Zineb El Filali Ech-Chafiq, Pierre Henry Labordère, Jérôme Lelong
{"title":"Pricing Bermudan Options Using Regression Trees/Random Forests","authors":"Zineb El Filali Ech-Chafiq, Pierre Henry Labordère, Jérôme Lelong","doi":"10.1137/21m1460648","DOIUrl":"https://doi.org/10.1137/21m1460648","url":null,"abstract":"","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135666711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bastien Baldacci, Philippe Bergault, Dylan Possamaï
We design a market-making model à la Avellaneda and Stoikov [Quant. Finance, 8 (2008), pp. 217–224] in which the market-takers act strategically, in the sense that they design their trading strategy based on an exogenous trading signal. The market-maker chooses her quotes based on the average market-takers’ behavior, modelled through a mean-field interaction. We derive, up to the resolution of a coupled HJB-Fokker–Planck system, the optimal controls of the market-maker and the representative market-taker. This approach is flexible enough to incorporate different behaviors for the market-takers and takes into account the impact of their strategies on the price process.
我们设计了一个做市模型(la Avellaneda和Stoikov [Quant. Finance, 8 (2008)， pp. 217-224])，在这个模型中，市场接受者采取战略行动，从某种意义上说，他们根据外生交易信号设计交易策略。做市商根据市场接受者的平均行为选择报价，并通过平均场相互作用建模。在HJB-Fokker-Planck耦合系统的分辨率下，我们导出了做市商和代表性市场接受者的最优控制。这种方法足够灵活，可以将市场参与者的不同行为结合起来，并考虑到他们的策略对价格过程的影响。
{"title":"A Mean-Field Game of Market-Making against Strategic Traders","authors":"Bastien Baldacci, Philippe Bergault, Dylan Possamaï","doi":"10.1137/22m1486492","DOIUrl":"https://doi.org/10.1137/22m1486492","url":null,"abstract":"We design a market-making model à la Avellaneda and Stoikov [Quant. Finance, 8 (2008), pp. 217–224] in which the market-takers act strategically, in the sense that they design their trading strategy based on an exogenous trading signal. The market-maker chooses her quotes based on the average market-takers’ behavior, modelled through a mean-field interaction. We derive, up to the resolution of a coupled HJB-Fokker–Planck system, the optimal controls of the market-maker and the representative market-taker. This approach is flexible enough to incorporate different behaviors for the market-takers and takes into account the impact of their strategies on the price process.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135823475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For time-inconsistent optimal control problems, a quite popular approach is the equilibrium approach, taken by sophisticated agents. In this short note, we construct a deterministic continuous-time example where the unique equilibrium is dominated by another control. Therefore, in this situation, it may not be wise to take the equilibrium strategy.
{"title":"Short Communication: Is a Sophisticated Agent Always a Wise One?","authors":"Jianfeng Zhang","doi":"10.1137/23m1569137","DOIUrl":"https://doi.org/10.1137/23m1569137","url":null,"abstract":"For time-inconsistent optimal control problems, a quite popular approach is the equilibrium approach, taken by sophisticated agents. In this short note, we construct a deterministic continuous-time example where the unique equilibrium is dominated by another control. Therefore, in this situation, it may not be wise to take the equilibrium strategy.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136033197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop a backward-in-time machine learning algorithm that uses a sequence of neural networks to solve optimal switching problems in energy production, where electricity and fossil fuel prices are subject to stochastic jumps. We then apply this algorithm to a variety of energy scheduling problems, including novel high-dimensional energy production problems. Our experimental results demonstrate that the algorithm performs with accuracy and experiences linear to sublinear slowdowns as dimension increases, demonstrating the value of the algorithm for solving high-dimensional switching problems.
