"arXiv - QuantFin - Mathematical Finance","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141548011","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
Pricing VIX options under the Heston-Hawkes stochastic volatility model 根据赫斯顿-霍克斯随机波动率模型为 VIX 期权定价
arXiv - QuantFin - Mathematical Finance Pub Date : 2024-06-19 DOI: arxiv-2406.13508
Oriol Zamora Font
{"title":"Pricing VIX options under the Heston-Hawkes stochastic volatility model","authors":"Oriol Zamora Font","doi":"arxiv-2406.13508","DOIUrl":"https://doi.org/arxiv-2406.13508","url":null,"abstract":"We derive a semi-analytical pricing formula for European VIX call options\u0000under the Heston-Hawkes stochastic volatility model introduced in\u0000arXiv:2210.15343. This arbitrage-free model incorporates the volatility\u0000clustering feature by adding an independent compound Hawkes process to the\u0000Heston volatility. Using the Markov property of the exponential Hawkes an\u0000explicit expression of $text{VIX}^2$ is derived as a linear combination of the\u0000variance and the Hawkes intensity. We apply qualitative ODE theory to study the\u0000existence of some generalized Riccati ODEs. Thereafter, we compute the joint\u0000characteristic function of the variance and the Hawkes intensity exploiting the\u0000exponential affine structure of the model. Finally, the pricing formula is\u0000obtained by applying standard Fourier techniques.","PeriodicalId":501084,"journal":{"name":"arXiv - QuantFin - Mathematical Finance","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141525933","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 dividend policy: Equivalence of Epstein-Zin and Maenhout preferences 稳健的股息政策：Epstein-Zin 和 Maenhout 偏好的等价性
arXiv - QuantFin - Mathematical Finance Pub Date : 2024-06-18 DOI: arxiv-2406.12305
Kexin Chen, Kyunghyun Park, Hoi Ying Wong
{"title":"Robust dividend policy: Equivalence of Epstein-Zin and Maenhout preferences","authors":"Kexin Chen, Kyunghyun Park, Hoi Ying Wong","doi":"arxiv-2406.12305","DOIUrl":"https://doi.org/arxiv-2406.12305","url":null,"abstract":"In a continuous-time economy, this study formulates the Epstein-Zin (EZ)\u0000preference for the discounted dividend (or cash payouts) of stockholders as an\u0000EZ singular control utility. We show that such a problem is well-defined and\u0000equivalent to the robust dividend policy set by the firm's executive in the\u0000sense of Maenhout's ambiguity-averse preference. While the firm's executive\u0000announces the expected future earnings in financial reports, they also signal\u0000the firm's confidence in the expected earnings through dividend or cash\u0000payouts. The robust dividend policy can then be characterized by a\u0000Hamilton-Jacobi-Bellman (HJB) variational inequality (VI). By constructing a\u0000novel shooting method for the HJB-VI, we theoretically prove that the robust\u0000dividend policy is a threshold strategy on the firm's surplus process.\u0000Therefore, dividend-caring investors can choose firms that match their\u0000preferences by examining stock's dividend policies and financial statements,\u0000whereas executives can make use of dividend to signal their confidence, in the\u0000form of ambiguity aversion, on realizing the earnings implied by their\u0000financial statements.","PeriodicalId":501084,"journal":{"name":"arXiv - QuantFin - Mathematical Finance","volume":"2013 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141525934","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
Local wealth condensation for yard-sale models with wealth-dependent biases 具有财富偏差的旧货市场模型的局部财富浓缩
arXiv - QuantFin - Mathematical Finance Pub Date : 2024-06-16 DOI: arxiv-2406.10978
Christoph Börgers, Claude Greengard
{"title":"Local wealth condensation for yard-sale models with wealth-dependent biases","authors":"Christoph Börgers, Claude Greengard","doi":"arxiv-2406.10978","DOIUrl":"https://doi.org/arxiv-2406.10978","url":null,"abstract":"In Chakraborti's yard-sale model of an economy, identical agents engage in\u0000pairwise trades, resulting in wealth exchanges that conserve each agent's\u0000expected wealth. Doob's martingale convergence theorem immediately implies\u0000almost sure wealth condensation, i.e., convergence to a state in which a single\u0000agent owns the entire economy. If some pairs of agents are not allowed to trade\u0000with each other, the martingale convergence theorem still implies local wealth\u0000condensation, i.e., convergence to a state in which some agents are wealthy,\u0000while all their trading partners are impoverished. In this note, we propose a\u0000new, more elementary proof of this result. Unlike the proof based on the\u0000martingale convergence theorem, our argument applies to models with a\u0000wealth-acquired advantage, and even to certain models with a poverty-acquired\u0000advantage.","PeriodicalId":501084,"journal":{"name":"arXiv - QuantFin - Mathematical Finance","volume":"363 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141548012","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
Computation of Robust Option Prices via Structured Multi-Marginal Martingale Optimal Transport 通过结构化多边际马丁格尔最优传输计算稳健期权价格
