Claudio Bellani, D. Brigo, Mikko S. Pakkanen, Leandro Sánchez-Betancourt
{"title":"Price Impact Without Averaging","authors":"Claudio Bellani, D. Brigo, Mikko S. Pakkanen, Leandro Sánchez-Betancourt","doi":"10.1080/1350486x.2024.2303078","DOIUrl":"https://doi.org/10.1080/1350486x.2024.2303078","url":null,"abstract":"","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"35 11","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139598101","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}
{"title":"Arbitrage-Free Neural-SDE Market Models","authors":"Samuel N. Cohen, Christoph Reisinger, Sheng Wang","doi":"10.1080/1350486x.2023.2257217","DOIUrl":"https://doi.org/10.1080/1350486x.2023.2257217","url":null,"abstract":"Modelling joint dynamics of liquid vanilla options is crucial for arbitrage-free pricing of illiquid derivatives and managing risks of option trade books. This paper develops a nonparametric model for the European options book respecting underlying financial constraints and while being practically implementable. We derive a state space for prices which are free from static (or model-independent) arbitrage and study the inference problem where a model is learnt from discrete time series data of stock and option prices. We use neural networks as function approximators for the drift and diffusion of the modelled SDE system, and impose constraints on the neural nets such that no-arbitrage conditions are preserved. In particular, we give methods to calibrate neural SDE models which are guaranteed to satisfy a set of linear inequalities. We validate our approach with numerical experiments using data generated from a Heston stochastic local volatility model.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"156 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135799214","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}
{"title":"On the Skew and Curvature of the Implied and Local Volatilities","authors":"Elisa Alòs, David García-Lorite, Makar Pravosud","doi":"10.1080/1350486x.2023.2261459","DOIUrl":"https://doi.org/10.1080/1350486x.2023.2261459","url":null,"abstract":"ABSTRACTIn this paper, we study the relationship between the short-end of the local and the implied volatility surfaces. Our results, based on Malliavin calculus techniques, recover the recent 1H+3/2 rule (where H denotes the Hurst parameter of the volatility process) for rough volatilities (see F. Bourgey, S. De Marco, P. Friz, and P. Pigato. 2022. “Local Volatility under Rough Volatility.” arXiv:2204.02376v1 [q-fin.MF] https://doi.org/10.48550/arXiv.2204.02376.), that states that the short-time skew slope of the at-the-money implied volatility is 1H+3/2 of the corresponding slope for local volatilities. Moreover, we see that the at-the-money short-end curvature of the implied volatility can be written in terms of the short-end skew and curvature of the local volatility and vice versa. Additionally, this relationship depends on H.KEYWORDS: Stochastic volatilitylocal volatilityrough volatilityMalliavin calculus Disclosure statementNo potential conflict of interest was reported by the author(s).Notes1 For example we can suppose a local volatility flat between 0 to first Monte Carlo, or even to use a discretization of the following asymptotic expression limT→0I0(T,K)=log(KST)∫STk1σ(T,u)du 2 Another way to get the skew is to take the derivative with respect to K in E((ST−K)+)=BS(T,K,I(T,K)). Then we get the following expression for the skew ∂kI(T,K)=−E(I(ST>K))−∂KBS(T,K,I(T,K)∂σBS(T,K,I(T,K)). Notice that the term E(I(ST>K)) can be estimated in the same simulation where we get the price of the option. We have checked both approaches and we have confirmed that they lead to identical resultsAdditional informationFundingThis work was supported by Agencia Estatal de Investigación [grant number PID2020-118339GB-I00].","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135798446","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}
{"title":"Policy Gradient Learning Methods for Stochastic Control with Exit Time and Applications to Share Repurchase Pricing","authors":"Mohamed Hamdouche, P. Henry-Labordère, H. Pham","doi":"10.1080/1350486X.2023.2239850","DOIUrl":"https://doi.org/10.1080/1350486X.2023.2239850","url":null,"abstract":"ABSTRACT We develop policy gradients methods for stochastic control with exit time in a model-free setting. We propose two types of algorithms for learning either directly the optimal policy or by learning alternately the value function (critic) and the optimal control (actor). The use of randomized policies is crucial for overcoming notably the issue related to the exit time in the gradient computation. We demonstrate the effectiveness of our approach by implementing our numerical schemes in the application to the problem of share repurchase pricing. Our results show that the proposed policy gradient methods outperform PDE or other neural networks techniques in a model-based setting. Furthermore, our algorithms are flexible enough to incorporate realistic market conditions like, e.g., price impact or transaction costs.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"7 1","pages":"439 - 456"},"PeriodicalIF":0.0,"publicationDate":"2022-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90336117","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}
