Holdings: Financial Mathematics

Financial Mathematics: A Comprehensive Treatment in Continuous Time Volume II.

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Bibliographic Details
Main Author:	Campolieti, Giuseppe (Author)
Format:	Electronic eBook
Language:	English
Published:	Milton CRC Press LLC 2022
Series:	Textbooks in Mathematics Ser
Subjects:	Electronic books
Online Access:	HWR01
Physical Description:	1 Online-Ressource (511 Seiten)
ISBN:	9780429889097 9781138603639

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505	8		\|a Cover -- Half Title -- Series Page -- Title Page -- Copyright Page -- Dedication -- Contents -- List of Figures -- Preface -- Authors -- I. Stochastic Calculus with Brownian Motion -- 1. One-Dimensional Brownian Motion and Related Processes -- 1.1. Multivariate Normal Distributions -- 1.1.1. Multivariate Normal Distribution -- 1.1.2. Conditional Normal Distributions -- 1.2. Standard Brownian Motion -- 1.2.1. One-Dimensional Symmetric Random Walk -- 1.2.2. Formal Definition and Basic Properties of Brownian Motion -- 1.2.3. Multivariate Distribution of Brownian Motion -- 1.2.4. The Markov Property and the Transition PDF -- 1.2.5. Quadratic Variation and Nondifferentiability of Paths -- 1.3. Some Processes Derived from Brownian Motion -- 1.3.1. Drifted Brownian Motion -- 1.3.2. Geometric Brownian Motion -- 1.3.3. Processes related by a monotonic mapping -- 1.3.4. Brownian Bridge -- 1.3.5. Gaussian Processes -- 1.4. First Hitting Times and Maximum and Minimum of Brownian Motion -- 1.4.1. The Reflection Principle: Standard Brownian Motion -- 1.4.2. Translated and Scaled Driftless Brownian Motion -- 1.4.3. Brownian Motion with Drift -- 1.5. Exercises -- 2. Introduction to Continuous-Time Stochastic Calculus -- 2.1. The Riemann Integral of Brownian Motion -- 2.1.1. The Riemann Integral -- 2.1.2. The Integral of a Brownian Path -- 2.2. The Riemann-Stieltjes Integral of Brownian Motion -- 2.2.1. The Riemann-Stieltjes Integral -- 2.2.2. Integrals w.r.t. Brownian Motion: Preliminary Discussion -- 2.3. The Itô Integral and Its Basic Properties -- 2.3.1. The Itô Integral for Simple Processes -- 2.3.2. The Itô Integral for General Processes -- 2.4. Itô Processes and Their Properties -- 2.4.1. Gaussian Processes Generated by Itô Integrals -- 2.4.2. Itô Processes -- 2.4.3. Quadratic and Co-Variation of Itô Processes
505	8		\|a 2.5. Itô's Formula for Functions of BM and Itô Processes -- 2.5.1. Itô's Formula for Functions of BM -- 2.5.2. An "Antiderivative" Formula for Evaluating Itô Integrals -- 2.5.3. Itô's Formula for Itô Processes -- 2.6. Stochastic Differential Equations -- 2.6.1. Solutions to Linear SDEs -- 2.6.2. Existence and Uniqueness of a Strong Solution to an SDE -- 2.7. The Markov Property, Martingales, Feynman-Kac Formulae, and Transition CDFs and PDFs -- 2.7.1. Forward Kolmogorov PDE -- 2.7.2. Transition CDF/PDF for Time-Homogeneous Diffusions -- 2.8. Radon-Nikodym Derivative Process and Girsanov's Theorem -- 2.8.1. Some Applications of Girsanov's Theorem -- 2.9. Brownian Martingale Representation Theorem -- 2.10. Stochastic Calculus for Multidimensional BM -- 