analyst report Numerix is leading the way in pricing and valuation for capi... Read more about Numerix is leading the way in pricing and valuation for capital markets
analyst report Numerix is leading the way in pricing and valuation for capi... Read more about Numerix is leading the way in pricing and valuation for capital markets
analyst report Numerix is leading the way in pricing and valuation for capi... Read more about Numerix is leading the way in pricing and valuation for capital markets
analyst report Numerix is leading the way in pricing and valuation for capi... Read more about Numerix is leading the way in pricing and valuation for capital markets
analyst report Numerix is leading the way in pricing and valuation for capi... Read more about Numerix is leading the way in pricing and valuation for capital markets
analyst report Numerix is leading the way in pricing and valuation for capi... Read more about Numerix is leading the way in pricing and valuation for capital markets
analyst report Numerix is leading the way in pricing and valuation for capi... Read more about Numerix is leading the way in pricing and valuation for capital markets
analyst report Numerix is leading the way in pricing and valuation for capi... Read more about Numerix is leading the way in pricing and valuation for capital markets
Resources Filter by Type All Resources Publications Webinars Case Studies Videos Podcasts Events Who We ServeMarket RiskQuantitative Research & DevelopmentSalesSoftware DevelopmentSoftware VendorsValuationsAccountingActuariesConsultingCounterparty Credit RiskIT & InfrastructureStructuringAudit/AdvisoryModel ValidationRegulationsTradingCollateral ManagementPortfolio ManagementXVA DeskFront Office & TradingRisk ManagementBack Offices & OperationsQuant & ActuariesTechnology & DevelopmentFinTech & Consultants What We OfferDevelopment PlatformExoticsValuationGraph FrameworkMarket RiskVanilla DerivativesFixed Income & Structured FinanceXVA & Counterparty RiskOTC TradingStructured Finance Trading & RiskBanking Asset Liability ManagementDerivatives & Fixed Income AnalyticsFront-to-Risk ApplicationsCapital Markets Development Platform webinar To Cloud or Not to Cloud: Shifting Attitudes Around Cloud Adoption Paul Sinthunont, Strategic Advisor at Aite-Novarica Group unveil key findings from one-on-one interviews with key stakeholders across the financial services sector uncovering what’s driving—or holding back—cloud migration decisions for trading and risk management functions. Register Now webinar Estimating Cross-Model Correlations for CCR & XVA In this webinar, Numerix’s Andrew McClelland, SVP of Quantitative Research, shares detailed insights on Estimating Cross-Model Correlations for CCR & XVA. Register Now article Digitalization and New Tech Bring Opportunity and Changing Approaches to Derivatives Markets In this article Numerix technologist, Linus Yu, joins Heads of Collateral and Liquidity Management, Treasury and Market Risk at some of the world’s leading banks to discuss where they see the path forward for building a more impactful future-state infrastructure for a derivatives business. Read article analyst report Aite-Novarica Report In new research report Audrey Blater of the Aite-Novarica Group shares key findings from her one-on-one interviews with key stakeholders across the financial services sector to uncover what’s driving cloud migration decisions for trading and risk management functions.The report analyzes the factors influencing banks when it comes to deciding whether or not to migrate to the cloud and discusses what is needed to bring a pro-cloud shift into the decision-making process. The report also provides views on why on-premise legacy systems may or may not be preferred. Read analyst report webinar Transforming Treasury and Derivatives management post Covid-19 In this webinar, Numerix and Risk.net, will examine new strategies to enhance your derivatives business and treasury management in the face of Covid-19, regulatory changes and counterparty risk. Register Now webinar Digitalisation of Derivatives Trading: Utilising Technology to Increase Profits In this webinar, learn how new technologies, such as AI and machine learning, can be leveraged to better manage several aspects of a derivatives business. Register Now webinar Deep Dive: Advances in Counterparty Credit Modelling in Energy Markets Numerix’s Andrew McClelland, SVP of Quantitative Research, addresses natural gas and electricity curves and the dynamics that complicate modelling, Register Now webinar XVAs and Counterparty Credit Risk for Energy Markets – Addressing the