Download or read book Credit Risk Modeling Valuation and Hedging written by Tomasz R. Bielecki and published by Springer Science & Business Media. This book was released on 2004-01-22 with total page 524 pages. Available in PDF, EPUB and Kindle. Book excerpt: The motivation for the mathematical modeling studied in this text on developments in credit risk research is the bridging of the gap between mathematical theory of credit risk and the financial practice. Mathematical developments are covered thoroughly and give the structural and reduced-form approaches to credit risk modeling. Included is a detailed study of various arbitrage-free models of default term structures with several rating grades.

Download or read book Econometric Methods and Their Applications in Finance Macro and Related Fields written by Kaddour Hadri and published by World Scientific. This book was released on 2014 with total page 616 pages. Available in PDF, EPUB and Kindle. Book excerpt: The volume aims at providing an outlet for some of the best papers presented at the 15th Annual Conference of the African Econometric Society, which is one of the OC chaptersOCO of the International Econometric Society. Many of these papers represent the state of the art in financial econometrics and applied econometric modeling, and some also provide useful simulations that shed light on the models'' ability to generate meaningful scenarios for forecasting and policy analysis. Contents: Financial Econometrics and International Finance: Modeling Interest Rates Using Reducible Stochastic Differential Equations: A Copula-Based Multivariate Approach (Ruijun Bu, Ludovic Giet, Kaddour Hadri and Michel Lubrano); Financial Risk Management Using Asymmetric Heavy-Tailed Distribution and Nonlinear Dependence Structures of Asset Returns Under Discontinuous Dynamics (Alaa El-Shazly); Modeling Time-Varying Dependence in the Term Structure of Interest Rates (Diaa Noureldin); Nonlinear Filtering and Market Implied Rating for a Jump-Diffusion Structural Model of Credit Risk (Alaa El-Shazly); Time-Varying Optimal Weights for International Asset Allocation in African and South Asian Markets (Dalia El-Edel); Econometric Theory and Methods: Econometric Methods for Ordered Responses: Some Recent Developments (Franco Peracchi); Which Quantile Is the Most Informative? Maximum Likelihood, Maximum Entropy and Quantile Regression (Anil K Bera, Antonio F Galvao Jr., Gabriel V Montes-Rojas, Sung Y Park); The Experimetrics of Fairness (Anna Conte and Peter Moffatt); Uniform in Bandwidth Tests of Specification for Conditional Moment Restrictions Models (Pascal Lavergne and Pierre Nguimkeu); Joint LM Test for Homoscedasticity in a Two Way Error Components Model (Eugene Kouassi, Joel Sango, J M BossonBrou and Kern O Kymn); An Approximation to the Distribution of the Pooled Estimator When the Time Series Equation Is One of a Complete System (Ghazal Amer and William Mikhail); Monetary, Labor, Environmental and Other Econometric Applications: Monetary Policy and the Role of the Exchange Rate in Egypt (Tarek Morsi and Mai El-Mossallamy); International Migration, Remittances and Household Poverty Status in Egypt (Rania Roushdy, Ragui Assaad and Ali Rashed); Determinants of Job Quality and Wages of the Working Poor: Evidence From 1998OCo2006 Egypt Labor Market Panel Survey (Mona Said); A Contract-Theoretic Model of Conservation Agreements (Heidi Gjertsen, Theodore Groves, David A Miller, Eduard Niesten, Dale Squires and Joel Watson); Household Environment and Child Health in Egypt (Mahmoud Hailat and Franco Peracchi); Modeling the Relationship between Natural Resource Abundance, Economic Growth, and the Environment: A Cross-Country Study (Hala Abou-Ali and Yasmine M Abdelfattah); Global Cement Industry: Competitive and Institutional Frameworks (Tarek H Selim and Ahmed S Salem); On the Occurrence of Ponzi Schemes in Presence of Credit Restrictions Penalizing Default (Abdelkrim Seghir); Is Targeted Advertising Always Beneficial? (Nada Ben Elhadj-Ben Brahim, Rim Lahmandi-Ayed and Didier Laussel). Readership: Graduate students and researchers in the fields of econometrics, economic theory, applied econometrics.