arXiv - QuantFin - Mathematical Finance Pub Date : 2024-06-14 DOI: arxiv-2406.09959
Linn Engström, Sigrid Källblad, Johan Karlsson
{"title":"Computation of Robust Option Prices via Structured Multi-Marginal Martingale Optimal Transport","authors":"Linn Engström, Sigrid Källblad, Johan Karlsson","doi":"arxiv-2406.09959","DOIUrl":"https://doi.org/arxiv-2406.09959","url":null,"abstract":"We introduce an efficient computational framework for solving a class of\u0000multi-marginal martingale optimal transport problems, which includes many\u0000robust pricing problems of large financial interest. Such problems are\u0000typically computationally challenging due to the martingale constraint,\u0000however, by extending the state space we can identify them with problems that\u0000exhibit a certain sequential martingale structure. Our method exploits such\u0000structures in combination with entropic regularisation, enabling fast\u0000computation of optimal solutions and allowing us to solve problems with a large\u0000number of marginals. We demonstrate the method by using it for computing robust\u0000price bounds for different options, such as lookback options and Asian options.","PeriodicalId":501084,"journal":{"name":"arXiv - QuantFin - Mathematical Finance","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141525936","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
Note on a Theoretical Justification for Approximations of Arithmetic Forwards 关于算术前向近似值理论依据的说明
arXiv - QuantFin - Mathematical Finance Pub Date : 2024-06-13 DOI: arxiv-2406.09488
Álvaro Romaniega
{"title":"Note on a Theoretical Justification for Approximations of Arithmetic Forwards","authors":"Álvaro Romaniega","doi":"arxiv-2406.09488","DOIUrl":"https://doi.org/arxiv-2406.09488","url":null,"abstract":"This brief note explores the theoretical justification for some\u0000approximations of arithmetic forwards ($F_a$) with weighted averages of\u0000overnight (ON) forwards ($F_k$). The central equation presented in this\u0000analysis is: begin{equation*}\u0000F_a(0;T_s,T_e)=frac{1}{tau(T_s,T_e)}sum_{k=1}^K tau_k mathcal{A}_k F_k,,\u0000end{equation*} with $mathcal{A}_k$ being explicit model-dependent quantities\u0000that, under certain market scenarios, are close to one. We will present\u0000computationally cheaper methods that approximate $F_a$, i.e., we will define\u0000some ${tilde{mathcal{A}}_k}_{k=1}^K$ such that begin{equation*}\u0000F_a(0;T_s,T_e)approx frac{1}{tau(T_s,T_e)}sum_{k=1}^K tilde{mathcal{A}}_k\u0000tau_k F_k,. end{equation*} We also demonstrate that one of these forms can\u0000be closely aligned with an approximation suggested by Katsumi Takada in his\u0000work on the valuation of arithmetic averages of Fed Funds rates.","PeriodicalId":501084,"journal":{"name":"arXiv - QuantFin - Mathematical Finance","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141525937","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
Convex ordering for stochastic control: the swing contracts case 随机控制的凸序：摆动合约案例
arXiv - QuantFin - Mathematical Finance Pub Date : 2024-06-11 DOI: arxiv-2406.07464
Gilles Pagès, Christian Yeo
{"title":"Convex ordering for stochastic control: the swing contracts case","authors":"Gilles Pagès, Christian Yeo","doi":"arxiv-2406.07464","DOIUrl":"https://doi.org/arxiv-2406.07464","url":null,"abstract":"We investigate propagation of convexity and convex ordering on a typical\u0000stochastic optimal control problem, namely the pricing of\u0000q{emph{Take-or-Pay}} swing option, a financial derivative product commonly\u0000traded on energy markets. The dynamics of the underlying asset is modelled by\u0000an emph{ARCH} model with convex coefficients. We prove that the value function\u0000associated to the stochastic optimal control problem is a convex function of\u0000the underlying asset price. We also introduce a domination criterion offering\u0000insights into the monotonicity of the value function with respect to parameters\u0000of the underlying emph{ARCH} coefficients. We particularly focus on the\u0000one-dimensional setting where, by means of Stein's formula and regularization\u0000techniques, we show that the convexity assumption for the emph{ARCH}\u0000coefficients can be relaxed with a semi-convexity assumption. To validate the\u0000results presented in this paper, we also conduct numerical illustrations.","PeriodicalId":501084,"journal":{"name":"arXiv - QuantFin - Mathematical Finance","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141548014","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
From rank-based models with common noise to pathwise entropy solutions of SPDEs 从具有共同噪声的基于秩的模型到 SPDE 的路径熵解
arXiv - QuantFin - Mathematical Finance Pub Date : 2024-06-11 DOI: arxiv-2406.07286
Mykhaylo Shkolnikov, Lane Chun Yeung