Challenges and Unravelling Complexity In this webinar, panel of quantitative researchers and risk practitioners from banks, energy firms, and a software vendor discuss the many practical challenges they’ve encountered in the modeling and risk management of XVAs/CCR in the energy markets, and how to overcome them. Register Now journal issue Numerix Journal Vol. 7 No. 1 The Vol 7. No. 1 Issue of the Numerix Journal represents some of Numerix's quantitative research and development achievements lately. Many of these achievements have been implemented as functionalities in our products. With the ongoing R&D effort at Numerix, this collection of papers showcases the quality of the research on various topics of interest in the field. Readers get an update on the latest progress in the theoretical research and product enhancements at Numerix. Read journal issue Subscribe to our monthly newsletter to get exclusive resources from Numerix. Marketo assets failed to be loaded white paper New Technology Is Redefining the Success of the Front Office Numerix Chief Product Officer, Satyam Kancharla, discusses how firms are rethinking trading infrastructures Read white paper white paper Risk.net: Next-generation technologies and the future of trading Panel of industry experts share their insights and observations regarding emerging technologies in trading Read white paper webinar Advances in Counterparty Credit Risk Modelling in Energy Markets In this webinar, Numerix SVP of Quantitative Research, Andrew McClelland Ph.D., looked at what is being done to improve energy models inside the counterparty credit risk setting. Register Now case study OCBC Bank Scales Business and Mitigates Risk with Numerix Oneview During the course of its nearly 10-year partnership with Numerix, OCBC Bank has faced a number of challenges it engaged Numerix to address. The bank required a platform that could help it meet demand for products with more innovative features within the structured products market and which integrated a sophisticated risk management process. View case study white paper Real-Time Risk Management in Practice: The Experts’ Views This whitepaper provides an overview of some of the key topics discussed by a group of market experts during a roundtable webinar hosted by CubeLogic in partnership with Numerix and PRMIA. Various applications of risk management in real time were examined, including, among others. Read white paper analyst report LIBOR Risk Q2 2021 As the deadline to Libor cessation approaches, Liang Wu, executive director of financial engineering and head of cross asset product management at Numerix, presents a series of market themes that warrant closer inspection. Read analyst report webinar Modelling Energy Curves for XVA Watch Numerix’s SVP of Quantitative Research Andrew McClelland present updates to his latest research, "Modelling Energy Curves for XVA." Register Now white paper The New Rules of Market Risk Management This paper highlights several of the key viewpoints expressed by the panel and explores some the current dynamics that are impacting approaches to and the complexities of market risk management. Read white paper webinar Next generation technologies and the future of trading In this webinar, Numerix and Risk.net panelists examine the future of trading technology, how companies will implement these new innovations, and explore the range of new skills that might be needed. Register Now Pagination First page « First Previous page Previous … Page 10 Page 11 Page 12 Page 13 Current page 14 Page 15 Page 16 Page 17 Page 18 … Next page Next Last page Last » Who We ServeMarket RiskQuantitative Research & DevelopmentSalesSoftware DevelopmentSoftware VendorsValuationsAccountingActuariesConsultingCounterparty Credit RiskIT & InfrastructureStructuringAudit/AdvisoryModel ValidationRegulationsTradingCollateral ManagementPortfolio ManagementXVA DeskFront Office & TradingRisk ManagementBack Offices & OperationsQuant & ActuariesTechnology & DevelopmentFinTech & Consultants What We OfferDevelopment PlatformExoticsValuationGraph FrameworkMarket RiskVanilla DerivativesFixed Income & Structured FinanceXVA & Counterparty RiskOTC TradingStructured Finance Trading & RiskBanking Asset Liability ManagementDerivatives & Fixed Income AnalyticsFront-to-Risk ApplicationsCapital Markets Development Platform Content type All Publications Analyst report Article Blog post Journal issue Newsletter Quantitative research White paper quantitative research USLV: Unspanned Stochastic Local Volatility Model In this article, we propose a new framework for modeling stochastic local volatility, with