Download or read book Credit Risk Valuation written by Manuel Ammann and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 259 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers an advanced introduction to models of credit risk valuation, concentrating on firm-value and reduced-form approaches and their application. Also included are new models for valuing derivative securities with credit risk. The book provides detailed descriptions of the state-of-the-art martingale methods and advanced numerical implementations based on multivariate trees used to price derivative credit risk. Numerical examples illustrate the effects of credit risk on the prices of financial derivatives.

Download or read book Financial Modelling with Jump Processes written by Peter Tankov and published by CRC Press. This book was released on 2003-12-30 with total page 552 pages. Available in PDF, EPUB and Kindle. Book excerpt: WINNER of a Riskbook.com Best of 2004 Book Award! During the last decade, financial models based on jump processes have acquired increasing popularity in risk management and option pricing. Much has been published on the subject, but the technical nature of most papers makes them difficult for nonspecialists to understand, and the mathematic

Download or read book Credit Risk Modeling written by David Lando and published by Princeton University Press. This book was released on 2009-12-13 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: Credit risk is today one of the most intensely studied topics in quantitative finance. This book provides an introduction and overview for readers who seek an up-to-date reference to the central problems of the field and to the tools currently used to analyze them. The book is aimed at researchers and students in finance, at quantitative analysts in banks and other financial institutions, and at regulators interested in the modeling aspects of credit risk. David Lando considers the two broad approaches to credit risk analysis: that based on classical option pricing models on the one hand, and on a direct modeling of the default probability of issuers on the other. He offers insights that can be drawn from each approach and demonstrates that the distinction between the two approaches is not at all clear-cut. The book strikes a fruitful balance between quickly presenting the basic ideas of the models and offering enough detail so readers can derive and implement the models themselves. The discussion of the models and their limitations and five technical appendixes help readers expand and generalize the models themselves or to understand existing generalizations. The book emphasizes models for pricing as well as statistical techniques for estimating their parameters. Applications include rating-based modeling, modeling of dependent defaults, swap- and corporate-yield curve dynamics, credit default swaps, and collateralized debt obligations.

Download or read book The Oxford Handbook of Credit Derivatives written by Alexander Lipton and published by OUP Oxford. This book was released on 2013-01-17 with total page 828 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the late 1990s, the spectacular growth of a secondary market for credit through derivatives has been matched by the emergence of mathematical modelling analysing the credit risk embedded in these contracts. This book aims to provide a broad and deep overview of this modelling, covering statistical analysis and techniques, modelling of default of both single and multiple entities, counterparty risk, Gaussian and non-Gaussian modelling, and securitisation. Both reduced-form and firm-value models for the default of single entities are considered in detail, with extensive discussion of both their theoretical underpinnings and practical usage in pricing and risk. For multiple entity modelling, the now notorious Gaussian copula is discussed with analysis of its shortcomings, as well as a wide range of alternative approaches including multivariate extensions to both firm-value and reduced form models, and continuous-time Markov chains. One important case of multiple entities modelling - counterparty risk in credit derivatives - is further explored in two dedicated chapters. Alternative non-Gaussian approaches to modelling are also discussed, including extreme-value theory and saddle-point approximations to deal with tail risk. Finally, the recent growth in securitisation is covered, including house price modelling and pricing models for asset-backed CDOs. The current credit crisis has brought modelling of the previously arcane credit markets into the public arena. Lipton and Rennie with their excellent team of contributors, provide a timely discussion of the mathematical modelling that underpins both credit derivatives and securitisation. Though technical in nature, the pros and cons of various approaches attempt to provide a balanced view of the role that mathematical modelling plays in the modern credit markets. This book will appeal to students and researchers in statistics, economics, and finance, as well as practitioners, credit traders, and quantitative analysts

Download or read book Systemic Contingent Claims Analysis written by Mr.Andreas A. Jobst and published by International Monetary Fund. This book was released on 2013-02-27 with total page 93 pages. Available in PDF, EPUB and Kindle. Book excerpt: The recent global financial crisis has forced a re-examination of risk transmission in the financial sector and how it affects financial stability. Current macroprudential policy and surveillance (MPS) efforts are aimed establishing a regulatory framework that helps mitigate the risk from systemic linkages with a view towards enhancing the resilience of the financial sector. This paper presents a forward-looking framework ("Systemic CCA") to measure systemic solvency risk based on market-implied expected losses of financial institutions with practical applications for the financial sector risk management and the system-wide capital assessment in top-down stress testing. The suggested approach uses advanced contingent claims analysis (CCA) to generate aggregate estimates of the joint default risk of multiple institutions as a conditional tail expectation using multivariate extreme value theory (EVT). In addition, the framework also helps quantify the individual contributions to systemic risk and contingent liabilities of the financial sector during times of stress.