2.10.1. The Itô Integral and Itô's Formula for Multiple Processes on Multidimensional BM -- 2.10.2. Multidimensional SDEs, Feynman-Kac Formulae, and Transition CDFs and PDFs -- 2.10.3. Girsanov's Theorem for Multidimensional BM -- 2.10.4. Martingale Representation Theorem for Multidimensional BM -- 2.11. Exercises -- II. Continuous-Time Modelling -- 3. Risk-Neutral Pricing in the (B, S) Economy: One Underlying Stock -- 3.1. From the CRR Model to the BSM Model -- 3.1.1. Portfolio Strategies in the Binomial Tree Model -- 3.1.2. The Cox-Ross-Rubinstein Model and its Continuous-Time Limit -- 3.2. Replication (Hedging) and Derivative Pricing in the Simplest Black-Scholes Economy -- 3.2.1. Pricing Standard European Calls and Puts -- 3.2.2. Hedging Standard European Calls and Puts -- 3.2.3. Europeans with Piecewise Linear Payoffs -- 3.2.4. Power Options -- 3.2.5. Dividend Paying Stock -- 3.2.6. Option Pricing with the Stock Numéraire -- 3.3. Forward Starting, Chooser, and Compound Options -- 3.4. Some European-Style Path-Dependent Derivatives -- 3.4.1. Risk-Neutral Pricing under GBM.
505	8		\|a 3.4.2. Pricing Single Barrier Options -- 3.4.3. Pricing Lookback Options -- 3.5. Structural Credit Risk Models -- 3.5.1. The Merton Model -- 3.5.2. The Black-Cox Model -- 3.6. Exercises -- 4. Risk-Neutral Pricing in a Multi-Asset Economy -- 4.1. General Multi-Asset Market Model: Replication and Risk-Neutral Pricing -- 4.2. Equivalent Martingale Measures: Derivative Pricing with General Numéraire Assets -- 4.3. Black-Scholes PDE and Delta Hedging for Standard Multi-Asset Derivatives -- 4.3.1. Standard European Option Pricing for Multi-Stock GBM -- 4.3.2. Explicit Pricing Formulae for the GBM Model -- 4.3.3. Cross-Currency Option Valuation -- 4.3.4. Option Valuation with General Numéraire Assets -- 4.4. Exercises -- 5. American Options -- 5.1. Basic Properties of Early-Exercise Options -- 5.2. Arbitrage-Free Pricing of American Options -- 5.2.1. Optimal Stopping Formulation and Early-Exercise Boundary -- 5.2.2. The Smooth Pasting Condition -- 5.2.3. Put-Call Symmetry Relation -- 5.2.4. Dynamic Programming Approach for Bermudan Options -- 5.3. Perpetual American Options -- 5.3.1. Pricing a Perpetual Put Option -- 5.3.2. Pricing a Perpetual Call Option -- 5.4. Finite-Expiration American Options -- 5.4.1. The PDE Formulation -- 5.4.2. The Integral Equation Formulation -- 5.5. Exercises -- 6. Interest-Rate Modelling and Derivative Pricing -- 6.1. Basic Fixed Income Instruments -- 6.1.1. Bonds -- 6.1.2. Forward Rates -- 6.1.3. Arbitrage-Free Pricing -- 6.1.4. Fixed Income Derivatives -- 6.2. Single-Factor Models -- 6.2.1. Diffusion Models for the Short Rate Process -- 6.2.2. PDE for the Zero-Coupon Bond Value -- 6.2.3. Affine Term Structure Models -- 6.2.4. The Ho-Lee Model -- 6.2.5. The Vasiček Model -- 6.2.6. The Cox-Ingersoll-Ross Model -- 6.3. Heath-Jarrow-Morton Formulation -- 6.3.1. HJM under Risk-Neutral Measure