potential applications to modeling derivatives on interest rates, commodities, credit, equity, FX etc., as well as hybrid derivatives. Read quantitative research quantitative research Exposure & CVA for Large Portfolios of Vanilla Swaps: The Thin-Out Optimization In this article we present an efficient optimization for calculating the exposure and CVA for large portfolios of vanilla swaps. Read quantitative research blog Advanced FX Hedging Strategies (and what NOT to do): A Case Study Learn advanced FX hedging strategies and what is most effective. Read Blog quantitative research Algorithmic Exposure and CVA for Exotic Derivatives In this article, we develop the algorithmic approach for Counterparty exposure calculation and automate its application to arbitrary complicated instruments. Read quantitative research blog Capturing Stochastic Volatility: Key to Trading Inflation Derivatives Modeling the smile and capturing the stochastic nature of volatility has become critically important for inflation derivatives trading. Read Blog quantitative research Analytical Approximations for Short Rate Models In this article, we present the analytical approximation of zero-coupon bonds and swaption prices for general short rate models. Read quantitative research blog Bates Model and Cliquet Pricing in Numerix Bates stochastic volatility jump-diffusion model is the market standard model for pricing exotic options that depend heavily on the forward skew, such as cliquets and other forward-starting trades. Read Blog quantitative research Generalized Vanna-Volga Method and Its Applications In this article, we give a general treatment of the Vanna-Volga mark-to-market volatility smile correction in application to pricing of contracts with European exercise on a single underlying. Read quantitative research quantitative research Decoupled American Option Pricing Method: Computation of Implied Volatilities and Further Applications In this article, we introduce a method for volatility computation from listed prices of American options on an underlying close to log-normal. Read quantitative research Subscribe to our monthly newsletter to get exclusive resources from Numerix. Marketo assets failed to be loaded quantitative research Dynamic Model for Pricing and Hedging Heterogenous CDOs In this article, we present a simple bottom-up dynamic credit model that can be calibrated simultaneously to the market quotes on CDO tranches and individual CDSs constituting the credit portfolio. Read quantitative research quantitative research Analytical Formulas for Pricing CMS Products in the LMM with Stochastic Volatility In this paper, we develop a series of approximations for a fast analytical pricing of European constant maturity swap (CMS) products, such as CMS swaps, CMS caps/floors, and CMS spread options, for the LIBOR Market Model (LMM) with stochastic volatility. Read quantitative research quantitative research Projection on a Quadratic Model by Asymptotic Expansion with an Application to LMM Swaption In this article, we develop a technique of parameter averaging and Markovian projection on a quadratic volatility model based on a term-by-term matching of the asymptotic expansions of option prices in volatilities. Read quantitative research quantitative research Markovian Projection to a Displaced Volatility Heston Model In this article, we generalize the results of Markovian Projection onto a Heston model to a wider class of approximating models, a Heston model with displaced volatility. Read quantitative research quantitative research Two-Dimensional Markovian Model for Dynamics of Aggregate Credit Loss In this article, we propose a new model for the dynamics of the aggregate credit portfolio loss. The model is Markovian in two dimensions with the state variables being the total accumulated loss Lt and the stochastic default intensity λt. Read quantitative research quantitative research Markovian Projection onto a Displaced Diffusion In this paper, we develop a systematic approach to Markovian projection onto an effective displaced diffusion, and work out a set of computationally efficient formulas valid for a large class of non-Markovian underlying processes. Read quantitative research quantitative research Markovian Projection Onto a Heston Model In this article, we develop a systematic approach to the reduction of dimensionality of smile-enabled models by projecting them onto a displaced version of the two-dimensional Heston process. Read quantitative research quantitative research Efficient Calibration to FX Options by Markovian Projection in Cross-Currency LIBOR Market Models In