Mathematical Finance最新文献

{"title":"Dynamics of market making algorithms in dealer markets: Learning and tacit collusion","authors":"Rama Cont,&nbsp;Wei Xiong","doi":"10.1111/mafi.12401","DOIUrl":"10.1111/mafi.12401","url":null,"abstract":"<p>The widespread use of market-making algorithms in electronic over-the-counter markets may give rise to unexpected effects resulting from the autonomous learning dynamics of these algorithms. In particular the possibility of “tacit collusion” among market makers has increasingly received regulatory scrutiny. We model the interaction of market makers in a dealer market as a stochastic differential game of intensity control with partial information and study the resulting dynamics of bid-ask spreads. Competition among dealers is modeled as a Nash equilibrium, while collusion is described in terms of Pareto optima. Using a decentralized multi-agent deep reinforcement learning algorithm to model how competing market makers learn to adjust their quotes, we show that the interaction of market making algorithms via market prices, without any sharing of information, may give rise to tacit collusion, with spread levels strictly above the competitive equilibrium level.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"34 2","pages":"467-521"},"PeriodicalIF":1.6,"publicationDate":"2023-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/mafi.12401","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42985217","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":"Trading under the proof-of-stake protocol – A continuous-time control approach","authors":"Wenpin Tang,&nbsp;David D. Yao","doi":"10.1111/mafi.12403","DOIUrl":"10.1111/mafi.12403","url":null,"abstract":"<p>We develop a continuous-time control approach to optimal trading in a Proof-of-Stake (PoS) blockchain, formulated as a consumption-investment problem that aims to strike the optimal balance between a participant's (or agent's) utility from holding/trading stakes and utility from consumption. We present solutions via dynamic programming and the Hamilton–Jacobi–Bellman (HJB) equations. When the utility functions are linear or convex, we derive close-form solutions and show that the bang-bang strategy is optimal (i.e., always buy or sell at full capacity). Furthermore, we bring out the explicit connection between the rate of return in trading/holding stakes and the participant's risk-adjusted valuation of the stakes. In particular, we show when a participant is risk-neutral or risk-seeking, corresponding to the risk-adjusted valuation being a martingale or a sub-martingale, the optimal strategy must be to either buy all the time, sell all the time, or first buy then sell, and with both buying and selling executed at full capacity. We also propose a risk-control version of the consumption-investment problem; and for a special case, the “stake-parity” problem, we show a mean-reverting strategy is optimal.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"33 4","pages":"979-1004"},"PeriodicalIF":1.6,"publicationDate":"2023-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44081069","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":"Local volatility under rough volatility","authors":"Florian Bourgey,&nbsp;Stefano De Marco,&nbsp;Peter K. Friz,&nbsp;Paolo Pigato","doi":"10.1111/mafi.12392","DOIUrl":"10.1111/mafi.12392","url":null,"abstract":"<p>Several asymptotic results for the implied volatility generated by a rough volatility model have been obtained in recent years (notably in the small-maturity regime), providing a better understanding of the shapes of the volatility surface induced by rough volatility models, supporting their calibration power to SP500 option data. Rough volatility models also generate a local volatility surface, via the so-called Markovian projection of the stochastic volatility. We complement the existing results on implied volatility by studying the asymptotic behavior of the local volatility surface generated by a class of rough stochastic volatility models, encompassing the rough Bergomi model. Notably, we observe that the celebrated “1/2 skew rule” linking the short-term at-the-money skew of the implied volatility to the short-term at-the-money skew of the local volatility, a consequence of the celebrated “harmonic mean formula” of [Berestycki et al. (2002). <i>Quantitative Finance, 2, 61–69</i>], is replaced by a new rule: the ratio of the at-the-money implied and local volatility skews tends to the constant <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>/</mo>\u0000 <mo>(</mo>\u0000 <mi>H</mi>\u0000 <mo>+</mo>\u0000 <mn>3</mn>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$1/(H + 3/2)$</annotation>\u0000 </semantics></math> (as opposed to the constant 1/2), where <i>H</i> is the regularity index of the underlying instantaneous volatility process.