We provide a general HJM framework for forward contracts written on abstract market indices with arbitrary fixing and payment adjustments. We allow for indices on any asset class, featuring collateralization in arbitrary currency denominations. The framework is pivotal for describing portfolios of interest rate products which are denominated in multiple currencies. The benchmark transition has created significant discrepancies among the market conventions of different currency areas: our framework simultaneously covers forward-looking risky IBOR rates, such as EURIBOR, and backward-looking rates based on overnight rates, such as SOFR. In view of this, we provide a thorough study of cross-currency markets in the presence of collateral, where the cash flows of the contract and the margin account can be denominated in arbitrary combinations of currencies. We finally consider cross-currency swap contracts as an example of a contract simultaneously depending on all the risk factors that we describe within our framework.
{"title":"Cross-Currency Heath-Jarrow-Morton Framework in the Multiple-Curve Setting","authors":"Alessandro Gnoatto, Silvia Lavagnini","doi":"arxiv-2312.13057","DOIUrl":"https://doi.org/arxiv-2312.13057","url":null,"abstract":"We provide a general HJM framework for forward contracts written on abstract\u0000market indices with arbitrary fixing and payment adjustments. We allow for\u0000indices on any asset class, featuring collateralization in arbitrary currency\u0000denominations. The framework is pivotal for describing portfolios of interest\u0000rate products which are denominated in multiple currencies. The benchmark\u0000transition has created significant discrepancies among the market conventions\u0000of different currency areas: our framework simultaneously covers\u0000forward-looking risky IBOR rates, such as EURIBOR, and backward-looking rates\u0000based on overnight rates, such as SOFR. In view of this, we provide a thorough\u0000study of cross-currency markets in the presence of collateral, where the cash\u0000flows of the contract and the margin account can be denominated in arbitrary\u0000combinations of currencies. We finally consider cross-currency swap contracts\u0000as an example of a contract simultaneously depending on all the risk factors\u0000that we describe within our framework.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138824488","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 robust option pricing problem is to find upper and lower bounds on fair prices of financial claims using only the most minimal assumptions. It contrasts with the classical, model-based approach and gained prominence in the wake of the 2008 financial crisis, and can be used to understand the extent to which a model-based price is sensitive to the underlying model assumptions. Common approaches involve pricing exotic derivatives such as variance options by incorporating market data through implied volatility. The existing literature focuses largely on incorporating implied volatility information corresponding to the maturity of the exotic option. In this paper, we aim to explain how intermediate data can and should be incorporated. It is natural to expect that this additional information will improve the robust pricing bounds. To investigate this question, we consider variance options, where the bounds of the informed robust pricing problem are known. We proceed to conduct an empirical study uncovering a surprising finding: Contrary to common belief, the incorporation of more information does not lead to an improvement of the robust pricing bounds.
{"title":"Robust option pricing with volatility term structure -- An empirical study for variance options","authors":"Alexander M. G. Cox, Annemarie M. Grass","doi":"arxiv-2312.09201","DOIUrl":"https://doi.org/arxiv-2312.09201","url":null,"abstract":"The robust option pricing problem is to find upper and lower bounds on fair\u0000prices of financial claims using only the most minimal assumptions. It\u0000contrasts with the classical, model-based approach and gained prominence in the\u0000wake of the 2008 financial crisis, and can be used to understand the extent to\u0000which a model-based price is sensitive to the underlying model assumptions.\u0000Common approaches involve pricing exotic derivatives such as variance options\u0000by incorporating market data through implied volatility. The existing\u0000literature focuses largely on incorporating implied volatility information\u0000corresponding to the maturity of the exotic option. In this paper, we aim to\u0000explain how intermediate data can and should be incorporated. It is natural to\u0000expect that this additional information will improve the robust pricing bounds.\u0000To investigate this question, we consider variance options, where the bounds of\u0000the informed robust pricing problem are known. We proceed to conduct an\u0000empirical study uncovering a surprising finding: Contrary to common belief, the\u0000incorporation of more information does not lead to an improvement of the robust\u0000pricing bounds.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138684567","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 apply a physics-informed deep-learning approach the PINN approach to the Black-Scholes equation for pricing American and European options. We test our approach on both simulated as well as real market data, compare it to analytical/numerical benchmarks. Our model is able to accurately capture the price behaviour on simulation data, while also exhibiting reasonable performance for market data. We also experiment with the architecture and learning process of our PINN model to provide more understanding of convergence and stability issues that impact performance.
