A. M. Ferreiro, J. A. García, J. G. López-Salas, C. Vázquez
In order to overcome the drawbacks of assuming deterministic volatility coefficients in the standard LIBOR market models to capture volatility smiles and skews in real markets, several extensions of LIBOR models to incorporate stochastic volatilities have been proposed. The efficient calibration to market data of these more complex models becomes a relevant target in practice. The main objective of the present work is to efficiently calibrate some recent SABR/LIBOR market models to real market prices of caplets and swaptions. For the calibration we propose a parallelized version of the simulated annealing algorithm for multi-GPUs. The numerical results clearly illustrate the advantages of using the proposed multi-GPUs tools when applied to real market data and popular SABR/LIBOR models.
{"title":"SABR/LIBOR market models: pricing and calibration for some interest rate derivatives","authors":"A. M. Ferreiro, J. A. García, J. G. López-Salas, C. Vázquez","doi":"arxiv-2408.01470","DOIUrl":"https://doi.org/arxiv-2408.01470","url":null,"abstract":"In order to overcome the drawbacks of assuming deterministic volatility\u0000coefficients in the standard LIBOR market models to capture volatility smiles\u0000and skews in real markets, several extensions of LIBOR models to incorporate\u0000stochastic volatilities have been proposed. The efficient calibration to market\u0000data of these more complex models becomes a relevant target in practice. The\u0000main objective of the present work is to efficiently calibrate some recent\u0000SABR/LIBOR market models to real market prices of caplets and swaptions. For\u0000the calibration we propose a parallelized version of the simulated annealing\u0000algorithm for multi-GPUs. The numerical results clearly illustrate the\u0000advantages of using the proposed multi-GPUs tools when applied to real market\u0000data and popular SABR/LIBOR models.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141938680","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 derive the short-maturity asymptotics for European and VIX option prices in local-stochastic volatility models where the volatility follows a continuous-path Markov process. Both out-of-the-money (OTM) and at-the-money (ATM) asymptotics are considered. Using large deviations theory methods, the asymptotics for the OTM options are expressed as a two-dimensional variational problem, which is reduced to an extremal problem for a function of two real variables. This extremal problem is solved explicitly in an expansion in log-moneyness. We derive series expansions for the implied volatility for European and VIX options which should be useful for model calibration. We give explicit results for two classes of local-stochastic volatility models relevant in practice, with Heston-type and SABR-type stochastic volatility. The leading-order asymptotics for at-the-money options are computed in closed-form. The asymptotic results reproduce known results in the literature for the Heston and SABR models and for the uncorrelated local-stochastic volatility model. The asymptotic results are tested against numerical simulations for a local-stochastic volatility model with bounded local volatility.
{"title":"Short-maturity asymptotics for VIX and European options in local-stochastic volatility models","authors":"Dan Pirjol, Xiaoyu Wang, Lingjiong Zhu","doi":"arxiv-2407.16813","DOIUrl":"https://doi.org/arxiv-2407.16813","url":null,"abstract":"We derive the short-maturity asymptotics for European and VIX option prices\u0000in local-stochastic volatility models where the volatility follows a\u0000continuous-path Markov process. Both out-of-the-money (OTM) and at-the-money\u0000(ATM) asymptotics are considered. Using large deviations theory methods, the\u0000asymptotics for the OTM options are expressed as a two-dimensional variational\u0000problem, which is reduced to an extremal problem for a function of two real\u0000variables. This extremal problem is solved explicitly in an expansion in\u0000log-moneyness. We derive series expansions for the implied volatility for\u0000European and VIX options which should be useful for model calibration. We give\u0000explicit results for two classes of local-stochastic volatility models relevant\u0000in practice, with Heston-type and SABR-type stochastic volatility. The\u0000leading-order asymptotics for at-the-money options are computed in closed-form.\u0000The asymptotic results reproduce known results in the literature for the Heston\u0000and SABR models and for the uncorrelated local-stochastic volatility model. The\u0000asymptotic results are tested against numerical simulations for a\u0000local-stochastic volatility model with bounded local volatility.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141770781","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 generalizations of the COGARCH model of Kl"uppelberg et al. from 2004 and the volatility and price model of Barndorff-Nielsen and Shephard from 2001 to a Markov-switching environment. These generalizations allow for exogeneous jumps of the volatility at times of a regime switch. Both models are studied within the framework of Markov-modulated generalized Ornstein-Uhlenbeck processes which allows to derive conditions for stationarity, formulas for moments, as well as the autocovariance structure of volatility and price process. It turns out that both models inherit various properties of the original models and therefore are able to capture basic stylized facts of financial time-series such as uncorrelated log-returns, correlated squared log-returns and non-existence of higher moments in the COGARCH case.