{"title":"A Neural Network Approach to High-Dimensional Optimal Switching Problems with Jumps in Energy Markets","authors":"Erhan Bayraktar, Asaf Cohen, April Nellis","doi":"10.1137/22m1527246","DOIUrl":"https://doi.org/10.1137/22m1527246","url":null,"abstract":"We develop a backward-in-time machine learning algorithm that uses a sequence of neural networks to solve optimal switching problems in energy production, where electricity and fossil fuel prices are subject to stochastic jumps. We then apply this algorithm to a variety of energy scheduling problems, including novel high-dimensional energy production problems. Our experimental results demonstrate that the algorithm performs with accuracy and experiences linear to sublinear slowdowns as dimension increases, demonstrating the value of the algorithm for solving high-dimensional switching problems.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136115606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Brian Ning, Sebastian Jaimungal, Xiaorong Zhang, Maxime Bergeron
We propose a hybrid method for generating arbitrage-free implied volatility (IV) surfaces consistent with historical data by combining model-free variational autoencoders (VAEs) with continuous time stochastic differential equation (SDE) driven models. We focus on two classes of SDE models: regime switching models and Lévy additive processes. By projecting historical surfaces onto the space of SDE model parameters, we obtain a distribution on the parameter subspace faithful to the data on which we then train a VAE. Arbitrage-free IV surfaces are then generated by sampling from the posterior distribution on the latent space, decoding to obtain SDE model parameters, and finally mapping those parameters to IV surfaces. We further refine the VAE model by including conditional features and demonstrate its superior generative out-of-sample performance. Finally, we showcase how our method can be used as a data augmentation tool to help practitioners manage the tail risk of option portfolios.
{"title":"Arbitrage-Free Implied Volatility Surface Generation with Variational Autoencoders","authors":"Brian Ning, Sebastian Jaimungal, Xiaorong Zhang, Maxime Bergeron","doi":"10.1137/21m1443546","DOIUrl":"https://doi.org/10.1137/21m1443546","url":null,"abstract":"We propose a hybrid method for generating arbitrage-free implied volatility (IV) surfaces consistent with historical data by combining model-free variational autoencoders (VAEs) with continuous time stochastic differential equation (SDE) driven models. We focus on two classes of SDE models: regime switching models and Lévy additive processes. By projecting historical surfaces onto the space of SDE model parameters, we obtain a distribution on the parameter subspace faithful to the data on which we then train a VAE. Arbitrage-free IV surfaces are then generated by sampling from the posterior distribution on the latent space, decoding to obtain SDE model parameters, and finally mapping those parameters to IV surfaces. We further refine the VAE model by including conditional features and demonstrate its superior generative out-of-sample performance. Finally, we showcase how our method can be used as a data augmentation tool to help practitioners manage the tail risk of option portfolios.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136058237","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce the cubature formula for stochastic Volterra integral equations. We first derive the stochastic Taylor expansion in this setting, by utilizing a functional Itô formula, and provide its tail estimates. We then introduce the cubature measure for such equations, and construct it explicitly in some special cases, including a long memory stochastic volatility model. We shall provide the error estimate rigorously. Our numerical examples show that the cubature method is much more efficient than the Euler scheme, provided certain conditions are satisfied.
{"title":"Cubature Method for Stochastic Volterra Integral Equations","authors":"Qi Feng, Jianfeng Zhang","doi":"10.1137/22m146889x","DOIUrl":"https://doi.org/10.1137/22m146889x","url":null,"abstract":"In this paper, we introduce the cubature formula for stochastic Volterra integral equations. We first derive the stochastic Taylor expansion in this setting, by utilizing a functional Itô formula, and provide its tail estimates. We then introduce the cubature measure for such equations, and construct it explicitly in some special cases, including a long memory stochastic volatility model. We shall provide the error estimate rigorously. Our numerical examples show that the cubature method is much more efficient than the Euler scheme, provided certain conditions are satisfied.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136254864","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this short note is to present a solution to the discrete time exponential utility maximization problem in a case where the underlying asset has a multivariate normal distribution. In addition to the usual setting considered in mathematical finance, we also consider an investor who is informed about the risky asset’s price changes with a delay . Our method of solution is based on the theory developed in [W. Barrett and P. Feinsilver, Linear Algebra Appl., 41 (1981), pp. 111–130] and guessing the optimal portfolio.
{"title":"Short Communication: Exponential Utility Maximization in a Discrete Time Gaussian Framework","authors":"Yan Dolinsky, Or Zuk","doi":"10.1137/23m1576074","DOIUrl":"https://doi.org/10.1137/23m1576074","url":null,"abstract":"The aim of this short note is to present a solution to the discrete time exponential utility maximization problem in a case where the underlying asset has a multivariate normal distribution. In addition to the usual setting considered in mathematical finance, we also consider an investor who is informed about the risky asset’s price changes with a delay . Our method of solution is based on the theory developed in [W. Barrett and P. Feinsilver, Linear Algebra Appl., 41 (1981), pp. 111–130] and guessing the optimal portfolio.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135254490","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Financial Mathematics, Volume 14, Issue 3, Page 751-776, September 2023. Abstract. The first half of the paper is devoted to description and implementation of statistical tests arguing for the presence of a Brownian component in the inventories and wealth processes of individual traders. We use intraday data from the Toronto Stock Exchange to provide empirical evidence of this claim. We work with regularly spaced time intervals, as well as with asynchronously observed data. The tests reveal with high significance the presence of a nonzero Brownian motion component. The second half of the paper is concerned with the analysis of trader behaviors throughout the day. We extend the theoretical analysis of an existing optimal execution model to accommodate the presence of Itô inventory processes, and we compare empirically the optimal behavior of traders in such fitted models to the actual behavior we read off the data.
