In this paper we propose a semi-analytic approach to pricing American options for some time-dependent jump-diffusions models. The idea of the method is to further generalize our approach developed for pricing barrier, [Itkin et al., 2021], and American, [Carr and Itkin, 2021; Itkin and Muravey, 2023], options in various time-dependent one factor and even stochastic volatility models. Our approach i) allows arbitrary dependencies of the model parameters on time; ii) reduces solution of the pricing problem for American options to a simpler problem of solving an algebraic nonlinear equation for the exercise boundary and a linear Fredholm-Volterra equation for the the option price; iii) the options Greeks solve a similar Fredholm-Volterra linear equation obtained by just differentiating Eq. (25) by the required parameter.
本文针对一些时变跳跃扩散模型，提出了一种求解美式期权定价的半解析方法。该方法的思想是进一步推广我们为定价障碍(Itkin et al.，2021)和美国(Carr and Itkin, 2021;Itkin and Muravey, 2023]，各种随时间变化的单因素甚至随机波动模型中的选项。我们的方法i)允许模型参数对时间的任意依赖;ii)将美式期权定价问题的求解简化为求解求解求解行使边界的代数非线性方程和求解期权价格的线性Fredholm-Volterra方程的问题;iii)希腊人解出一个类似的Fredholm-Volterra线性方程，只需将Eq.(25)微分即可得到所需参数。
{"title":"Semi-analytic pricing of American options in some time-dependent jump-diffusion models","authors":"Andrey Itkin","doi":"arxiv-2308.08760","DOIUrl":"https://doi.org/arxiv-2308.08760","url":null,"abstract":"In this paper we propose a semi-analytic approach to pricing American options\u0000for some time-dependent jump-diffusions models. The idea of the method is to\u0000further generalize our approach developed for pricing barrier, [Itkin et al.,\u00002021], and American, [Carr and Itkin, 2021; Itkin and Muravey, 2023], options\u0000in various time-dependent one factor and even stochastic volatility models. Our\u0000approach i) allows arbitrary dependencies of the model parameters on time; ii)\u0000reduces solution of the pricing problem for American options to a simpler\u0000problem of solving an algebraic nonlinear equation for the exercise boundary\u0000and a linear Fredholm-Volterra equation for the the option price; iii) the\u0000options Greeks solve a similar Fredholm-Volterra linear equation obtained by\u0000just differentiating Eq. (25) by the required parameter.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522493","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}
Tanut Treetanthiploet, Yufei Zhang, Lukasz Szpruch, Isaac Bowers-Barnard, Henrietta Ridley, James Hickey, Chris Pearce
The emergence of price comparison websites (PCWs) has presented insurers with unique challenges in formulating effective pricing strategies. Operating on PCWs requires insurers to strike a delicate balance between competitive premiums and profitability, amidst obstacles such as low historical conversion rates, limited visibility of competitors' actions, and a dynamic market environment. In addition to this, the capital intensive nature of the business means pricing below the risk levels of customers can result in solvency issues for the insurer. To address these challenges, this paper introduces reinforcement learning (RL) framework that learns the optimal pricing policy by integrating model-based and model-free methods. The model-based component is used to train agents in an offline setting, avoiding cold-start issues, while model-free algorithms are then employed in a contextual bandit (CB) manner to dynamically update the pricing policy to maximise the expected revenue. This facilitates quick adaptation to evolving market dynamics and enhances algorithm efficiency and decision interpretability. The paper also highlights the importance of evaluating pricing policies using an offline dataset in a consistent fashion and demonstrates the superiority of the proposed methodology over existing off-the-shelf RL/CB approaches. We validate our methodology using synthetic data, generated to reflect private commercially available data within real-world insurers, and compare against 6 other benchmark approaches. Our hybrid agent outperforms these benchmarks in terms of sample efficiency and cumulative reward with the exception of an agent that has access to perfect market information which would not be available in a real-world set-up.