this article, we revisit the cross-currency LIBOR Market Model armed with the technique of Markovian projection. Read quantitative research quantitative research Overlapping Credit Portfolios In this article, we present an accurate analytical approximation for a joint distribution function of loss of two overlapping credit portfolios using the multidimensional saddlepoint method. Read quantitative research Pagination First page « First Previous page Previous … Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Current page 14 Page 15 Next page Next Last page Last » Who We ServeMarket RiskQuantitative Research & DevelopmentSalesSoftware DevelopmentSoftware VendorsValuationsAccountingActuariesConsultingCounterparty Credit RiskIT & InfrastructureStructuringAudit/AdvisoryModel ValidationRegulationsTradingCollateral ManagementPortfolio ManagementXVA DeskFront Office & TradingRisk ManagementBack Offices & OperationsQuant & ActuariesTechnology & DevelopmentFinTech & Consultants What We OfferDevelopment PlatformExoticsValuationGraph FrameworkMarket RiskVanilla DerivativesFixed Income & Structured FinanceXVA & Counterparty RiskOTC TradingStructured Finance Trading & RiskBanking Asset Liability ManagementDerivatives & Fixed Income AnalyticsFront-to-Risk ApplicationsCapital Markets Development Platform No resources found. Pagination First page « First Previous page Previous Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Current page 8 Who We ServeMarket RiskQuantitative Research & DevelopmentSalesSoftware DevelopmentSoftware VendorsValuationsAccountingActuariesConsultingCounterparty Credit RiskIT & InfrastructureStructuringAudit/AdvisoryModel ValidationRegulationsTradingCollateral ManagementPortfolio ManagementXVA DeskFront Office & TradingRisk ManagementBack Offices & OperationsQuant & ActuariesTechnology & DevelopmentFinTech & Consultants What We OfferDevelopment PlatformExoticsValuationGraph FrameworkMarket RiskVanilla DerivativesFixed Income & Structured FinanceXVA & Counterparty RiskOTC TradingStructured Finance Trading & RiskBanking Asset Liability ManagementDerivatives & Fixed Income AnalyticsFront-to-Risk ApplicationsCapital Markets Development Platform No resources found. Pagination First page « First Previous page Previous Page 1 Current page 2 Who We ServeMarket RiskQuantitative Research & DevelopmentSalesSoftware DevelopmentSoftware VendorsValuationsAccountingActuariesConsultingCounterparty Credit RiskIT & InfrastructureStructuringAudit/AdvisoryModel ValidationRegulationsTradingCollateral ManagementPortfolio ManagementXVA DeskFront Office & TradingRisk ManagementBack Offices & OperationsQuant & ActuariesTechnology & DevelopmentFinTech & Consultants What We OfferDevelopment PlatformExoticsValuationGraph FrameworkMarket RiskVanilla DerivativesFixed Income & Structured FinanceXVA & Counterparty RiskOTC TradingStructured Finance Trading & RiskBanking Asset Liability ManagementDerivatives & Fixed Income AnalyticsFront-to-Risk ApplicationsCapital Markets Development Platform No resources found. Who We ServeMarket RiskQuantitative Research & DevelopmentSalesSoftware DevelopmentSoftware VendorsValuationsAccountingActuariesConsultingCounterparty Credit RiskIT & InfrastructureStructuringAudit/AdvisoryModel ValidationRegulationsTradingCollateral ManagementPortfolio ManagementXVA DeskFront Office & TradingRisk ManagementBack Offices & OperationsQuant & ActuariesTechnology & DevelopmentFinTech & Consultants What We OfferDevelopment PlatformExoticsValuationGraph FrameworkMarket RiskVanilla DerivativesFixed Income & Structured FinanceXVA & Counterparty RiskOTC TradingStructured Finance Trading & RiskBanking Asset Liability ManagementDerivatives & Fixed Income AnalyticsFront-to-Risk ApplicationsCapital Markets Development Platform No resources found. Pagination First page « First Previous page Previous Page 1 Current page 2 Who We ServeMarket RiskQuantitative Research & DevelopmentSalesSoftware DevelopmentSoftware VendorsValuationsAccountingActuariesConsultingCounterparty Credit RiskIT & InfrastructureStructuringAudit/AdvisoryModel ValidationRegulationsTradingCollateral ManagementPortfolio ManagementXVA DeskFront Office & TradingRisk ManagementBack Offices & OperationsQuant & ActuariesTechnology & DevelopmentFinTech & Consultants What We OfferDevelopment PlatformExoticsValuationGraph FrameworkMarket RiskVanilla DerivativesFixed Income & Structured FinanceXVA & Counterparty RiskOTC TradingStructured Finance Trading & RiskBanking Asset Liability ManagementDerivatives & Fixed Income AnalyticsFront-to-Risk ApplicationsCapital Markets Development Platform No resources found.