{"title":"Physics Informed Neural Network for Option Pricing","authors":"Ashish Dhiman, Yibei Hu","doi":"arxiv-2312.06711","DOIUrl":"https://doi.org/arxiv-2312.06711","url":null,"abstract":"We apply a physics-informed deep-learning approach the PINN approach to the\u0000Black-Scholes equation for pricing American and European options. We test our\u0000approach on both simulated as well as real market data, compare it to\u0000analytical/numerical benchmarks. Our model is able to accurately capture the\u0000price behaviour on simulation data, while also exhibiting reasonable\u0000performance for market data. We also experiment with the architecture and\u0000learning process of our PINN model to provide more understanding of convergence\u0000and stability issues that impact performance.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138630373","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}
This article describes a case study concerned with modelling the price of wholesale diamonds, as part of a project to develop an online diamond auction platform. The work was extended to exploring how to develop an index that could be used to track market trends of wholesale diamond prices. The approach we used is readily generalised to defining market indices for so-called Collectables, and can provide the basis for construction of derivatives. With the burgeoning interest in new markets of collectables such as those generated by the concept of a Non-Fungible Token, it is reasonable to suppose that there will be concomitant increasing interest in developing derivatives for these markets.
{"title":"A Hedonic Index for Collectables Arising from Modelling Diamond Prices","authors":"Nicholas I Fisher, Alan J Lee","doi":"arxiv-2312.11496","DOIUrl":"https://doi.org/arxiv-2312.11496","url":null,"abstract":"This article describes a case study concerned with modelling the price of\u0000wholesale diamonds, as part of a project to develop an online diamond auction\u0000platform. The work was extended to exploring how to develop an index that could\u0000be used to track market trends of wholesale diamond prices. The approach we\u0000used is readily generalised to defining market indices for so-called\u0000Collectables, and can provide the basis for construction of derivatives. With\u0000the burgeoning interest in new markets of collectables such as those generated\u0000by the concept of a Non-Fungible Token, it is reasonable to suppose that there\u0000will be concomitant increasing interest in developing derivatives for these\u0000markets.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138821243","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}
Yongzhuo Chen, Yixuan Liang, Yiran Liu, Brian Hobbs, Michael Kane
This research paper addresses the critical challenge of accurately valuing post-revenue drug assets in the biotechnology and pharmaceutical sectors, a key factor influencing a wide range of strategic operations and investment decisions. Recognizing the importance of reliable valuations for stakeholders such as pharmaceutical companies, venture capitalists, and private equity firms, this study introduces a novel model for forecasting future sales of post-revenue biopharmaceutical assets. The proposed model leverages historical sales data, a resource known for its high quality and availability in company financial records, to produce distributional estimates of cumulative sales for individual assets. These estimates are instrumental in calculating the Net Present Value of each asset, thereby facilitating more informed and strategic investment decisions. A practical application of this model is demonstrated through its implementation in analyzing Pfizer's portfolio of post-revenue assets. This precision highlights the model's potential as a valuable tool in the financial assessment and decision-making processes within the biotech and pharmaceutical industries, offering a methodical approach to identifying investment opportunities and optimizing capital allocation.
{"title":"Valuing Post-Revenue Biopharmaceutical Assets with Pfizer's Current Portfolio as a Case Study","authors":"Yongzhuo Chen, Yixuan Liang, Yiran Liu, Brian Hobbs, Michael Kane","doi":"arxiv-2312.02250","DOIUrl":"https://doi.org/arxiv-2312.02250","url":null,"abstract":"This research paper addresses the critical challenge of accurately valuing\u0000post-revenue drug assets in the biotechnology and pharmaceutical sectors, a key\u0000factor influencing a wide range of strategic operations and investment\u0000decisions. Recognizing the importance of reliable valuations for stakeholders\u0000such as pharmaceutical companies, venture capitalists, and private equity\u0000firms, this study introduces a novel model for forecasting future sales of\u0000post-revenue biopharmaceutical assets. The proposed model leverages historical\u0000sales data, a resource known for its high quality and availability in company\u0000financial records, to produce distributional estimates of cumulative sales for\u0000individual assets. These estimates are instrumental in calculating the Net\u0000Present Value of each asset, thereby facilitating more informed and strategic\u0000investment decisions. A practical application of this model is demonstrated\u0000through its implementation in analyzing Pfizer's portfolio of post-revenue\u0000assets. This precision highlights the model's potential as a valuable tool in\u0000the financial assessment and decision-making processes within the biotech and\u0000pharmaceutical industries, offering a methodical approach to identifying\u0000investment opportunities and optimizing capital allocation.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138561416","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}
Matteo Gambara, Giulia Livieri, Andrea Pallavicini
In the present work, we introduce and compare state-of-the-art algorithms, that are now classified under the name of machine learning, to price Asian and look-back products with early-termination features. These include randomized feed-forward neural networks, randomized recurrent neural networks, and a novel method based on signatures of the underlying price process. Additionally, we explore potential applications on callable certificates. Furthermore, we present an innovative approach for calculating sensitivities, specifically Delta and Gamma, leveraging Chebyshev interpolation techniques.