{"title":"Insurance pricing on price comparison websites via reinforcement learning","authors":"Tanut Treetanthiploet, Yufei Zhang, Lukasz Szpruch, Isaac Bowers-Barnard, Henrietta Ridley, James Hickey, Chris Pearce","doi":"arxiv-2308.06935","DOIUrl":"https://doi.org/arxiv-2308.06935","url":null,"abstract":"The emergence of price comparison websites (PCWs) has presented insurers with\u0000unique challenges in formulating effective pricing strategies. Operating on\u0000PCWs requires insurers to strike a delicate balance between competitive\u0000premiums and profitability, amidst obstacles such as low historical conversion\u0000rates, limited visibility of competitors' actions, and a dynamic market\u0000environment. In addition to this, the capital intensive nature of the business\u0000means pricing below the risk levels of customers can result in solvency issues\u0000for the insurer. To address these challenges, this paper introduces\u0000reinforcement learning (RL) framework that learns the optimal pricing policy by\u0000integrating model-based and model-free methods. The model-based component is\u0000used to train agents in an offline setting, avoiding cold-start issues, while\u0000model-free algorithms are then employed in a contextual bandit (CB) manner to\u0000dynamically update the pricing policy to maximise the expected revenue. This\u0000facilitates quick adaptation to evolving market dynamics and enhances algorithm\u0000efficiency and decision interpretability. The paper also highlights the\u0000importance of evaluating pricing policies using an offline dataset in a\u0000consistent fashion and demonstrates the superiority of the proposed methodology\u0000over existing off-the-shelf RL/CB approaches. We validate our methodology using\u0000synthetic data, generated to reflect private commercially available data within\u0000real-world insurers, and compare against 6 other benchmark approaches. Our\u0000hybrid agent outperforms these benchmarks in terms of sample efficiency and\u0000cumulative reward with the exception of an agent that has access to perfect\u0000market information which would not be available in a real-world set-up.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522645","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}
Ghost, or fictitious points allow to capture boundary conditions that are not located on the finite difference grid discretization. We explore in this paper the impact of ghost points on the stability of the explicit Euler finite difference scheme in the context of the diffusion equation. In particular, we consider the case of a one-touch option under the Black-Scholes model. The observations and results are however valid for a much wider range of financial contracts and models.
{"title":"Instabilities of explicit finite difference schemes with ghost points on the diffusion equation","authors":"Fabien Le Floc'h","doi":"arxiv-2308.04629","DOIUrl":"https://doi.org/arxiv-2308.04629","url":null,"abstract":"Ghost, or fictitious points allow to capture boundary conditions that are not\u0000located on the finite difference grid discretization. We explore in this paper\u0000the impact of ghost points on the stability of the explicit Euler finite\u0000difference scheme in the context of the diffusion equation. In particular, we\u0000consider the case of a one-touch option under the Black-Scholes model. The\u0000observations and results are however valid for a much wider range of financial\u0000contracts and models.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522644","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}
We introduce a Path Shadowing Monte-Carlo method, which provides prediction of future paths, given any generative model. At any given date, it averages future quantities over generated price paths whose past history matches, or `shadows', the actual (observed) history. We test our approach using paths generated from a maximum entropy model of financial prices, based on a recently proposed multi-scale analogue of the standard skewness and kurtosis called `Scattering Spectra'. This model promotes diversity of generated paths while reproducing the main statistical properties of financial prices, including stylized facts on volatility roughness. Our method yields state-of-the-art predictions for future realized volatility and allows one to determine conditional option smiles for the S&P500 that outperform both the current version of the Path-Dependent Volatility model and the option market itself.
{"title":"Path Shadowing Monte-Carlo","authors":"Rudy Morel, Stéphane Mallat, Jean-Philippe Bouchaud","doi":"arxiv-2308.01486","DOIUrl":"https://doi.org/arxiv-2308.01486","url":null,"abstract":"We introduce a Path Shadowing Monte-Carlo method, which provides prediction\u0000of future paths, given any generative model. At any given date, it averages\u0000future quantities over generated price paths whose past history matches, or\u0000`shadows', the actual (observed) history. We test our approach using paths\u0000generated from a maximum entropy model of financial prices, based on a recently\u0000proposed multi-scale analogue of the standard skewness and kurtosis called\u0000`Scattering Spectra'. This model promotes diversity of generated paths while\u0000reproducing the main statistical properties of financial prices, including\u0000stylized facts on volatility roughness. Our method yields state-of-the-art\u0000predictions for future realized volatility and allows one to determine\u0000conditional option smiles for the S&P500 that outperform both the current\u0000version of the Path-Dependent Volatility model and the option market itself.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522655","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}
In this paper we examine the problem of valuing an exotic derivative known as the American passport option where the underlying is driven by a L'evy process. The passport option is a call option on a trading account. We derive the pricing equation, using the dynamic programming principle, and prove that the option value is a viscosity solution of variational inequality. We also establish the comparison principle, which yields uniqueness and the convexity of the viscosity solution.