{"title":"Volatility modeling in a Markovian environment: Two Ornstein-Uhlenbeck-related approaches","authors":"Anita Behme","doi":"arxiv-2407.05866","DOIUrl":"https://doi.org/arxiv-2407.05866","url":null,"abstract":"We introduce generalizations of the COGARCH model of Kl\"uppelberg et al.\u0000from 2004 and the volatility and price model of Barndorff-Nielsen and Shephard\u0000from 2001 to a Markov-switching environment. These generalizations allow for\u0000exogeneous jumps of the volatility at times of a regime switch. Both models are\u0000studied within the framework of Markov-modulated generalized Ornstein-Uhlenbeck\u0000processes which allows to derive conditions for stationarity, formulas for\u0000moments, as well as the autocovariance structure of volatility and price\u0000process. It turns out that both models inherit various properties of the\u0000original models and therefore are able to capture basic stylized facts of\u0000financial time-series such as uncorrelated log-returns, correlated squared\u0000log-returns and non-existence of higher moments in the COGARCH case.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":"62 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141567454","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 path-dependent volatility (PDV) model of Guyon and Lekeufack (2023), where the instantaneous volatility is a linear combination of a weighted sum of past returns and the square root of a weighted sum of past squared returns. We discuss the influence of an additional parameter that unlocks enough volatility on the upside to reproduce the implied volatility smiles of S&P 500 and VIX options. This PDV model, motivated by empirical studies, comes with computational challenges, especially in relation to VIX options pricing and calibration. We propose an accurate neural network approximation of the VIX which leverages on the Markovianity of the 4-factor version of the model. The VIX is learned as a function of the Markovian factors and the model parameters. We use this approximation to tackle the joint calibration of S&P 500 and VIX options.
{"title":"Pricing and calibration in the 4-factor path-dependent volatility model","authors":"Guido Gazzani, Julien Guyon","doi":"arxiv-2406.02319","DOIUrl":"https://doi.org/arxiv-2406.02319","url":null,"abstract":"We consider the path-dependent volatility (PDV) model of Guyon and Lekeufack\u0000(2023), where the instantaneous volatility is a linear combination of a\u0000weighted sum of past returns and the square root of a weighted sum of past\u0000squared returns. We discuss the influence of an additional parameter that\u0000unlocks enough volatility on the upside to reproduce the implied volatility\u0000smiles of S&P 500 and VIX options. This PDV model, motivated by empirical\u0000studies, comes with computational challenges, especially in relation to VIX\u0000options pricing and calibration. We propose an accurate neural network\u0000approximation of the VIX which leverages on the Markovianity of the 4-factor\u0000version of the model. The VIX is learned as a function of the Markovian factors\u0000and the model parameters. We use this approximation to tackle the joint\u0000calibration of S&P 500 and VIX options.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141255895","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 present a functional generative approach to extract risk-neutral densities from market prices of options. Specifically, we model the log-returns on the time-to-maturity continuum as a stochastic curve driven by standard normal. We then use neural nets to represent the term structures of the location, the scale, and the higher-order moments, and impose stringent conditions on the learning process to ensure the neural net-based curve representation is free of static arbitrage. This specification is structurally clear in that it separates the modeling of randomness from the modeling of the term structures of the parameters. It is data adaptive in that we use neural nets to represent the shape of the stochastic curve. It is also generative in that the functional form of the stochastic curve, although parameterized by neural nets, is an explicit and deterministic function of the standard normal. This explicitness allows for the efficient generation of samples to price options across strikes and maturities, without compromising data adaptability. We have validated the effectiveness of this approach by benchmarking it against a comprehensive set of baseline models. Experiments show that the extracted risk-neutral densities accommodate a diverse range of shapes. Its accuracy significantly outperforms the extensive set of baseline models--including three parametric models and nine stochastic process models--in terms of accuracy and stability. The success of this approach is attributed to its capacity to offer flexible term structures for risk-neutral skewness and kurtosis.