{"title":"From rank-based models with common noise to pathwise entropy solutions of SPDEs","authors":"Mykhaylo Shkolnikov, Lane Chun Yeung","doi":"arxiv-2406.07286","DOIUrl":"https://doi.org/arxiv-2406.07286","url":null,"abstract":"We study the mean field limit of a rank-based model with common noise, which\u0000arises as an extension to models for the market capitalization of firms in\u0000stochastic portfolio theory. We show that, under certain conditions on the\u0000drift and diffusion coefficients, the empirical cumulative distribution\u0000function converges to the solution of a stochastic PDE. A key step in the\u0000proof, which is of independent interest, is to show that any solution to an\u0000associated martingale problem is also a pathwise entropy solution to the\u0000stochastic PDE, a notion introduced in a recent series of papers [32, 33, 19,\u000016, 17].","PeriodicalId":501084,"journal":{"name":"arXiv - QuantFin - Mathematical Finance","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141525939","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
Change of numeraire for weak martingale transport 弱马丁格尔输运的数字变化
arXiv - QuantFin - Mathematical Finance Pub Date : 2024-06-11 DOI: arxiv-2406.07523
Mathias Beiglböck, Gudmund Pammer, Lorenz Riess
{"title":"Change of numeraire for weak martingale transport","authors":"Mathias Beiglböck, Gudmund Pammer, Lorenz Riess","doi":"arxiv-2406.07523","DOIUrl":"https://doi.org/arxiv-2406.07523","url":null,"abstract":"Change of numeraire is a classical tool in mathematical finance.\u0000Campi-Laachir-Martini established its applicability to martingale optimal\u0000transport. We note that the results of Campi-Laachir-Martini extend to the case\u0000of weak martingale transport. We apply this to shadow couplings, continuous\u0000time martingale transport problems in the framework of Huesmann-Trevisan and in\u0000particular to establish the correspondence between stretched Brownian motion\u0000with its geometric counterpart. Note: We emphasize that we learned about the geometric stretched Brownian\u0000motion gSBM (defined in PDE terms) in a presentation of Loeper cite{Lo23}\u0000before our work on this topic started. We noticed that a change of numeraire\u0000transformation in the spirit of cite{CaLaMa14} allows for an alternative\u0000viewpoint in the weak optimal transport framework. We make our work public\u0000following the publication of Backhoff-Loeper-Obloj's work cite{BaLoOb24} on\u0000arxiv.org. The article cite{BaLoOb24} derives gSBM using PDE techniques as\u0000well as through an independent probabilistic approach which is close to the one\u0000we give in the present article.","PeriodicalId":501084,"journal":{"name":"arXiv - QuantFin - Mathematical Finance","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141525938","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
Geometric Martingale Benamou-Brenier transport and geometric Bass martingales 几何马丁格尔贝纳摩-布雷尼尔输运和几何巴斯马丁格尔
arXiv - QuantFin - Mathematical Finance Pub Date : 2024-06-06 DOI: arxiv-2406.04016
Julio Backhoff-Veraguas, Gregoire Loeper, Jan Obloj
{"title":"Geometric Martingale Benamou-Brenier transport and geometric Bass martingales","authors":"Julio Backhoff-Veraguas, Gregoire Loeper, Jan Obloj","doi":"arxiv-2406.04016","DOIUrl":"https://doi.org/arxiv-2406.04016","url":null,"abstract":"We introduce and study geometric Bass martingales. Bass martingales were\u0000introduced in cite{Ba83} and studied recently in a series of works, including\u0000cite{BaBeHuKa20,BaBeScTs23}, where they appear as solutions to the martingale\u0000version of the Benamou-Brenier optimal transport formulation. These arithmetic,\u0000as well as our novel geometric, Bass martingales are continuous martingale on\u0000$[0,1]$ with prescribed initial and terminal distributions. An arithmetic Bass\u0000martingale is the one closest to Brownian motion: its quadratic variation is as\u0000close as possible to being linear in the averaged $L^2$ sense. Its geometric\u0000counterpart we develop here, is the one closest to a geometric Brownian motion:\u0000the quadratic variation of its logarithm is as close as possible to being\u0000linear. By analogy between Bachelier and Black-Scholes models in mathematical\u0000finance, the newly obtained geometric Bass martingales} have the potential to\u0000be of more practical importance in a number of applications. Our main contribution is to exhibit an explicit bijection between geometric\u0000Bass martingales and their arithmetic counterparts. This allows us, in\u0000particular, to translate fine properties of the latter into the new geometric\u0000setting. We obtain an explicit representation for a geometric Bass martingale\u0000for given initial and terminal marginals, we characterise it as a solution to\u0000an SDE, and we show that geometric Brownian motion is the only process which is\u0000both an arithmetic and a geometric Bass martingale. Finally, we deduce a dual\u0000formulation for our geometric martingale Benamou-Brenier problem. Our main\u0000proof is probabilistic in nature and uses a suitable change of measure, but we\u0000also provide PDE arguments relying on the dual formulation of the problem,\u0000which offer a rigorous proof under suitable regularity assumptions.","PeriodicalId":501084,"journal":{"name":"arXiv - QuantFin - Mathematical Finance","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141547952","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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