Download or read book Term Structure Models written by Damir Filipovic and published by Springer Science & Business Media. This book was released on 2009-07-28 with total page 259 pages. Available in PDF, EPUB and Kindle. Book excerpt: Changing interest rates constitute one of the major risk sources for banks, insurance companies, and other financial institutions. Modeling the term-structure movements of interest rates is a challenging task. This volume gives an introduction to the mathematics of term-structure models in continuous time. It includes practical aspects for fixed-income markets such as day-count conventions, duration of coupon-paying bonds and yield curve construction; arbitrage theory; short-rate models; the Heath-Jarrow-Morton methodology; consistent term-structure parametrizations; affine diffusion processes and option pricing with Fourier transform; LIBOR market models; and credit risk. The focus is on a mathematically straightforward but rigorous development of the theory. Students, researchers and practitioners will find this volume very useful. Each chapter ends with a set of exercises, that provides source for homework and exam questions. Readers are expected to be familiar with elementary Itô calculus, basic probability theory, and real and complex analysis.

Download or read book Mathematical and Statistical Methods for Actuarial Sciences and Finance written by Marco Corazza and published by Springer. This book was released on 2014-08-06 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: The interaction between mathematicians and statisticians has been shown to be an effective approach for dealing with actuarial, insurance and financial problems, both from an academic perspective and from an operative one. The collection of original papers presented in this volume pursues precisely this purpose. It covers a wide variety of subjects in actuarial, insurance and finance fields, all treated in the light of the successful cooperation between the above two quantitative approaches. The papers published in this volume present theoretical and methodological contributions and their applications to real contexts. With respect to the theoretical and methodological contributions, some of the considered areas of investigation are: actuarial models; alternative testing approaches; behavioral finance; clustering techniques; coherent and non-coherent risk measures; credit scoring approaches; data envelopment analysis; dynamic stochastic programming; financial contagion models; financial ratios; intelligent financial trading systems; mixture normality approaches; Monte Carlo-based methods; multicriteria methods; nonlinear parameter estimation techniques; nonlinear threshold models; particle swarm optimization; performance measures; portfolio optimization; pricing methods for structured and non-structured derivatives; risk management; skewed distribution analysis; solvency analysis; stochastic actuarial valuation methods; variable selection models; time series analysis tools. As regards the applications, they are related to real problems associated, among the others, to: banks; collateralized fund obligations; credit portfolios; defined benefit pension plans; double-indexed pension annuities; efficient-market hypothesis; exchange markets; financial time series; firms; hedge funds; non-life insurance companies; returns distributions; socially responsible mutual funds; unit-linked contracts. This book is aimed at academics, Ph.D. students, practitioners, professionals and researchers. But it will also be of interest to readers with some quantitative background knowledge.

Download or read book Specification Analysis of Structural Credit Risk Models written by Jing-zhi Huang and published by . This book was released on 2008 with total page 56 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Credit Risk written by Darrell Duffie and published by Princeton University Press. This book was released on 2012-01-12 with total page 415 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, two of America's leading economists provide the first integrated treatment of the conceptual, practical, and empirical foundations for credit risk pricing and risk measurement. Masterfully applying theory to practice, Darrell Duffie and Kenneth Singleton model credit risk for the purpose of measuring portfolio risk and pricing defaultable bonds, credit derivatives, and other securities exposed to credit risk. The methodological rigor, scope, and sophistication of their state-of-the-art account is unparalleled, and its singularly in-depth treatment of pricing and credit derivatives further illuminates a problem that has drawn much attention in an era when financial institutions the world over are revising their credit management strategies. Duffie and Singleton offer critical assessments of alternative approaches to credit-risk modeling, while highlighting the strengths and weaknesses of current practice. Their approach blends in-depth discussions of the conceptual foundations of modeling with extensive analyses of the empirical properties of such credit-related time series as default probabilities, recoveries, ratings transitions, and yield spreads. Both the "structura" and "reduced-form" approaches to pricing defaultable securities are presented, and their comparative fits to historical data are assessed. The authors also provide a comprehensive treatment of the pricing of credit derivatives, including credit swaps, collateralized debt obligations, credit guarantees, lines of credit, and spread options. Not least, they describe certain enhancements to current pricing and management practices that, they argue, will better position financial institutions for future changes in the financial markets. Credit Risk is an indispensable resource for risk managers, traders or regulators dealing with financial products with a significant credit risk component, as well as for academic researchers and students.