{"title":"American Passport options in an exponential Lévy model","authors":"Zakaria Marah","doi":"arxiv-2307.16649","DOIUrl":"https://doi.org/arxiv-2307.16649","url":null,"abstract":"In this paper we examine the problem of valuing an exotic derivative known as\u0000the American passport option where the underlying is driven by a L'evy\u0000process. The passport option is a call option on a trading account. We derive\u0000the pricing equation, using the dynamic programming principle, and prove that\u0000the option value is a viscosity solution of variational inequality. We also\u0000establish the comparison principle, which yields uniqueness and the convexity\u0000of the viscosity solution.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522648","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}
Introduced in the late 90s, the passport option gives its holder the right to trade in a market and receive any positive gain in the resulting traded account at maturity. Pricing the option amounts to solving a stochastic control problem that for $d>1$ risky assets remains an open problem. Even in a correlated Black-Scholes (BS) market with $d=2$ risky assets, no optimal trading strategy has been derived in closed form. In this paper, we derive a discrete-time solution for multi-dimensional BS markets with uncorrelated assets. Moreover, inspired by the success of deep reinforcement learning in, e.g., board games, we propose two machine learning-powered approaches to pricing general options on a portfolio value in general markets. These approaches prove to be successful for pricing the passport option in one-dimensional and multi-dimensional uncorrelated BS markets.
{"title":"Machine Learning-powered Pricing of the Multidimensional Passport Option","authors":"Josef Teichmann, Hanna Wutte","doi":"arxiv-2307.14887","DOIUrl":"https://doi.org/arxiv-2307.14887","url":null,"abstract":"Introduced in the late 90s, the passport option gives its holder the right to\u0000trade in a market and receive any positive gain in the resulting traded account\u0000at maturity. Pricing the option amounts to solving a stochastic control problem\u0000that for $d>1$ risky assets remains an open problem. Even in a correlated\u0000Black-Scholes (BS) market with $d=2$ risky assets, no optimal trading strategy\u0000has been derived in closed form. In this paper, we derive a discrete-time\u0000solution for multi-dimensional BS markets with uncorrelated assets. Moreover,\u0000inspired by the success of deep reinforcement learning in, e.g., board games,\u0000we propose two machine learning-powered approaches to pricing general options\u0000on a portfolio value in general markets. These approaches prove to be\u0000successful for pricing the passport option in one-dimensional and\u0000multi-dimensional uncorrelated BS markets.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":"119 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522646","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}
Earnings announcements (EADs) are corporate events that provide investors with fundamentally important information. The prospect of stock price rises may also contribute to EADs increased volatility. Using data on extremely short term options, we study that bimodality in the risk neutral distribution and concavity in the IV smiles are ubiquitous characteristics before an earnings announcement day. This study compares the returns between concave and non concave IV smiles to see if the concavity in the IV curve leads to any information about the risk in the market and showcases how investors hedge against extreme volatility during earnings announcements. In fact, our paper shows in the presence of concave IV smiles; investors pay a significant premium to hedge against the uncertainty caused by the forthcoming announcement.