webinar To Cloud or Not to Cloud: Shifting Attitudes Around Cloud Adoption Paul Sinthunont, Strategic Advisor at Aite-Novarica Group unveil key findings from one-on-one interviews with key stakeholders across the financial services sector uncovering what’s driving—or holding back—cloud migration decisions for trading and risk management functions. Register Now
webinar Estimating Cross-Model Correlations for CCR & XVA In this webinar, Numerix’s Andrew McClelland, SVP of Quantitative Research, shares detailed insights on Estimating Cross-Model Correlations for CCR & XVA. Register Now
article Digitalization and New Tech Bring Opportunity and Changing Approaches to Derivatives Markets In this article Numerix technologist, Linus Yu, joins Heads of Collateral and Liquidity Management, Treasury and Market Risk at some of the world’s leading banks to discuss where they see the path forward for building a more impactful future-state infrastructure for a derivatives business. Read article
analyst report Aite-Novarica Report In new research report Audrey Blater of the Aite-Novarica Group shares key findings from her one-on-one interviews with key stakeholders across the financial services sector to uncover what’s driving cloud migration decisions for trading and risk management functions.The report analyzes the factors influencing banks when it comes to deciding whether or not to migrate to the cloud and discusses what is needed to bring a pro-cloud shift into the decision-making process. The report also provides views on why on-premise legacy systems may or may not be preferred. Read analyst report
webinar Transforming Treasury and Derivatives management post Covid-19 In this webinar, Numerix and Risk.net, will examine new strategies to enhance your derivatives business and treasury management in the face of Covid-19, regulatory changes and counterparty risk. Register Now
webinar Digitalisation of Derivatives Trading: Utilising Technology to Increase Profits In this webinar, learn how new technologies, such as AI and machine learning, can be leveraged to better manage several aspects of a derivatives business. Register Now
webinar Deep Dive: Advances in Counterparty Credit Modelling in Energy Markets Numerix’s Andrew McClelland, SVP of Quantitative Research, addresses natural gas and electricity curves and the dynamics that complicate modelling, Register Now
webinar XVAs and Counterparty Credit Risk for Energy Markets – Addressing the Challenges and Unravelling Complexity In this webinar, panel of quantitative researchers and risk practitioners from banks, energy firms, and a software vendor discuss the many practical challenges they’ve encountered in the modeling and risk management of XVAs/CCR in the energy markets, and how to overcome them. Register Now
journal issue Numerix Journal Vol. 7 No. 1 The Vol 7. No. 1 Issue of the Numerix Journal represents some of Numerix's quantitative research and development achievements lately. Many of these achievements have been implemented as functionalities in our products. With the ongoing R&D effort at Numerix, this collection of papers showcases the quality of the research on various topics of interest in the field. Readers get an update on the latest progress in the theoretical research and product enhancements at Numerix. Read journal issue
white paper New Technology Is Redefining the Success of the Front Office Numerix Chief Product Officer, Satyam Kancharla, discusses how firms are rethinking trading infrastructures Read white paper
white paper Risk.net: Next-generation technologies and the future of trading Panel of industry experts share their insights and observations regarding emerging technologies in trading Read white paper
webinar Advances in Counterparty Credit Risk Modelling in Energy Markets In this webinar, Numerix SVP of Quantitative Research, Andrew McClelland Ph.D., looked at what is being done to improve energy models inside the counterparty credit risk setting. Register Now
case study OCBC Bank Scales Business and Mitigates Risk with Numerix Oneview During the course of its nearly 10-year partnership with Numerix, OCBC Bank has faced a number of challenges it engaged Numerix to address. The bank required a platform that could help it meet demand for products with more innovative features within the structured products market and which integrated a sophisticated risk management process. View case study
white paper Real-Time Risk Management in Practice: The Experts’ Views This whitepaper provides an overview of some of the key topics discussed by a group of market experts during a roundtable webinar hosted by CubeLogic in partnership with Numerix and PRMIA. Various applications of risk management in real time were examined, including, among others. Read white paper
analyst report LIBOR Risk Q2 2021 As the deadline to Libor cessation approaches, Liang Wu, executive director of financial engineering and head of cross asset product management at Numerix, presents a series of market themes that warrant closer inspection. Read analyst report
webinar Modelling Energy Curves for XVA Watch Numerix’s SVP of Quantitative Research Andrew McClelland present updates to his latest research, "Modelling Energy Curves for XVA." Register Now
white paper The New Rules of Market Risk Management This paper highlights several of the key viewpoints expressed by the panel and explores some the current dynamics that are impacting approaches to and the complexities of market risk management. Read white paper