{"title":"Machine learning methods for American-style path-dependent contracts","authors":"Matteo Gambara, Giulia Livieri, Andrea Pallavicini","doi":"arxiv-2311.16762","DOIUrl":"https://doi.org/arxiv-2311.16762","url":null,"abstract":"In the present work, we introduce and compare state-of-the-art algorithms,\u0000that are now classified under the name of machine learning, to price Asian and\u0000look-back products with early-termination features. These include randomized\u0000feed-forward neural networks, randomized recurrent neural networks, and a novel\u0000method based on signatures of the underlying price process. Additionally, we\u0000explore potential applications on callable certificates. Furthermore, we\u0000present an innovative approach for calculating sensitivities, specifically\u0000Delta and Gamma, leveraging Chebyshev interpolation techniques.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522488","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}
Credit Valuation Adjustment is a balance sheet item which is nowadays subject to active risk management by specialized traders. However, one of the most important risk factors, which is the vector of default intensities of the counterparty, affects in a non-differentiable way the most general Monte Carlo estimator of the adjustment, through simulation of default times. Thus the computation of first and second order (pure and mixed) sensitivities involving these inputs cannot rely on direct path-wise differentiation, while any approach involving finite differences shows very high statistical noise. We present ad hoc analytical estimators which overcome these issues while offering very low runtime overheads over the baseline computation of the price adjustment. We also discuss the conversion of the so-obtained sensitivities to model parameters (e.g. default intensities) into sensitivities to market quotes (e.g. Credit Default Swap spreads).
{"title":"Fast and Stable Credit Gamma of CVA","authors":"Roberto Daluiso","doi":"arxiv-2311.11672","DOIUrl":"https://doi.org/arxiv-2311.11672","url":null,"abstract":"Credit Valuation Adjustment is a balance sheet item which is nowadays subject\u0000to active risk management by specialized traders. However, one of the most\u0000important risk factors, which is the vector of default intensities of the\u0000counterparty, affects in a non-differentiable way the most general Monte Carlo\u0000estimator of the adjustment, through simulation of default times. Thus the\u0000computation of first and second order (pure and mixed) sensitivities involving\u0000these inputs cannot rely on direct path-wise differentiation, while any\u0000approach involving finite differences shows very high statistical noise. We\u0000present ad hoc analytical estimators which overcome these issues while offering\u0000very low runtime overheads over the baseline computation of the price\u0000adjustment. We also discuss the conversion of the so-obtained sensitivities to\u0000model parameters (e.g. default intensities) into sensitivities to market quotes\u0000(e.g. Credit Default Swap spreads).","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522491","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}
Dorsaf Cherif, Meriam El Mansour, Emmanuel Lepinette
The usual theory of asset pricing in finance assumes that the financial strategies, i.e. the quantity of risky assets to invest, are real-valued so that they are not integer-valued in general, see the Black and Scholes model for instance. This is clearly contrary to what it is possible to do in the real world. Surprisingly, it seems that there is no many contributions in that direction in the literature, except for a finite number of states. In this paper, for arbitrary {Omega}, we show that, in discrete-time, it is possible to evaluate the minimal super-hedging price when we restrict ourselves to integer-valued strategies. To do so, we only consider terminal claims that are continuous piecewise affine functions of the underlying asset. We formulate a dynamic programming principle that can be directly implemented on an historical data and which also provides the optimal integer-valued strategy. The problem with general payoffs remains open but should be solved with the same approach.