505	8		\|a 6.3.2. Relationship between HJM and Affine Yield Models -- 6.4. Multifactor Affine Term Structure Models -- 6.4.1. Gaussian Multifactor Models -- 6.4.2. Equivalent Classes of Affine Models -- 6.5. Pricing Derivatives under Forward Measures -- 6.5.1. Forward Measures -- 6.5.2. Pricing Stock Options under Stochastic Interest Rates -- 6.5.3. Pricing Options on Zero-Coupon Bonds -- 6.6. LIBOR Model -- 6.6.1. LIBOR Rates -- 6.6.2. The Brace-Gatarek-Musiela Model of LIBOR Rates -- 6.6.3. Pricing Caplets, Caps, and Swaps -- 6.7. Exercises -- 7. Alternative Models of Asset Price Dynamics -- 7.1. Characteristic Functions -- 7.1.1. Definition and Properties -- 7.1.2. Recovering the Distribution Function -- 7.1.3. Pricing Standard European Options -- 7.1.4. The Carr-Madan Method for Pricing Vanilla Options -- 7.2. Stochastic Volatility Diffusion Models -- 7.2.1. Local Volatility Models -- 7.2.2. Constant Elasticity of Variance Model -- 7.3. The Heston model -- 7.3.1. Solution to the Ricatti Equation -- 7.3.2. Implied Volatility for the Heston Model -- 7.4. Models with Jumps -- 7.4.1. The Poisson Process -- 7.4.2. Jump Diffusion Models with a Compound Poisson Component -- 7.4.3. The Merton Jump Diffusion Model -- 7.4.4. Characteristic Function for a Jump Diffusion Process -- 7.4.5. Change of Measure for Jump Diffusion Processes -- 7.4.6. The Variance Gamma Model -- 7.5. Exercises -- A. Essentials of General Probability Theory -- A.1. Random Variables and Lebesgue Integration -- A.2. Multidimensional Lebesgue Integration -- A.3. Multiple Random Variables and Joint Distributions -- A.4. Conditioning -- A.5. Changing Probability Measures -- B. Some Useful Integral (Expectation) Identities and Symmetry Properties of Normal Random Variables -- C. Answers and Hints to Exercises -- C.1. Chapter 1 -- C.2. Chapter 2 -- C.3. Chapter 3 -- C.4. Chapter 4 -- C.5. Chapter 5
505	8		\|a C.6. Chapter 6 -- C.7. Chapter 7 -- D. Glossary of Symbols and Abbreviations -- References -- Index
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contents	Cover -- Half Title -- Series Page -- Title Page -- Copyright Page -- Dedication -- Contents -- List of Figures -- Preface -- Authors -- I. Stochastic Calculus with Brownian Motion -- 1. One-Dimensional Brownian Motion and Related Processes -- 1.1. Multivariate Normal Distributions -- 1.1.1. Multivariate Normal Distribution -- 1.1.2. Conditional Normal Distributions -- 1.2. Standard Brownian Motion -- 1.2.1. One-Dimensional Symmetric Random Walk -- 1.2.2. Formal Definition and Basic Properties of Brownian Motion -- 1.2.3. Multivariate Distribution of Brownian Motion -- 1.2.4. The Markov Property and the Transition PDF -- 1.2.5. Quadratic Variation and Nondifferentiability of Paths -- 1.3. Some Processes Derived from Brownian Motion -- 1.3.1. Drifted Brownian Motion -- 1.3.2. Geometric Brownian Motion -- 1.3.3. Processes related by a monotonic mapping -- 1.3.4. Brownian Bridge -- 1.3.5. Gaussian Processes -- 1.4. First Hitting Times and Maximum and Minimum of Brownian Motion -- 1.4.1. The Reflection Principle: Standard Brownian Motion -- 1.4.2. Translated and Scaled Driftless Brownian Motion -- 1.4.3. Brownian Motion with Drift -- 1.5. Exercises -- 2. Introduction to Continuous-Time Stochastic Calculus -- 2.1. The Riemann Integral of Brownian Motion -- 2.1.1. The Riemann