Download or read book Financial Surveillance written by Marianne Frisen and published by John Wiley & Sons. This book was released on 2008-02-28 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first book-length treatment of statistical surveillance methods used in financial analysis. It contains carefully selected chapters written by specialists from both fields and strikes a balance between the financial and statistical worlds, enhancing future collaborations between the two areas, and enabling more successful prediction of financial market trends. The book discusses, in detail, schemes for different control charts and different linear and nonlinear time series models and applies methods to real data from worldwide markets, as well as including simulation studies.

Download or read book From Probability to Finance written by Ying Jiao and published by Springer Nature. This book was released on 2020-03-20 with total page 253 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume presents a collection of lecture notes of mini-courses taught at BICMR Summer School of Financial Mathematics, from May 29 to June 9, 2017. Each chapter is self-contained and corresponds to one mini-course which deals with a distinguished topic, such as branching processes, enlargement of filtrations, Hawkes processes, copula models and valuation adjustment analysis, whereas the global topics cover a wide range of advanced subjects in financial mathematics, from both theoretical and practical points of view. The authors include world-leading specialists in the domain and also young active researchers. This book will be helpful for students and those who work on probability and financial mathematics.

Download or read book Sustainable Statistical and Data Science Methods and Practices written by O. Olawale Awe and published by Springer Nature. This book was released on 2024-01-05 with total page 433 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume gathers papers presented at the LISA 2020 Sustainability Symposium in Kumasi, Ghana, May 2–6, 2022. They focus on sustainable methods and practices of using statistics and data science to address real-world problems. From utilizing social media for statistical collaboration to predicting obesity among rural women, and from analyzing inflation in Nigeria using machine learning to teaching data science in Africa, this book explores the intersection of data, statistics, and sustainability. With practical applications, code snippets, and case studies, this book offers valuable insights for researchers, policymakers, and data enthusiasts alike. The LISA 2020 Global Network aims to enhance statistical and data science capability in developing countries through the creation of a network of collaboration laboratories (also known as “stat labs”). These stat labs are intended to serve as engines for development by training the next generation of collaborative statisticians and data scientists, providing research infrastructure for researchers, data producers, and decision-makers, and enabling evidence-based decision-making that has a positive impact on society. The research conducted at LISA 2020 focuses on practical methods and applications for sustainable growth of statistical capacity in developing nations.