{"title":"Multi-Period Mean Expected-Shortfall Strategies: ‘Cut Your Losses and Ride Your Gains’","authors":"P. Forsyth, K. Vetzal","doi":"10.1080/1350486X.2023.2224354","DOIUrl":"https://doi.org/10.1080/1350486X.2023.2224354","url":null,"abstract":"Dynamic mean-variance (MV) optimal strategies are inherently contrarian. Following periods of strong equity returns, there is a tendency to de-risk the portfolio by shifting into risk-free investments. On the other hand, if the portfolio still has some equity exposure, the weight on equities will increase following stretches of poor equity returns. This is essentially due to using variance as a risk measure, which penalizes both upside and downside deviations relative to a satiation point. As an alternative, we propose a dynamic trading strategy based on an expected wealth (EW), expected shortfall (ES) objective function. ES is defined as the mean of the worst β fraction of the outcomes, hence the EW-ES objective directly targets left tail risk. We use stochastic control methods to determine the optimal trading strategy. Our numerical method allows us to impose realistic constraints: no leverage, no shorting, infrequent rebalancing. For 5 year investment horizons, this strategy generates an annualized alpha of 180 bps compared to a 60:40 stock-bond constant weight policy. Bootstrap resampling with historical data shows that these results are robust to parametric model misspecification. The optimal EW-ES strategy is generally a momentum-type policy, in contrast to the contrarian MV optimal strategy.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"115 1","pages":"402 - 438"},"PeriodicalIF":0.0,"publicationDate":"2022-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79818022","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}
{"title":"Accelerated Share Repurchases Under Stochastic Volatility","authors":"N. Krishnan, R. Sircar","doi":"10.1080/1350486X.2023.2210290","DOIUrl":"https://doi.org/10.1080/1350486X.2023.2210290","url":null,"abstract":"Accelerated share repurchases (ASRs) are a type of stock buyback wherein the repurchasing firm contracts a financial intermediary to acquire the shares on its behalf. The intermediary purchases the shares from the open market and is compensated by the firm according to the average of the stock price over the repurchasing interval, whose end can be chosen by the intermediary. Hence, the intermediary needs to decide both how to minimize the cost of acquiring the shares, and when to exercise its contract to maximize its payment. Studies of ASRs typically assume a constant volatility, but the longer time horizon of ASRs, on the order of months, indicates that the variation of the volatility should be considered. We analyze the optimal strategy of the intermediary within the continuous-time framework of the Heston model for the evolution of the stock price and volatility, which is described by a free-boundary problem which we derive here. To solve this system numerically, we make use of deep learning. Through simulations, we find that the intermediary can acquire shares at lower cost and lower risk if it takes into account the stochasticity of the volatility.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"79 1","pages":"331 - 365"},"PeriodicalIF":0.0,"publicationDate":"2022-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87596284","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}