{"title":"Option Smile Volatility and Implied Probabilities: Implications of Concavity in IV Curves","authors":"Darsh Kachhara, John K. E Markin, Astha Singh","doi":"arxiv-2307.15718","DOIUrl":"https://doi.org/arxiv-2307.15718","url":null,"abstract":"Earnings announcements (EADs) are corporate events that provide investors\u0000with fundamentally important information. The prospect of stock price rises may\u0000also contribute to EADs increased volatility. Using data on extremely short\u0000term options, we study that bimodality in the risk neutral distribution and\u0000concavity in the IV smiles are ubiquitous characteristics before an earnings\u0000announcement day. This study compares the returns between concave and non\u0000concave IV smiles to see if the concavity in the IV curve leads to any\u0000information about the risk in the market and showcases how investors hedge\u0000against extreme volatility during earnings announcements. In fact, our paper\u0000shows in the presence of concave IV smiles; investors pay a significant premium\u0000to hedge against the uncertainty caused by the forthcoming announcement.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522649","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}
The contributions of this paper are twofold: we define and investigate the properties of a short rate model driven by a general Gaussian Volterra process and, after defining precisely a notion of convexity adjustment, derive explicit formulae for it.
{"title":"Interest rate convexity in a Gaussian framework","authors":"Antoine Jacquier, Mugad Oumgari","doi":"arxiv-2307.14218","DOIUrl":"https://doi.org/arxiv-2307.14218","url":null,"abstract":"The contributions of this paper are twofold: we define and investigate the\u0000properties of a short rate model driven by a general Gaussian Volterra process\u0000and, after defining precisely a notion of convexity adjustment, derive explicit\u0000formulae for it.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522647","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}
Semi-analytical pricing of American options in a time-dependent Ornstein-Uhlenbeck model was presented in [Carr, Itkin, 2020]. It was shown that to obtain these prices one needs to solve (numerically) a nonlinear Volterra integral equation of the second kind to find the exercise boundary (which is a function of the time only). Once this is done, the option prices follow. It was also shown that computationally this method is as efficient as the forward finite difference solver while providing better accuracy and stability. Later this approach called "the Generalized Integral transform" method has been significantly extended by the authors (also, in cooperation with Peter Carr and Alex Lipton) to various time-dependent one factor, and stochastic volatility models as applied to pricing barrier options. However, for American options, despite possible, this was not explicitly reported anywhere. In this paper our goal is to fill this gap and also discuss which numerical method (including those in machine learning) could be efficient to solve the corresponding Volterra integral equations.
{"title":"American options in time-dependent one-factor models: Semi-analytic pricing, numerical methods and ML support","authors":"Andrey Itkin, Dmitry Muravey","doi":"arxiv-2307.13870","DOIUrl":"https://doi.org/arxiv-2307.13870","url":null,"abstract":"Semi-analytical pricing of American options in a time-dependent\u0000Ornstein-Uhlenbeck model was presented in [Carr, Itkin, 2020]. It was shown\u0000that to obtain these prices one needs to solve (numerically) a nonlinear\u0000Volterra integral equation of the second kind to find the exercise boundary\u0000(which is a function of the time only). Once this is done, the option prices\u0000follow. It was also shown that computationally this method is as efficient as\u0000the forward finite difference solver while providing better accuracy and\u0000stability. Later this approach called \"the Generalized Integral transform\"\u0000method has been significantly extended by the authors (also, in cooperation\u0000with Peter Carr and Alex Lipton) to various time-dependent one factor, and\u0000stochastic volatility models as applied to pricing barrier options. However,\u0000for American options, despite possible, this was not explicitly reported\u0000anywhere. In this paper our goal is to fill this gap and also discuss which\u0000numerical method (including those in machine learning) could be efficient to\u0000solve the corresponding Volterra integral equations.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522641","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}
We consider the problem of pricing American Exchange options driven by a L'evy process. We study the properties of American Exchange options, we represented it as the sum of the price of the corresponding European exchange option price and an early exercise premium. Secondly, we show some properties of the free boundary and give an approximative formula of an American Exchange option.
{"title":"American Exchange option driven by a Lévy process","authors":"Zakaria Marah","doi":"arxiv-2307.10900","DOIUrl":"https://doi.org/arxiv-2307.10900","url":null,"abstract":"We consider the problem of pricing American Exchange options driven by a\u0000L'evy process. We study the properties of American Exchange options, we\u0000represented it as the sum of the price of the corresponding European exchange\u0000option price and an early exercise premium. Secondly, we show some properties\u0000of the free boundary and give an approximative formula of an American Exchange\u0000option.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522652","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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