Integral -- 2.1.2. The Integral of a Brownian Path -- 2.2. The Riemann-Stieltjes Integral of Brownian Motion -- 2.2.1. The Riemann-Stieltjes Integral -- 2.2.2. Integrals w.r.t. Brownian Motion: Preliminary Discussion -- 2.3. The Itô Integral and Its Basic Properties -- 2.3.1. The Itô Integral for Simple Processes -- 2.3.2. The Itô Integral for General Processes -- 2.4. Itô Processes and Their Properties -- 2.4.1. Gaussian Processes Generated by Itô Integrals -- 2.4.2. Itô Processes -- 2.4.3. Quadratic and Co-Variation of Itô Processes 2.5. Itô's Formula for Functions of BM and Itô Processes -- 2.5.1. Itô's Formula for Functions of BM -- 2.5.2. An "Antiderivative" Formula for Evaluating Itô Integrals -- 2.5.3. Itô's Formula for Itô Processes -- 2.6. Stochastic Differential Equations -- 2.6.1. Solutions to Linear SDEs -- 2.6.2. Existence and Uniqueness of a Strong Solution to an SDE -- 2.7. The Markov Property, Martingales, Feynman-Kac Formulae, and Transition CDFs and PDFs -- 2.7.1. Forward Kolmogorov PDE -- 2.7.2. Transition CDF/PDF for Time-Homogeneous Diffusions -- 2.8. Radon-Nikodym Derivative Process and Girsanov's Theorem -- 2.8.1. Some Applications of Girsanov's Theorem -- 2.9. Brownian Martingale Representation Theorem -- 2.10. Stochastic Calculus for Multidimensional BM -- 2.10.1. The Itô Integral and Itô's Formula for Multiple Processes on Multidimensional BM -- 2.10.2. Multidimensional SDEs, Feynman-Kac Formulae, and Transition CDFs and PDFs -- 2.10.3. Girsanov's Theorem for Multidimensional BM -- 2.10.4. Martingale Representation Theorem for Multidimensional BM -- 2.11. Exercises -- II. Continuous-Time Modelling -- 3. Risk-Neutral Pricing in the (B, S) Economy: One Underlying Stock -- 3.1. From the CRR Model to the BSM Model -- 3.1.1. Portfolio Strategies in the Binomial Tree Model -- 3.1.2. The Cox-Ross-Rubinstein Model and its Continuous-Time Limit -- 3.2. Replication (Hedging) and Derivative Pricing in the Simplest Black-Scholes Economy -- 3.2.1. Pricing Standard European Calls and Puts -- 3.2.2. Hedging Standard European Calls and Puts -- 3.2.3. Europeans with Piecewise Linear Payoffs -- 3.2.4. Power Options -- 3.2.5. Dividend Paying Stock -- 3.2.6. Option Pricing with the Stock Numéraire -- 3.3. Forward Starting, Chooser, and Compound Options -- 3.4. Some European-Style Path-Dependent Derivatives -- 3.4.1. Risk-Neutral Pricing under GBM. 3.4.2. Pricing Single Barrier Options -- 3.4.3. Pricing Lookback Options -- 3.5. Structural Credit Risk Models -- 3.5.1. The Merton Model -- 3.5.2. The Black-Cox Model -- 3.6. Exercises -- 4. Risk-Neutral Pricing in a Multi-Asset Economy -- 4.1. General Multi-Asset Market Model: Replication and Risk-Neutral Pricing -- 4.2. Equivalent Martingale Measures: Derivative Pricing with General Numéraire Assets -- 4.3. Black-Scholes PDE and Delta Hedging for Standard Multi-Asset Derivatives -- 4.3.1. Standard European Option Pricing for Multi-Stock GBM -- 4.3.2. Explicit Pricing Formulae for the GBM Model -- 4.3.3. Cross-Currency Option Valuation -- 4.3.4. Option Valuation with General Numéraire Assets -- 4.4. Exercises -- 5. American Options -- 5.1. Basic