{"title":"Polar Coordinates for the 3/2 Stochastic Volatility Model","authors":"Paul Nekoranik","doi":"10.1111/mafi.12455","DOIUrl":"https://doi.org/10.1111/mafi.12455","url":null,"abstract":"<p>The 3/2 stochastic volatility model is a continuous positive process <i>s</i> with a correlated infinitesimal variance process <span></span><math>\u0000 <semantics>\u0000 <mi>ν</mi>\u0000 <annotation>$nu $</annotation>\u0000 </semantics></math>. The exact definition is provided in the Introduction immediately below. By inspecting the geometry associated with this model, we discover an explicit smooth map <span></span><math>\u0000 <semantics>\u0000 <mi>ψ</mi>\u0000 <annotation>$ psi $</annotation>\u0000 </semantics></math> from <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mo>+</mo>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$({mathbb{R}}^+)^2 $</annotation>\u0000 </semantics></math> to the punctured plane <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>−</mo>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <mn>0</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>${mathbb{R}}^2-(0,0)$</annotation>\u0000 </semantics></math> for which the process <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>u</mi>\u0000 <mo>,</mo>\u0000 <mi>v</mi>\u0000 <mo>)</mo>\u0000 <mo>=</mo>\u0000 <mi>ψ</mi>\u0000 <mo>(</mo>\u0000 <mi>ν</mi>\u0000 <mo>,</mo>\u0000 <mi>s</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(u,v)=psi(nu,s)$</annotation>\u0000 </semantics></math> satisfies an SDE of a simpler form, with independent Brownian motions and the identity matrix as diffusion coefficient. Moreover, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>ν</mi>\u0000 <mi>t</mi>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>s</mi>\u0000 <mi>t</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(nu_t,s_t)$</annotation>\u0000 </semantics></math> is r","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"35 3","pages":"708-723"},"PeriodicalIF":1.6,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/mafi.12455","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144524683","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":"Spanning Multi-Asset Payoffs With ReLUs","authors":"Sébastien Bossu, Stéphane Crépey, Hoang-Dung Nguyen","doi":"10.1111/mafi.12454","DOIUrl":"https://doi.org/10.1111/mafi.12454","url":null,"abstract":"<p>We propose a distributional formulation of the spanning problem of a multi-asset payoff by vanilla basket options. This problem is shown to have a unique solution if and only if the payoff function is even and absolutely homogeneous, and we establish a Fourier-based formula to calculate the solution. Financial payoffs are typically piecewise linear, resulting in a solution that may be derived explicitly, yet may also be hard to exploit numerically. One-hidden-layer feedforward neural networks instead provide a natural and efficient numerical alternative for discrete spanning. We test this approach for a selection of archetypal payoffs and obtain better hedging results with vanilla basket options compared to industry-favored approaches based on single-asset vanilla hedges.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"35 3","pages":"682-707"},"PeriodicalIF":1.6,"publicationDate":"2025-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/mafi.12454","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144524795","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}
Peter Bank, Christian Bayer, Peter K. Friz, Luca Pelizzari
{"title":"Rough PDEs for Local Stochastic Volatility Models","authors":"Peter Bank, Christian Bayer, Peter K. Friz, Luca Pelizzari","doi":"10.1111/mafi.12458","DOIUrl":"https://doi.org/10.1111/mafi.12458","url":null,"abstract":"<p>In this work, we introduce a novel pricing methodology in general, possibly non-Markovian local stochastic volatility (LSV) models. We observe that by conditioning the LSV dynamics on the Brownian motion that drives the volatility, one obtains a time-inhomogeneous Markov process. Using tools from rough path theory, we describe how to precisely understand the conditional LSV dynamics and reveal their Markovian nature. The latter allows us to connect the conditional dynamics to so-called rough partial differential equations (RPDEs), through a Feynman–Kac type of formula. In terms of European pricing, conditional on realizations of one Brownian motion, we can compute conditional option prices by solving the corresponding linear RPDEs, and then average over all samples to find unconditional prices. Our approach depends only minimally on the specification of the volatility, making it applicable for a wide range of classical and rough LSV models, and it establishes a PDE pricing method for non-Markovian models. Finally, we present a first glimpse at numerical methods for RPDEs and apply them to price European options in several rough LSV models.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"35 3","pages":"661-681"},"PeriodicalIF":1.6,"publicationDate":"2025-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/mafi.12458","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144524777","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}
Beatrice Acciaio, Julio Backhoff-Veraguas, Gudmund Pammer