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"33 4","pages":"1119-1145"},"PeriodicalIF":1.6,"publicationDate":"2023-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46921786","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":"Proximity with affinity: How M&A top executives could exacerbate agency conflicts?","authors":"Jean-Gabriel Cousin, Marion Dupire, Jean-Yves Filbien","doi":"10.3917/fina.pr.023","DOIUrl":"https://doi.org/10.3917/fina.pr.023","url":null,"abstract":"","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"75 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2023-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83781789","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":"Asymptotic subadditivity/superadditivity of Value-at-Risk under tail dependence","authors":"Wenhao Zhu,&nbsp;Lujun Li,&nbsp;Jingping Yang,&nbsp;Jiehua Xie,&nbsp;Liulei Sun","doi":"10.1111/mafi.12393","DOIUrl":"10.1111/mafi.12393","url":null,"abstract":"<p>This paper presents a new method for discussing the asymptotic subadditivity/superadditivity of Value-at-Risk (VaR) for multiple risks. We consider the asymptotic subadditivity and superadditivity properties of VaR for multiple risks whose copula admits a stable tail dependence function (STDF). For the purpose, a marginal region is defined by the marginal distributions of the multiple risks, and a stochastic order named tail concave order is presented for comparing individual tail risks. We prove that asymptotic subadditivity of VaR holds when individual risks are smaller than regularly varying (RV) random variables with index −1 under the tail concave order. We also provide sufficient conditions for VaR being asymptotically superadditive. For two multiple risks sharing the same copula function and satisfying the tail concave order, a comparison result on the asymptotic subadditivity/superadditivity of VaR is given. Asymptotic diversification ratios for RV and log regularly varying (LRV) margins with specific copula structures are obtained. Empirical analysis on financial data is provided for highlighting our results.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"33 4","pages":"1314-1369"},"PeriodicalIF":1.6,"publicationDate":"2023-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41367368","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":"The log-moment formula for implied volatility","authors":"Vimal Raval,&nbsp;Antoine Jacquier","doi":"10.1111/mafi.12396","DOIUrl":"https://doi.org/10.1111/mafi.12396","url":null,"abstract":"<p>We revisit the foundational Moment Formula proved by Roger Lee fifteen years ago. We show that in the absence of arbitrage, if the underlying stock price at time <i>T</i> admits finite log-moments <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>E</mi>\u0000 <mo>[</mo>\u0000 <mo>|</mo>\u0000 <mi>log</mi>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mi>T</mi>\u0000 </msub>\u0000 <msup>\u0000 <mo>|</mo>\u0000 <mi>q</mi>\u0000 </msup>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <annotation>$mathbb {E}[|log S_T|^q]$</annotation>\u0000 </semantics></math> for some positive <i>q</i>, the arbitrage-free growth in the left wing of the implied volatility smile for <i>T</i> is less constrained than Lee's bound. The result is rationalized by a market trading discretely monitored variance swaps wherein the payoff is a function of squared log-returns, and requires no assumption for the underlying price to admit any negative moment. In this respect, the result can be derived from a model-independent setup. As a byproduct, we relax the moment assumptions on the stock price to provide a new proof of the notorious Gatheral–Fukasawa formula expressing variance swaps in terms of the implied volatility.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"33 4","pages":"1146-1165"},"PeriodicalIF":1.6,"publicationDate":"2023-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/mafi.12396","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50129610","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 Leland model for delta hedging in central risk books","authors":"Johannes Muhle-Karbe,&nbsp;Zexin Wang,&nbsp;Kevin Webster","doi":"10.1111/mafi.12395","DOIUrl":"10.1111/mafi.12395","url":null,"abstract":"<p>Using a tractable extension of the model of Leland (1985), we study how a delta-hedging strategy can realistically be implemented using market <i>and</i> limit orders in a centralized, automated market-making desk that integrates trading and liquidity provision for both options and their underlyings. In the continuous-time limit, the optimal limit-order exposure can be computed explicitly by a <i>pointwise</i> maximization. It is determined by the relative magnitudes of adverse selection, bid–ask spreads, and volatilities. The corresponding option price—from which the option can be replicated using market and limit orders—is characterized via a nonlinear PDE. Our results highlight the benefit of tactical liquidity provision for contrarian trading strategies, even for a trading desk that is not a competitive market maker. More generally, the paper also showcases how reduced-form models are competitive with “brute force” numerical approaches to market microstructure. Both the estimation of microstructure parameters and the simulation of the optimal trading strategy are made concrete and reconciled with real-life high frequency data.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"33 3","pages":"504-547"},"PeriodicalIF":1.6,"publicationDate":"2023-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/mafi.12395","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43718101","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":"Equilibrium investment with random risk aversion","authors":"Sascha Desmettre,&nbsp;Mogens Steffensen","doi":"10.1111/mafi.12394","DOIUrl":"10.1111/mafi.12394","url":null,"abstract":"<p>We solve the problem of an investor who maximizes utility but faces random preferences. We propose a problem formulation based on expected certainty equivalents. We tackle the time-consistency issues arising from that formulation by applying the equilibrium theory approach. To this end, we provide the proper definitions and prove a rigorous verification theorem. We complete the calculations for the cases of power and exponential utility. For power utility, we illustrate in a numerical example that the equilibrium stock proportion is independent of wealth, but decreasing in time, which we also supplement by a theoretical discussion. For exponential utility, the usual constant absolute risk aversion is replaced by its expectation.