Download or read book Optimization Based Models for Measuring and Hedging Risk in Fixed Income Markets written by Johan Hagenbjörk and published by Linköping University Electronic Press. This book was released on 2019-12-09 with total page 156 pages. Available in PDF, EPUB and Kindle. Book excerpt: The global fixed income market is an enormous financial market whose value by far exceeds that of the public stock markets. The interbank market consists of interest rate derivatives, whose primary purpose is to manage interest rate risk. The credit market primarily consists of the bond market, which links investors to companies, institutions, and governments with borrowing needs. This dissertation takes an optimization perspective upon modeling both these areas of the fixed-income market. Legislators on the national markets require financial actors to value their financial assets in accordance with market prices. Thus, prices of many assets, which are not publicly traded, must be determined mathematically. The financial quantities needed for pricing are not directly observable but must be measured through solving inverse optimization problems. These measurements are based on the available market prices, which are observed with various degrees of measurement noise. For the interbank market, the relevant financial quantities consist of term structures of interest rates, which are curves displaying the market rates for different maturities. For the bond market, credit risk is an additional factor that can be modeled through default intensity curves and term structures of recovery rates in case of default. By formulating suitable optimization models, the different underlying financial quantities can be measured in accordance with observable market prices, while conditions for economic realism are imposed. Measuring and managing risk is closely connected to the measurement of the underlying financial quantities. Through a data-driven method, we can show that six systematic risk factors can be used to explain almost all variance in the interest rate curves. By modeling the dynamics of these six risk factors, possible outcomes can be simulated in the form of term structure scenarios. For short-term simulation horizons, this results in a representation of the portfolio value distribution that is consistent with the realized outcomes from historically observed term structures. This enables more accurate measurements of interest rate risk, where our proposed method exhibits both lower risk and lower pricing errors compared to traditional models. We propose a method for decomposing changes in portfolio values for an arbitrary portfolio into the risk factors that affect the value of each instrument. By demonstrating the method for the six systematic risk factors identified for the interbank market, we show that almost all changes in portfolio value and portfolio variance can be attributed to these risk factors. Additional risk factors and approximation errors are gathered into two terms, which can be studied to ensure the quality of the performance attribution, and possibly improve it. To eliminate undesired risk within trading books, banks use hedging. Traditional methods do not take transaction costs into account. We, therefore, propose a method for managing the risks in the interbank market through a stochastic optimization model that considers transaction costs. This method is based on a scenario approximation of the optimization problem where the six systematic risk factors are simulated, and the portfolio variance is weighted against the transaction costs. This results in a method that is preferred over the traditional methods for all risk-averse investors. For the credit market, we use data from the bond market in combination with the interbank market to make accurate measurements of the financial quantities. We address the notoriously difficult problem of separating default risk from recovery risk. In addition to the previous identified six systematic risk factors for risk-free interests, we identify four risk factors that explain almost all variance in default intensities, while a single risk factor seems sufficient to model the recovery risk. Overall, this is a higher number of risk factors than is usually found in the literature. Through a simple model, we can measure the variance in bond prices in terms of these systematic risk factors, and through performance attribution, we relate these values to the empirically realized variances from the quoted bond prices. De globala ränte- och kreditmarknaderna är enorma finansiella marknader vars sammanlagda värden vida överstiger de publika aktiemarknadernas. Räntemarknaden består av räntederivat vars främsta användningsområde är hantering av ränterisker. Kreditmarknaden utgörs i första hand av obligationsmarknaden som syftar till att förmedla pengar från investerare till företag, institutioner och stater med upplåningsbehov. Denna avhandling fokuserar på att utifrån ett optimeringsperspektiv modellera både ränte- och obligationsmarknaden. Lagstiftarna på de nationella marknaderna kräver att de finansiella aktörerna värderar sina finansiella tillgångar i enlighet med marknadspriser. Därmed måste priserna på många instrument, som inte handlas publikt, beräknas matematiskt. De finansiella storheter som krävs för denna prissättning är inte direkt observerbara, utan måste mätas genom att lösa inversa optimeringsproblem. Dessa mätningar görs utifrån tillgängliga marknadspriser, som observeras med varierande grad av mätbrus. För räntemarknaden utgörs de relevanta finansiella storheterna av räntekurvor som åskådliggör marknadsräntorna för olika löptider. För obligationsmarknaden utgör kreditrisken en ytterligare faktor som modelleras via fallissemangsintensitetskurvor och kurvor kopplade till förväntat återvunnet kapital vid eventuellt fallissemang. Genom att formulera lämpliga optimeringsmodeller kan de olika underliggande finansiella storheterna mätas i enlighet med observerbara marknadspriser samtidigt som ekonomisk realism eftersträvas. Mätning och hantering av risker är nära kopplat till mätningen av de underliggande finansiella storheterna. Genom en datadriven metod kan vi visa att sex systematiska riskfaktorer kan användas för att förklara nästan all varians i räntekurvorna. Genom att modellera dynamiken i dessa sex riskfaktorer kan tänkbara utfall för räntekurvor simuleras. För kortsiktiga simuleringshorisonter resulterar detta i en representation av fördelningen av portföljvärden som väl överensstämmer med de realiserade utfallen från historiskt observerade räntekurvor. Detta möjliggör noggrannare mätningar av ränterisk där vår föreslagna metod uppvisar såväl lägre risk som mindre prissättningsfel jämfört med traditionella modeller. Vi föreslår en metod för att dekomponera portföljutvecklingen för en godtycklig portfölj till de riskfaktorer som påverkar värdet för respektive instrument. Genom att demonstrera metoden för de sex systematiska riskfaktorerna som identifierats för räntemarknaden visar vi att nästan all portföljutveckling och portföljvarians kan härledas till dessa riskfaktorer. Övriga riskfaktorer och approximationsfel samlas i två termer, vilka kan användas för att säkerställa och eventuellt förbättra kvaliteten i prestationshärledningen. För att eliminera oönskad risk i sina tradingböcker använder banker sig av hedging. Traditionella metoder tar ingen hänsyn till transaktionskostnader. Vi föreslår därför en metod för att hantera riskerna på räntemarknaden genom en stokastisk optimeringsmodell som också tar hänsyn till transaktionskostnader. Denna metod bygger på en scenarioapproximation av optimeringsproblemet där de sex systematiska riskfaktorerna simuleras och portföljvariansen vägs mot transaktionskostnaderna. Detta resulterar i en metod som, för alla riskaverta investerare, är att föredra framför de traditionella metoderna. På kreditmarknaden använder vi data från obligationsmarknaden i kombination räntemarknaden för att göra noggranna mätningar av de finansiella storheterna. Vi angriper det erkänt svåra problemet att separera fallissemangsrisk från återvinningsrisk. Förutom de tidigare sex systematiska riskfaktorerna för riskfri ränta, identifierar vi fyra riskfaktorer som förklarar nästan all varians i fallissemangsintensiteter, medan en enda riskfaktor tycks räcka för att modellera återvinningsrisken. Sammanlagt är detta ett större antal riskfaktorer än vad som brukar användas i litteraturen. Via en enkel modell kan vi mäta variansen i obligationspriser i termer av dessa systematiska riskfaktorer och genom prestationshärledningen relatera dessa värden till de empiriskt realiserade varianserna från kvoterade obligationspriser.