{"title":"Risk-Neutral Generative Networks","authors":"Zhonghao Xian, Xing Yan, Cheuk Hang Leung, Qi Wu","doi":"arxiv-2405.17770","DOIUrl":"https://doi.org/arxiv-2405.17770","url":null,"abstract":"We present a functional generative approach to extract risk-neutral densities\u0000from market prices of options. Specifically, we model the log-returns on the\u0000time-to-maturity continuum as a stochastic curve driven by standard normal. We\u0000then use neural nets to represent the term structures of the location, the\u0000scale, and the higher-order moments, and impose stringent conditions on the\u0000learning process to ensure the neural net-based curve representation is free of\u0000static arbitrage. This specification is structurally clear in that it separates\u0000the modeling of randomness from the modeling of the term structures of the\u0000parameters. It is data adaptive in that we use neural nets to represent the\u0000shape of the stochastic curve. It is also generative in that the functional\u0000form of the stochastic curve, although parameterized by neural nets, is an\u0000explicit and deterministic function of the standard normal. This explicitness\u0000allows for the efficient generation of samples to price options across strikes\u0000and maturities, without compromising data adaptability. We have validated the\u0000effectiveness of this approach by benchmarking it against a comprehensive set\u0000of baseline models. Experiments show that the extracted risk-neutral densities\u0000accommodate a diverse range of shapes. Its accuracy significantly outperforms\u0000the extensive set of baseline models--including three parametric models and\u0000nine stochastic process models--in terms of accuracy and stability. The success\u0000of this approach is attributed to its capacity to offer flexible term\u0000structures for risk-neutral skewness and kurtosis.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141166935","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}
W. Brent Lindquist, Svetlozar T. Rachev, Jagdish Gnawali, Frank J. Fabozzi
We develop asset pricing under a unified Bachelier and Black-Scholes-Merton (BBSM) market model. We derive option pricing via the Feynman-Kac formula as well as through deflator-driven risk-neutral valuation. We show a necessary condition for the unified model to support a perpetual derivative. We develop discrete binomial pricing under the unified model. Finally, we investigate the term structure of interest rates by considering the pricing of zero-coupon bonds, forward and futures contracts. In all cases, we show that the unified model reduces to standard Black-Scholes-Merton pricing (in the appropriate parameter limit) and derive (also under the appropriate limit) pricing for a Bachelier model. The Bachelier limit of our unified model allows for positive riskless rates.
{"title":"Dynamic Asset Pricing in a Unified Bachelier-Black-Scholes-Merton Model","authors":"W. Brent Lindquist, Svetlozar T. Rachev, Jagdish Gnawali, Frank J. Fabozzi","doi":"arxiv-2405.12479","DOIUrl":"https://doi.org/arxiv-2405.12479","url":null,"abstract":"We develop asset pricing under a unified Bachelier and Black-Scholes-Merton\u0000(BBSM) market model. We derive option pricing via the Feynman-Kac formula as\u0000well as through deflator-driven risk-neutral valuation. We show a necessary\u0000condition for the unified model to support a perpetual derivative. We develop\u0000discrete binomial pricing under the unified model. Finally, we investigate the\u0000term structure of interest rates by considering the pricing of zero-coupon\u0000bonds, forward and futures contracts. In all cases, we show that the unified\u0000model reduces to standard Black-Scholes-Merton pricing (in the appropriate\u0000parameter limit) and derive (also under the appropriate limit) pricing for a\u0000Bachelier model. The Bachelier limit of our unified model allows for positive\u0000riskless rates.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141152871","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, a general framework is developed for continuous-time financial market models defined from simple strategies through conditional topologies that avoid stochastic calculus and do not necessitate semimartingale models. We then compare the usual no-arbitrage conditions of the literature, e.g. the usual no-arbitrage conditions NFL, NFLVR and NUPBR and the recent AIP condition. With appropriate pseudo-distance topologies, we show that they hold in continuous time if and only if they hold in discrete time. Moreover, the super-hedging prices in continuous time coincide with the discrete-time super-hedging prices, even without any no-arbitrage condition.
{"title":"No-arbitrage conditions and pricing from discrete-time to continuous-time strategies","authors":"Dorsaf Cherif, Emmanuel Lepinette","doi":"arxiv-2405.07713","DOIUrl":"https://doi.org/arxiv-2405.07713","url":null,"abstract":"In this paper, a general framework is developed for continuous-time financial\u0000market models defined from simple strategies through conditional topologies\u0000that avoid stochastic calculus and do not necessitate semimartingale models. We\u0000then compare the usual no-arbitrage conditions of the literature, e.g. the\u0000usual no-arbitrage conditions NFL, NFLVR and NUPBR and the recent AIP\u0000condition. With appropriate pseudo-distance topologies, we show that they hold\u0000in continuous time if and only if they hold in discrete time. Moreover, the\u0000super-hedging prices in continuous time coincide with the discrete-time\u0000super-hedging prices, even without any no-arbitrage condition.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140927469","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 formulate quantum computing solutions to a large class of dynamic nonlinear asset pricing models using algorithms, in theory exponentially more efficient than classical ones, which leverage the quantum properties of superposition and entanglement. The equilibrium asset pricing solution is a quantum state. We introduce quantum decision-theoretic foundations of ambiguity and model/parameter uncertainty to deal with model selection.