webinar Next generation technologies and the future of trading In this webinar, Numerix and Risk.net panelists examine the future of trading technology, how companies will implement these new innovations, and explore the range of new skills that might be needed. Register Now
quantitative research USLV: Unspanned Stochastic Local Volatility Model In this article, we propose a new framework for modeling stochastic local volatility, with potential applications to modeling derivatives on interest rates, commodities, credit, equity, FX etc., as well as hybrid derivatives. Read quantitative research
quantitative research Exposure & CVA for Large Portfolios of Vanilla Swaps: The Thin-Out Optimization In this article we present an efficient optimization for calculating the exposure and CVA for large portfolios of vanilla swaps. Read quantitative research
blog Advanced FX Hedging Strategies (and what NOT to do): A Case Study Learn advanced FX hedging strategies and what is most effective. Read Blog
quantitative research Algorithmic Exposure and CVA for Exotic Derivatives In this article, we develop the algorithmic approach for Counterparty exposure calculation and automate its application to arbitrary complicated instruments. Read quantitative research
blog Capturing Stochastic Volatility: Key to Trading Inflation Derivatives Modeling the smile and capturing the stochastic nature of volatility has become critically important for inflation derivatives trading. Read Blog
quantitative research Analytical Approximations for Short Rate Models In this article, we present the analytical approximation of zero-coupon bonds and swaption prices for general short rate models. Read quantitative research
blog Bates Model and Cliquet Pricing in Numerix Bates stochastic volatility jump-diffusion model is the market standard model for pricing exotic options that depend heavily on the forward skew, such as cliquets and other forward-starting trades. Read Blog
quantitative research Generalized Vanna-Volga Method and Its Applications In this article, we give a general treatment of the Vanna-Volga mark-to-market volatility smile correction in application to pricing of contracts with European exercise on a single underlying. Read quantitative research
quantitative research Decoupled American Option Pricing Method: Computation of Implied Volatilities and Further Applications In this article, we introduce a method for volatility computation from listed prices of American options on an underlying close to log-normal. Read quantitative research
quantitative research Dynamic Model for Pricing and Hedging Heterogenous CDOs In this article, we present a simple bottom-up dynamic credit model that can be calibrated simultaneously to the market quotes on CDO tranches and individual CDSs constituting the credit portfolio. Read quantitative research
quantitative research Analytical Formulas for Pricing CMS Products in the LMM with Stochastic Volatility In this paper, we develop a series of approximations for a fast analytical pricing of European constant maturity swap (CMS) products, such as CMS swaps, CMS caps/floors, and CMS spread options, for the LIBOR Market Model (LMM) with stochastic volatility. Read quantitative research
quantitative research Projection on a Quadratic Model by Asymptotic Expansion with an Application to LMM Swaption In this article, we develop a technique of parameter averaging and Markovian projection on a quadratic volatility model based on a term-by-term matching of the asymptotic expansions of option prices in volatilities. Read quantitative research
quantitative research Markovian Projection to a Displaced Volatility Heston Model In this article, we generalize the results of Markovian Projection onto a Heston model to a wider class of approximating models, a Heston model with displaced volatility. Read quantitative research
quantitative research Two-Dimensional Markovian Model for Dynamics of Aggregate Credit Loss In this article, we propose a new model for the dynamics of the aggregate credit portfolio loss. The model is Markovian in two dimensions with the state variables being the total accumulated loss Lt and the stochastic default intensity λt. Read quantitative research
quantitative research Markovian Projection onto a Displaced Diffusion In this paper, we develop a systematic approach to Markovian projection onto an effective displaced diffusion, and work out a set of computationally efficient formulas valid for a large class of non-Markovian underlying processes. Read quantitative research
quantitative research Markovian Projection Onto a Heston Model In this article, we develop a systematic approach to the reduction of dimensionality of smile-enabled models by projecting them onto a displaced version of the two-dimensional Heston process. Read quantitative research
quantitative research Efficient Calibration to FX Options by Markovian Projection in Cross-Currency LIBOR Market Models In this article, we revisit the cross-currency LIBOR Market Model armed with the technique of Markovian projection. Read quantitative research
quantitative research Overlapping Credit Portfolios In this article, we present an accurate analytical approximation for a joint distribution function of loss of two overlapping credit portfolios using the multidimensional saddlepoint method. Read quantitative research