Download or read book Advanced Bond Portfolio Management written by Frank J. Fabozzi and published by John Wiley & Sons. This book was released on 2006-03-08 with total page 578 pages. Available in PDF, EPUB and Kindle. Book excerpt: In order to effectively employ portfolio strategies that can control interest rate risk and/or enhance returns, you must understand the forces that drive bond markets, as well as the valuation and risk management practices of these complex securities. In Advanced Bond Portfolio Management, Frank Fabozzi, Lionel Martellini, and Philippe Priaulet have brought together more than thirty experienced bond market professionals to help you do just that. Divided into six comprehensive parts, Advanced Bond Portfolio Management will guide you through the state-of-the-art techniques used in the analysis of bonds and bond portfolio management. Topics covered include: General background information on fixed-income markets and bond portfolio strategies The design of a strategy benchmark Various aspects of fixed-income modeling that will provide key ingredients in the implementation of an efficient portfolio and risk management process Interest rate risk and credit risk management Risk factors involved in the management of an international bond portfolio Filled with in-depth insight and expert advice, Advanced Bond Portfolio Management is a valuable resource for anyone involved or interested in this important industry.

Download or read book Handbook of Computational Finance written by Jin-Chuan Duan and published by Springer Science & Business Media. This book was released on 2011-10-25 with total page 791 pages. Available in PDF, EPUB and Kindle. Book excerpt: Any financial asset that is openly traded has a market price. Except for extreme market conditions, market price may be more or less than a “fair” value. Fair value is likely to be some complicated function of the current intrinsic value of tangible or intangible assets underlying the claim and our assessment of the characteristics of the underlying assets with respect to the expected rate of growth, future dividends, volatility, and other relevant market factors. Some of these factors that affect the price can be measured at the time of a transaction with reasonably high accuracy. Most factors, however, relate to expectations about the future and to subjective issues, such as current management, corporate policies and market environment, that could affect the future financial performance of the underlying assets. Models are thus needed to describe the stochastic factors and environment, and their implementations inevitably require computational finance tools.