{"title":"Quantitative Fundamental Theorem of Asset Pricing","authors":"Beatrice Acciaio, Julio Backhoff-Veraguas, Gudmund Pammer","doi":"10.1111/mafi.12457","DOIUrl":"https://doi.org/10.1111/mafi.12457","url":null,"abstract":"<p>In this paper, we provide a quantitative analysis of the concept of arbitrage, that allows us to deal with model uncertainty without imposing the no-arbitrage condition. In markets that admit “small arbitrage,” we can still make sense of the problems of pricing and hedging. The pricing measures here will be such that asset price processes are close to being martingales, and the hedging strategies will need to cover some additional costs. We show a quantitative version of the fundamental theorem of asset pricing (FTAP) and of the super-replication theorem. Finally, we study robustness of the amount of arbitrage and existence of respective pricing measures, showing stability of these concepts with respect to a strongly adapted Wasserstein distance.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"35 3","pages":"636-660"},"PeriodicalIF":1.6,"publicationDate":"2025-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/mafi.12457","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144524887","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":"On Time-Inconsistency in Mean-Field Games","authors":"Erhan Bayraktar, Zhenhua Wang","doi":"10.1111/mafi.12456","DOIUrl":"https://doi.org/10.1111/mafi.12456","url":null,"abstract":"<div>\u0000 \u0000 <p>We investigate an infinite-horizon time-inconsistent mean-field game (MFG) in a discrete time setting. We first present a classic equilibrium for the MFG and its associated existence result. This classic equilibrium aligns with the conventional equilibrium concept studied in MFG literature when the context is time-consistent. Then we demonstrate that while this equilibrium produces an approximate optimal strategy when applied to the related <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math>-agent games, it does so solely in a precommitment sense. Therefore, it cannot function as a genuinely approximate equilibrium strategy from the perspective of a sophisticated agent within the <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math>-agent game. To address this limitation, we propose a new <i>consistent</i> equilibrium concept in both the MFG and the <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math>-agent game. We show that a consistent equilibrium in the MFG can indeed function as an approximate consistent equilibrium in the <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math>-agent game. Additionally, we analyze the convergence of consistent equilibria for <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math>-agent games toward a consistent MFG equilibrium as <span></span><math>\u0000 <semantics>\u0000 <mi>N</mi>\u0000 <annotation>$N$</annotation>\u0000 </semantics></math> tends to infinity.</p></div>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"35 3","pages":"613-635"},"PeriodicalIF":1.6,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144525151","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}
Andres Max Reppen, Halil Mete Soner, Valentin Tissot-Daguette
{"title":"Neural optimal stopping boundary","authors":"Andres Max Reppen, Halil Mete Soner, Valentin Tissot-Daguette","doi":"10.1111/mafi.12450","DOIUrl":"https://doi.org/10.1111/mafi.12450","url":null,"abstract":"<p>A method based on deep artificial neural networks and empirical risk minimization is developed to calculate the boundary separating the stopping and continuation regions in optimal stopping. The algorithm parameterizes the stopping boundary as the graph of a function and introduces relaxed stopping rules based on fuzzy boundaries to facilitate efficient optimization. Several financial instruments, some in high dimensions, are analyzed through this method, demonstrating its effectiveness. The existence of the stopping boundary is also proved under natural structural assumptions.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"35 2","pages":"441-469"},"PeriodicalIF":1.6,"publicationDate":"2024-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143638865","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":"Golden parachutes under the threat of accidents","authors":"Dylan Possamaï, Chiara Rossato","doi":"10.1111/mafi.12448","DOIUrl":"https://doi.org/10.1111/mafi.12448","url":null,"abstract":"<p>This paper addresses a continuous-time contracting model that extends Sannikov's problem. In our model, a principal hires a risk-averse agent to carry out a project. Specifically, the agent can perform two different tasks, namely to increase the instantaneous growth rate of the project's value, and to reduce the likelihood of accidents occurring. In order to compensate for these costly actions, the principal offers a continuous stream of payments throughout the entire duration of a contract, which concludes at a random time, potentially resulting in a lump-sum payment. We examine the consequences stemming from the introduction of accidents, modeled by a compound Poisson process that negatively impact the project's value. Furthermore, we investigate whether certain economic scenarii are still characterized by a golden parachute as in Sannikov's model. A golden parachute refers to a situation where the agent stops working and subsequently receives a compensation, which may be either a lump-sum payment leading to termination of the contract or a continuous stream of payments, thereby corresponding to a pension.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"35 2","pages":"337-421"},"PeriodicalIF":1.6,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143638923","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}