Properties of Early-Exercise Options -- 5.2. Arbitrage-Free Pricing of American Options -- 5.2.1. Optimal Stopping Formulation and Early-Exercise Boundary -- 5.2.2. The Smooth Pasting Condition -- 5.2.3. Put-Call Symmetry Relation -- 5.2.4. Dynamic Programming Approach for Bermudan Options -- 5.3. Perpetual American Options -- 5.3.1. Pricing a Perpetual Put Option -- 5.3.2. Pricing a Perpetual Call Option -- 5.4. Finite-Expiration American Options -- 5.4.1. The PDE Formulation -- 5.4.2. The Integral Equation Formulation -- 5.5. Exercises -- 6. Interest-Rate Modelling and Derivative Pricing -- 6.1. Basic Fixed Income Instruments -- 6.1.1. Bonds -- 6.1.2. Forward Rates -- 6.1.3. Arbitrage-Free Pricing -- 6.1.4. Fixed Income Derivatives -- 6.2. Single-Factor Models -- 6.2.1. Diffusion Models for the Short Rate Process -- 6.2.2. PDE for the Zero-Coupon Bond Value -- 6.2.3. Affine Term Structure Models -- 6.2.4. The Ho-Lee Model -- 6.2.5. The Vasiček Model -- 6.2.6. The Cox-Ingersoll-Ross Model -- 6.3. Heath-Jarrow-Morton Formulation -- 6.3.1. HJM under Risk-Neutral Measure 6.3.2. Relationship between HJM and Affine Yield Models -- 6.4. Multifactor Affine Term Structure Models -- 6.4.1. Gaussian Multifactor Models -- 6.4.2. Equivalent Classes of Affine Models -- 6.5. Pricing Derivatives under Forward Measures -- 6.5.1. Forward Measures -- 6.5.2. Pricing Stock Options under Stochastic Interest Rates -- 6.5.3. Pricing Options on Zero-Coupon Bonds -- 6.6. LIBOR Model -- 6.6.1. LIBOR Rates -- 6.6.2. The Brace-Gatarek-Musiela Model of LIBOR Rates -- 6.6.3. Pricing Caplets, Caps, and Swaps -- 6.7. Exercises -- 7. Alternative Models of Asset Price Dynamics -- 7.1. Characteristic Functions -- 7.1.1. Definition and Properties -- 7.1.2. Recovering the Distribution Function -- 7.1.3. Pricing Standard European Options -- 7.1.4. The Carr-Madan Method for Pricing Vanilla Options -- 7.2. Stochastic Volatility Diffusion Models -- 7.2.1. Local Volatility Models -- 7.2.2. Constant Elasticity of Variance Model -- 7.3. The Heston model -- 7.3.1. Solution to the Ricatti Equation -- 7.3.2. Implied Volatility for the Heston Model -- 7.4. Models with Jumps -- 7.4.1. The Poisson Process -- 7.4.2. Jump Diffusion Models with a Compound Poisson Component -- 7.4.3. The Merton Jump Diffusion Model -- 7.4.4. Characteristic Function for a Jump Diffusion Process -- 7.4.5. Change of Measure for Jump Diffusion Processes -- 7.4.6. The Variance Gamma Model -- 7.5. Exercises -- A. Essentials of General Probability Theory -- A.1. Random Variables and Lebesgue Integration -- A.2. Multidimensional Lebesgue Integration -- A.3. Multiple Random Variables and Joint Distributions -- A.4. Conditioning -- A.5. Changing Probability Measures -- B. Some Useful Integral (Expectation) Identities and Symmetry Properties of Normal Random Variables -- C. Answers and Hints to Exercises -- C.1. Chapter 1 -- C.2. Chapter 2 -- C.3. Chapter 3 -- C.4. Chapter 4 -- C.5. Chapter 5 C.6. Chapter 6 -- C.7. Chapter 7 -- D. Glossary of Symbols and Abbreviations -- References -- Index