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"33 3","pages":"946-975"},"PeriodicalIF":1.6,"publicationDate":"2023-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/mafi.12394","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41756346","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":"Weak equilibria for time-inconsistent control: With applications to investment-withdrawal decisions","authors":"Zongxia Liang,&nbsp;Fengyi Yuan","doi":"10.1111/mafi.12391","DOIUrl":"10.1111/mafi.12391","url":null,"abstract":"<p>This paper considers time-inconsistent problems when control and stopping strategies are required to be made <i>simultaneously</i> (called stopping control problems by us). We first formulate the time-inconsistent stopping control problems under general multidimensional controlled diffusion model and propose a formal definition of their equilibria. We show that an admissible pair <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mover>\u0000 <mi>u</mi>\u0000 <mo>̂</mo>\u0000 </mover>\u0000 <mo>,</mo>\u0000 <mi>C</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(hat{u},C)$</annotation>\u0000 </semantics></math> of control-stopping policy is equilibrium if and only if the auxiliary function associated with it solves the extended HJB system, providing a methodology to <i>verify</i> or <i>exclude</i> equilibrium solutions. We provide several examples to illustrate applications to mathematical finance and control theory. For a problem whose reward function endogenously depends on the current wealth, the equilibrium is explicitly obtained. For another model with a nonexponential discount, we prove that any constant proportion strategy can not be equilibrium. We further show that general nonconstant equilibrium exists and is described by singular boundary value problems. This example shows that considering our combined problems is essentially different from investigating them separately. In the end, we also provide a two-dimensional example with a hyperbolic discount.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"33 3","pages":"891-945"},"PeriodicalIF":1.6,"publicationDate":"2023-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43800406","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":"Designing universal causal deep learning models: The geometric (Hyper)transformer","authors":"Beatrice Acciaio,&nbsp;Anastasis Kratsios,&nbsp;Gudmund Pammer","doi":"10.1111/mafi.12389","DOIUrl":"10.1111/mafi.12389","url":null,"abstract":"<p>Several problems in stochastic analysis are defined through their geometry, and preserving that geometric structure is essential to generating meaningful predictions. Nevertheless, how to design principled deep learning (DL) models capable of encoding these geometric structures remains largely unknown. We address this open problem by introducing a universal causal geometric DL framework in which the user specifies a suitable pair of metric spaces <math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$mathcal {X}$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mi>Y</mi>\u0000 <annotation>$mathcal {Y}$</annotation>\u0000 </semantics></math> and our framework returns a DL model capable of causally approximating any “regular” map sending time series in <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>X</mi>\u0000 <mi>Z</mi>\u0000 </msup>\u0000 <annotation>$mathcal {X}^{mathbb {Z}}$</annotation>\u0000 </semantics></math> to time series in <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>Y</mi>\u0000 <mi>Z</mi>\u0000 </msup>\u0000 <annotation>$mathcal {Y}^{mathbb {Z}}$</annotation>\u0000 </semantics></math> while respecting their forward flow of information throughout time. Suitable geometries on <math>\u0000 <semantics>\u0000 <mi>Y</mi>\u0000 <annotation>$mathcal {Y}$</annotation>\u0000 </semantics></math> include various (adapted) Wasserstein spaces arising in optimal stopping problems, a variety of statistical manifolds describing the conditional distribution of continuous-time finite state Markov chains, and all Fréchet spaces admitting a Schauder basis, for example, as in classical finance. Suitable spaces <math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$mathcal {X}$</annotation>\u0000 </semantics></math> are compact subsets of any Euclidean space. Our results all quantitatively express the number of parameters needed for our DL model to achieve a given approximation error as a function of the target map's regularity and the geometric structure both of <math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$mathcal {X}$</annotation>\u0000 </semantics></math> and of <math>\u0000 <semantics>\u0000 <mi>Y</mi>\u0000 <annotation>$mathcal {Y}$</annotation>\u0000 </semantics></math>. Even when omitting any temporal structure, our universal approximation theorems are the first guarantees that Hölder functions, defined between such <math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$mathcal {X}$</annotation>\u0000 </semantics></math> and <math","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"34 2","pages":"671-735"},"PeriodicalIF":1.6,"publicationDate":"2023-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/mafi.12389","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42860230","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}