{"title":"Optimal Execution with Quadratic Variation Inventories","authors":"Rene Carmona, Laura Leal","doi":"10.1137/21m1416564","DOIUrl":"https://doi.org/10.1137/21m1416564","url":null,"abstract":"SIAM Journal on Financial Mathematics, Volume 14, Issue 3, Page 751-776, September 2023. <br/> Abstract. The first half of the paper is devoted to description and implementation of statistical tests arguing for the presence of a Brownian component in the inventories and wealth processes of individual traders. We use intraday data from the Toronto Stock Exchange to provide empirical evidence of this claim. We work with regularly spaced time intervals, as well as with asynchronously observed data. The tests reveal with high significance the presence of a nonzero Brownian motion component. The second half of the paper is concerned with the analysis of trader behaviors throughout the day. We extend the theoretical analysis of an existing optimal execution model to accommodate the presence of Itô inventory processes, and we compare empirically the optimal behavior of traders in such fitted models to the actual behavior we read off the data.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":"138 10","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bahman Angoshtari, Erhan Bayraktar, Virginia R. Young
We formulate and solve a deterministic optimal consumption problem to maximize the discounted constant relative risk aversion utility of an individual’s consumption-to-habit process assuming they only invest in a riskless market and that they are unwilling to consume at a rate below a certain proportion of their consumption habit. Increasing increases the degree of addictiveness of habit formation, with (respectively, ) corresponding to nonaddictive (respectively, completely addictive) model. We derive the optimal consumption policies explicitly in terms of the solution of a nonlinear free-boundary problem, which we analyze in detail. Impatient individuals (or, equivalently, those with more addictive habits) always consume above the minimum rate; thus, they eventually attain the minimum wealth-to-habit ratio. Patient individuals (or, equivalently, those with less addictive habits) consume at the minimum rate if their wealth-to-habit ratio is below a threshold and above it otherwise. By consuming patiently, these individuals maintain a wealth-to-habit ratio that is greater than the minimum acceptable level. Additionally, we prove that the optimal consumption path is hump-shaped if the initial wealth-to-habit ratio is either (1) larger than a high threshold or (2) below a low threshold and the agent is more risk seeking (that is, less risk averse). Thus, we provide a simple explanation for the consumption hump observed by various empirical studies.
{"title":"Optimal Consumption Under a Habit-Formation Constraint: The Deterministic Case","authors":"Bahman Angoshtari, Erhan Bayraktar, Virginia R. Young","doi":"10.1137/22m1471560","DOIUrl":"https://doi.org/10.1137/22m1471560","url":null,"abstract":"We formulate and solve a deterministic optimal consumption problem to maximize the discounted constant relative risk aversion utility of an individual’s consumption-to-habit process assuming they only invest in a riskless market and that they are unwilling to consume at a rate below a certain proportion of their consumption habit. Increasing increases the degree of addictiveness of habit formation, with (respectively, ) corresponding to nonaddictive (respectively, completely addictive) model. We derive the optimal consumption policies explicitly in terms of the solution of a nonlinear free-boundary problem, which we analyze in detail. Impatient individuals (or, equivalently, those with more addictive habits) always consume above the minimum rate; thus, they eventually attain the minimum wealth-to-habit ratio. Patient individuals (or, equivalently, those with less addictive habits) consume at the minimum rate if their wealth-to-habit ratio is below a threshold and above it otherwise. By consuming patiently, these individuals maintain a wealth-to-habit ratio that is greater than the minimum acceptable level. Additionally, we prove that the optimal consumption path is hump-shaped if the initial wealth-to-habit ratio is either (1) larger than a high threshold or (2) below a low threshold and the agent is more risk seeking (that is, less risk averse). Thus, we provide a simple explanation for the consumption hump observed by various empirical studies.","PeriodicalId":48880,"journal":{"name":"SIAM Journal on Financial Mathematics","volume":"341 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135996977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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