Christa Cuchiero, Luca Di Persio, Francesco Guida, Sara Svaluto-Ferro
{"title":"Measure-valued processes for energy markets","authors":"Christa Cuchiero, Luca Di Persio, Francesco Guida, Sara Svaluto-Ferro","doi":"10.1111/mafi.12452","DOIUrl":"https://doi.org/10.1111/mafi.12452","url":null,"abstract":"<p>We introduce a framework that allows to employ (non-negative) measure-valued processes for energy market modeling, in particular for electricity and gas futures. Interpreting the process' spatial structure as time to maturity, we show how the Heath–Jarrow–Morton approach can be translated to this framework, thus guaranteeing arbitrage free modeling in infinite dimensions, while allowing for the incorporation of important stylized facts, in particular stochastic discontinuities, that is, jumps or spikes at pre-specified (deterministic) dates. We derive an analog to the HJM-drift condition and then treat in a Markovian setting existence of non-negative measure-valued diffusions that satisfy this condition. To analyze mathematically convenient classes we consider measure-valued polynomial and affine diffusions, where we can precisely specify the diffusion part in terms of continuous functions satisfying certain admissibility conditions. For calibration purposes these functions can then be parameterized by neural networks yielding measure-valued analogs of neural SPDEs. By combining Fourier approaches or the moment formula with stochastic gradient descent methods, this then allows for tractable calibration procedures which we also test by way of example on market data.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"35 2","pages":"520-566"},"PeriodicalIF":1.6,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/mafi.12452","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143638924","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":"The fundamental theorem of asset pricing with and without transaction costs","authors":"Christoph Kühn","doi":"10.1111/mafi.12453","DOIUrl":"https://doi.org/10.1111/mafi.12453","url":null,"abstract":"<p>We prove a version of the fundamental theorem of asset pricing (FTAP) in continuous time that is based on the strict no-arbitrage condition and that is applicable to both frictionless markets and markets with proportional transaction costs. We consider a market with a single risky asset whose ask price process is higher than or equal to its bid price process. Neither the concatenation property of the set of wealth processes, that is used in the proof of the frictionless FTAP, nor some boundedness property of the trading volume of admissible strategies usually argued within models with a nonvanishing bid–ask spread need to be satisfied in our model.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"35 2","pages":"567-609"},"PeriodicalIF":1.6,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/mafi.12453","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143638925","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}
Eduardo Abi Jaber, Camille Illand, Shaun (Xiaoyuan) Li
{"title":"Joint SPX & VIX calibration with Gaussian polynomial volatility models: Deep pricing with quantization hints","authors":"Eduardo Abi Jaber, Camille Illand, Shaun (Xiaoyuan) Li","doi":"10.1111/mafi.12451","DOIUrl":"https://doi.org/10.1111/mafi.12451","url":null,"abstract":"<p>We consider the joint SPX & VIX calibration within a general class of <i>Gaussian polynomial volatility models</i> in which the volatility of the SPX is assumed to be a polynomial function of a Gaussian Volterra process defined as a stochastic convolution between a kernel and a Brownian motion. By performing joint calibration to daily SPX & VIX implied volatility surface data between 2011 and 2022, we compare the empirical performance of different kernels and their associated Markovian and non-Markovian models, such as rough and non-rough path-dependent volatility models. To ensure an efficient calibration and fair comparison between the models, we develop a generic unified method in our class of models for fast and accurate pricing of SPX & VIX derivatives based on functional quantization and neural networks. For the first time, we identify a <i>conventional one-factor Markovian continuous stochastic volatility model</i> that can achieve remarkable fits of the implied volatility surfaces of the SPX & VIX together with the term structure of VIX Futures. What is even more remarkable is that our <i>conventional one-factor Markovian continuous stochastic volatility model</i> outperforms, in all market conditions, its rough and non-rough path-dependent counterparts with the same number of parameters.</p>","PeriodicalId":49867,"journal":{"name":"Mathematical Finance","volume":"35 2","pages":"470-519"},"PeriodicalIF":1.6,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143639206","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}
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