{"title":"Optimal Execution with Identity Optionality","authors":"R. Carmona, Claire Zeng","doi":"10.1080/1350486X.2023.2193343","DOIUrl":"https://doi.org/10.1080/1350486X.2023.2193343","url":null,"abstract":"ABSTRACT This paper investigates the impact of anonymous trading on the agents' strategy in an optimal execution framework. It mainly explores the specificity of order attribution on the Toronto Stock Exchange, where brokers can choose to either trade with their own identity or under a generic anonymous code that is common to all the brokers. We formulate a stochastic differential game for the optimal execution problem of a population of N brokers and incorporate permanent and temporary price impacts for both the identity-revealed and anonymous trading processes. We then formulate the limiting mean-field game of controls with common noise and obtain a solution in closed-form via the probablistic approach for the Almgren-Chris price impact framework. Finally, we perform a sensitivity analysis to explore the impact of the model parameters on the optimal strategy.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"17 1","pages":"261 - 287"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88067390","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}
{"title":"Exchange Option Pricing Under Variance Gamma-Like Models","authors":"M. Gardini, P. Sabino","doi":"10.1080/1350486X.2023.2248791","DOIUrl":"https://doi.org/10.1080/1350486X.2023.2248791","url":null,"abstract":"In this article, we focus on the pricing of exchange options when the risk-neutral dynamic of log-prices follows either the well-known variance gamma or the recent variance gamma++ process introduced in Gardini et al. (2022. “The Variance Gamma++ Process and Applications to Energy Markets.” Applied Stochastic Models in Business and Industry 38 (2): 391–418. https://doi.org/10.1002/asmb.v38.2.). In particular, for the former model we can derive a Margrabe's type formula whereas for the latter one we can write an ‘integral free’ formula. Furthermore, we show how to construct a general multidimensional versions of the variance gamma++ processes preserving both the mathematical and numerical tractabilities. Finally we apply the derived models to German and French energy power markets: we calibrate their parameters using real market data and we accordingly evaluate exchange options with the derived closed formulas, Fourier based methods and Monte Carlo techniques.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"16 1","pages":"494 - 521"},"PeriodicalIF":0.0,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84405516","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}
{"title":"Hedging Option Books Using Neural-SDE Market Models","authors":"Samuel N. Cohen, C. Reisinger, Sheng Wang","doi":"10.1080/1350486X.2023.2221448","DOIUrl":"https://doi.org/10.1080/1350486X.2023.2221448","url":null,"abstract":"We study the capability of arbitrage-free neural-SDE market models to yield effective strategies for hedging options. In particular, we derive sensitivity-based and minimum-variance-based hedging strategies using these models and examine their performance when applied to various option portfolios using real-world data. Through backtesting analysis over typical and stressed market periods, we show that neural-SDE market models achieve lower hedging errors than Black–Scholes delta and delta-vega hedging consistently over time, and are less sensitive to the tenor choice of hedging instruments. In addition, hedging using market models leads to similar performance to hedging using Heston models, while the former tends to be more robust during stressed market periods.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"313 1","pages":"366 - 401"},"PeriodicalIF":0.0,"publicationDate":"2022-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82847029","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}
{"title":"Optimal Execution: A Review","authors":"R. Donnelly","doi":"10.1080/1350486X.2022.2161588","DOIUrl":"https://doi.org/10.1080/1350486X.2022.2161588","url":null,"abstract":"This review article is intended to collect and summarize many of the results in the field of optimal execution over the last twenty years. In doing so, we describe the general workings of the limit order book so that the sources of costs and risks which need to be optimized are understood. The initial models considered propose simple dynamics for prices which allow easily computable strategies which maximize risk-adjusted profits. Subsequently, the review is divided into two major parts. The first explores several works which investigate how optimal liquidation strategies are modified to account for more complex dynamics, namely other stochastic or non-linear factors. The second presents optimal trading strategies when the agent utilizes benchmarks in addition to risk-adjusted wealth, or when she has objectives beyond optimal liquidation.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"96 1","pages":"181 - 212"},"PeriodicalIF":0.0,"publicationDate":"2022-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88955800","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}
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