{"title":"A short note on super-hedging an arbitrary number of European options with integer-valued strategies","authors":"Dorsaf Cherif, Meriam El Mansour, Emmanuel Lepinette","doi":"arxiv-2311.08871","DOIUrl":"https://doi.org/arxiv-2311.08871","url":null,"abstract":"The usual theory of asset pricing in finance assumes that the financial\u0000strategies, i.e. the quantity of risky assets to invest, are real-valued so\u0000that they are not integer-valued in general, see the Black and Scholes model\u0000for instance. This is clearly contrary to what it is possible to do in the real\u0000world. Surprisingly, it seems that there is no many contributions in that\u0000direction in the literature, except for a finite number of states. In this\u0000paper, for arbitrary {Omega}, we show that, in discrete-time, it is possible\u0000to evaluate the minimal super-hedging price when we restrict ourselves to\u0000integer-valued strategies. To do so, we only consider terminal claims that are\u0000continuous piecewise affine functions of the underlying asset. We formulate a\u0000dynamic programming principle that can be directly implemented on an historical\u0000data and which also provides the optimal integer-valued strategy. The problem\u0000with general payoffs remains open but should be solved with the same approach.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522553","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 solve the problem of super-hedging European or Asian options for discrete-time financial market models where executable prices are uncertain. The risky asset prices are not described by single-valued processes but measurable selections of random sets that allows to consider a large variety of models including bid-ask models with order books, but also models with a delay in the execution of the orders. We provide a numerical procedure to compute the infimum price under a weak no-arbitrage condition, the so-called AIP condition, under which the prices of the non negative European options are non negative. This condition is weaker than the existence of a risk-neutral martingale measure but it is sufficient to numerically solve the super-hedging problem. We illustrate our method by a numerical example.
{"title":"Robust discrete-time super-hedging strategies under AIP condition and under price uncertainty","authors":"Meriam El Mansour, Emmanuel Lepinette","doi":"arxiv-2311.08847","DOIUrl":"https://doi.org/arxiv-2311.08847","url":null,"abstract":"We solve the problem of super-hedging European or Asian options for\u0000discrete-time financial market models where executable prices are uncertain.\u0000The risky asset prices are not described by single-valued processes but\u0000measurable selections of random sets that allows to consider a large variety of\u0000models including bid-ask models with order books, but also models with a delay\u0000in the execution of the orders. We provide a numerical procedure to compute the\u0000infimum price under a weak no-arbitrage condition, the so-called AIP condition,\u0000under which the prices of the non negative European options are non negative.\u0000This condition is weaker than the existence of a risk-neutral martingale\u0000measure but it is sufficient to numerically solve the super-hedging problem. We\u0000illustrate our method by a numerical example.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522558","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 study an optimal stopping problem with an unbounded, time-dependent and discontinuous reward function. This problem is motivated by the pricing of a variable annuity (VA) contract with guaranteed minimum maturity benefit, under the assumption that the policyholder's surrender behaviour maximizes the contract's risk-neutral value. We consider a general fee and surrender charge function, and give a condition under which optimal stopping always occurs at maturity. Using an alternative representation for the value function of the optimization problem, we study its analytical properties and the resulting surrender (or exercise) region. In particular, we show that the non-emptiness and the shape of the surrender region are fully characterized by the fee and the surrender charge functions, which provides a powerful tool for understanding the link between fees and surrender functions and how they affect early surrender and the optimal surrender boundary. When the fee and surrender charge only depend on time, we develop three different representations of the value function; two are analogous to their American option counterpart, and one is new to the actuarial and American option pricing literature. Our results allow for the development of new algorithms for the valuation of variable annuity contracts. We provide three such algorithms, based on continuous-time Markov chain approximations. The efficiency of these three algorithms is studied numerically and compared to other commonly used approaches.
{"title":"On an Optimal Stopping Problem with a Discontinuous Reward","authors":"Anne Mackay, Marie-Claude Vachon","doi":"arxiv-2311.03538","DOIUrl":"https://doi.org/arxiv-2311.03538","url":null,"abstract":"We study an optimal stopping problem with an unbounded, time-dependent and\u0000discontinuous reward function. This problem is motivated by the pricing of a\u0000variable annuity (VA) contract with guaranteed minimum maturity benefit, under\u0000the assumption that the policyholder's surrender behaviour maximizes the\u0000contract's risk-neutral value. We consider a general fee and surrender charge\u0000function, and give a condition under which optimal stopping always occurs at\u0000maturity. Using an alternative representation for the value function of the\u0000optimization problem, we study its analytical properties and the resulting\u0000surrender (or exercise) region. In particular, we show that the non-emptiness\u0000and the shape of the surrender region are fully characterized by the fee and\u0000the surrender charge functions, which provides a powerful tool for\u0000understanding the link between fees and surrender functions and how they affect\u0000early surrender and the optimal surrender boundary. When the fee and surrender\u0000charge only depend on time, we develop three different representations of the\u0000value function; two are analogous to their American option counterpart, and one\u0000is new to the actuarial and American option pricing literature. Our results allow for the development of new algorithms for the valuation of\u0000variable annuity contracts. We provide three such algorithms, based on\u0000continuous-time Markov chain approximations. The efficiency of these three\u0000algorithms is studied numerically and compared to other commonly used\u0000approaches.","PeriodicalId":501355,"journal":{"name":"arXiv - QuantFin - Pricing of Securities","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138522492","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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