{"title":"On Quantum Ambiguity and Potential Exponential Computational Speed-Ups to Solving","authors":"Eric Ghysels, Jack Morgan","doi":"arxiv-2405.01479","DOIUrl":"https://doi.org/arxiv-2405.01479","url":null,"abstract":"We formulate quantum computing solutions to a large class of dynamic\u0000nonlinear asset pricing models using algorithms, in theory exponentially more\u0000efficient than classical ones, which leverage the quantum properties of\u0000superposition and entanglement. The equilibrium asset pricing solution is a\u0000quantum state. We introduce quantum decision-theoretic foundations of ambiguity\u0000and model/parameter uncertainty to deal with model selection.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140828643","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, the pricing of financial derivatives and the calculation of their delta Greek are investigated as the underlying asset is a jump-diffusion process in which the stochastic intensity component follows the CIR process. Utilizing Malliavin derivatives for pricing financial derivatives and challenging to find the Malliavin weight for accurately calculating delta will be established in such models. Due to the dependence of asset price on the information of the intensity process, conditional expectation attribute to show boundedness of moments of Malliavin weights and the underlying asset is essential. Our approach is validated through numerical experiments, highlighting its effectiveness and potential for risk management and hedging strategies in markets characterized by jump and stochastic intensity dynamics.
{"title":"Pricing and delta computation in jump-diffusion models with stochastic intensity by Malliavin calculus","authors":"Ayub Ahmadi, Mahdieh Tahmasebi","doi":"arxiv-2405.00473","DOIUrl":"https://doi.org/arxiv-2405.00473","url":null,"abstract":"In this paper, the pricing of financial derivatives and the calculation of\u0000their delta Greek are investigated as the underlying asset is a jump-diffusion\u0000process in which the stochastic intensity component follows the CIR process.\u0000Utilizing Malliavin derivatives for pricing financial derivatives and\u0000challenging to find the Malliavin weight for accurately calculating delta will\u0000be established in such models. Due to the dependence of asset price on the\u0000information of the intensity process, conditional expectation attribute to show\u0000boundedness of moments of Malliavin weights and the underlying asset is\u0000essential. Our approach is validated through numerical experiments,\u0000highlighting its effectiveness and potential for risk management and hedging\u0000strategies in markets characterized by jump and stochastic intensity dynamics.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140828408","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 Chicago Board Options Exchange Volatility Index (VIX) is calculated from SPX options and derivatives of VIX are also traded in market, which leads to the so-called "consistent modeling" problem. This paper proposes a time-changed L'evy model for log price with a composite change of time structure to capture both features of the implied SPX volatility and the implied volatility of volatility. Consistent modeling is achieved naturally via flexible choices of jumps and leverage effects, as well as the composition of time changes. Many celebrated models are covered as special cases. From this model, we derive an explicit form of the characteristic function for the asset price (SPX) and the pricing formula for European options as well as VIX options. The empirical results indicate great competence of the proposed model in the problem of joint calibration of the SPX/VIX Markets.
{"title":"Joint calibration to SPX and VIX Derivative Markets with Composite Change of Time Models","authors":"Liexin Cheng, Xue Cheng, Xianhua Peng","doi":"arxiv-2404.16295","DOIUrl":"https://doi.org/arxiv-2404.16295","url":null,"abstract":"The Chicago Board Options Exchange Volatility Index (VIX) is calculated from\u0000SPX options and derivatives of VIX are also traded in market, which leads to\u0000the so-called \"consistent modeling\" problem. This paper proposes a time-changed\u0000L'evy model for log price with a composite change of time structure to capture\u0000both features of the implied SPX volatility and the implied volatility of\u0000volatility. Consistent modeling is achieved naturally via flexible choices of\u0000jumps and leverage effects, as well as the composition of time changes. Many\u0000celebrated models are covered as special cases. From this model, we derive an\u0000explicit form of the characteristic function for the asset price (SPX) and the\u0000pricing formula for European options as well as VIX options. The empirical\u0000results indicate great competence of the proposed model in the problem of joint\u0000calibration of the SPX/VIX Markets.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140804357","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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