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Stochastic Calculus with Brownian Motion -- 1. One-Dimensional Brownian Motion and Related Processes -- 1.1. Multivariate Normal Distributions -- 1.1.1. Multivariate Normal Distribution -- 1.1.2. Conditional Normal Distributions -- 1.2. Standard Brownian Motion -- 1.2.1. One-Dimensional Symmetric Random Walk -- 1.2.2. Formal Definition and Basic Properties of Brownian Motion -- 1.2.3. Multivariate Distribution of Brownian Motion -- 1.2.4. The Markov Property and the Transition PDF -- 1.2.5. Quadratic Variation and Nondifferentiability of Paths -- 1.3. Some Processes Derived from Brownian Motion -- 1.3.1. Drifted Brownian Motion -- 1.3.2. Geometric Brownian Motion -- 1.3.3. Processes related by a monotonic mapping -- 1.3.4. Brownian Bridge -- 1.3.5. Gaussian Processes -- 1.4. First Hitting Times and Maximum and Minimum of Brownian Motion -- 1.4.1. The Reflection Principle: Standard Brownian Motion -- 1.4.2. Translated and Scaled Driftless Brownian Motion -- 1.4.3. Brownian Motion with Drift -- 1.5. Exercises -- 2. Introduction to Continuous-Time Stochastic Calculus -- 2.1. The Riemann Integral of Brownian Motion -- 2.1.1. The Riemann Integral -- 2.1.2. The Integral of a Brownian Path -- 2.2. The Riemann-Stieltjes Integral of Brownian Motion -- 2.2.1. The Riemann-Stieltjes Integral -- 2.2.2. Integrals w.r.t. Brownian Motion: Preliminary Discussion -- 2.3. The Itô Integral and Its Basic Properties -- 2.3.1. The Itô Integral for Simple Processes -- 2.3.2. The Itô Integral for General Processes -- 2.4. Itô Processes and Their Properties -- 2.4.1. Gaussian Processes Generated by Itô Integrals -- 2.4.2. Itô Processes -- 2.4.3. Quadratic and Co-Variation of Itô Processes</subfield></datafield><datafield tag="505" ind1="8" ind2=" "><subfield code="a">2.5. Itô's Formula for Functions of BM and Itô Processes -- 2.5.1. Itô's Formula for Functions of BM -- 2.5.2. An "Antiderivative" Formula for Evaluating Itô Integrals -- 2.5.3. Itô's Formula for Itô Processes -- 2.6. Stochastic Differential Equations -- 2.6.1. Solutions to Linear SDEs -- 2.6.2. Existence and Uniqueness of a Strong Solution to an SDE -- 2.7. The Markov Property, Martingales, Feynman-Kac Formulae, and Transition CDFs and PDFs -- 2.7.1. Forward Kolmogorov PDE -- 2.7.2. Transition CDF/PDF for Time-Homogeneous Diffusions -- 2.8. Radon-Nikodym Derivative Process and Girsanov's Theorem -- 2.8.1. Some Applications of Girsanov's Theorem -- 2.9. Brownian Martingale Representation Theorem -- 2.10. Stochastic Calculus for Multidimensional BM -- 2.10.1. The Itô Integral and Itô's Formula for Multiple Processes on Multidimensional BM -- 2.10.2. Multidimensional SDEs, Feynman-Kac Formulae, and Transition CDFs and PDFs -- 2.10.3. Girsanov's Theorem for Multidimensional BM -- 2.10.4. Martingale Representation Theorem for Multidimensional BM -- 2.11. Exercises -- II. Continuous-Time Modelling -- 3. Risk-Neutral Pricing in the (B, S) Economy: One Underlying Stock -- 3.1. From the CRR Model to the BSM Model -- 3.1.1. Portfolio Strategies in the Binomial Tree Model -- 3.1.2. The Cox-Ross-Rubinstein Model and its Continuous-Time Limit -- 3.2. Replication (Hedging) and Derivative Pricing in the Simplest Black-Scholes Economy -- 3.2.1. Pricing Standard European Calls and Puts -- 3.2.2. Hedging Standard European Calls and Puts -- 3.2.3. Europeans with Piecewise Linear Payoffs -- 3.2.4. Power Options -- 3.2.5. Dividend Paying Stock -- 3.2.6. Option Pricing with the Stock Numéraire -- 3.3. Forward Starting, Chooser, and Compound Options -- 3.4. Some European-Style Path-Dependent Derivatives -- 3.4.1. Risk-Neutral Pricing under GBM.</subfield></datafield><datafield tag="505" ind1="8" ind2=" "><subfield code="a">3.4.2. Pricing Single Barrier Options -- 3.4.3. Pricing Lookback Options -- 3.5. Structural Credit Risk Models -- 3.5.1. The Merton Model -- 3.5.2. The Black-Cox Model -- 3.6. Exercises -- 4. Risk-Neutral Pricing in a Multi-Asset Economy -- 4.1. General Multi-Asset Market Model: Replication and Risk-Neutral Pricing -- 4.2. Equivalent Martingale Measures: Derivative Pricing with General Numéraire Assets -- 4.3. Black-Scholes PDE and Delta Hedging for Standard Multi-Asset Derivatives -- 4.3.1. Standard European Option Pricing for Multi-Stock GBM -- 4.3.2. Explicit Pricing Formulae for the GBM Model -- 4.3.3. Cross-Currency Option Valuation -- 4.3.4. Option Valuation with General Numéraire Assets -- 4.4. Exercises -- 5. American Options -- 5.1. Basic Properties of Early-Exercise Options -- 5.2. Arbitrage-Free Pricing of American Options -- 5.2.1. Optimal Stopping Formulation and Early-Exercise Boundary -- 5.2.2. The Smooth Pasting Condition -- 5.2.3. Put-Call Symmetry Relation -- 5.2.4. Dynamic Programming Approach for Bermudan Options -- 5.3. Perpetual American Options -- 5.3.1. Pricing a Perpetual Put Option -- 5.3.2. Pricing a Perpetual Call Option -- 5.4. Finite-Expiration American Options -- 5.4.1. The PDE Formulation -- 5.4.2. The Integral Equation Formulation -- 5.5. Exercises -- 6. Interest-Rate Modelling and Derivative Pricing -- 6.1. Basic Fixed Income Instruments -- 6.1.1. Bonds -- 6.1.2. Forward Rates -- 6.1.3. Arbitrage-Free Pricing -- 6.1.4. Fixed Income Derivatives -- 6.2. Single-Factor Models -- 6.2.1. Diffusion Models for the Short Rate Process -- 6.2.2. PDE for the Zero-Coupon Bond Value -- 6.2.3. Affine Term Structure Models -- 6.2.4. The Ho-Lee Model -- 6.2.5. The Vasiček Model -- 6.2.6. The Cox-Ingersoll-Ross Model -- 6.3. Heath-Jarrow-Morton Formulation -- 6.3.1. HJM under Risk-Neutral Measure</subfield></datafield><datafield tag="505" ind1="8" ind2=" "><subfield code="a">6.3.2. Relationship between HJM and Affine Yield Models -- 6.4. Multifactor Affine Term Structure Models -- 6.4.1. Gaussian Multifactor Models -- 6.4.2. Equivalent Classes of Affine Models -- 6.5. Pricing Derivatives under Forward Measures -- 6.5.1. Forward Measures -- 6.5.2. Pricing Stock Options under Stochastic Interest Rates -- 6.5.3. Pricing Options on Zero-Coupon Bonds -- 6.6. LIBOR Model -- 6.6.1. LIBOR Rates -- 6.6.2. The Brace-Gatarek-Musiela Model of LIBOR Rates -- 6.6.3. Pricing Caplets, Caps, and Swaps -- 6.7. Exercises -- 7. Alternative Models of Asset Price Dynamics -- 7.1. Characteristic Functions -- 7.1.1. Definition and Properties -- 7.1.2. Recovering the Distribution Function -- 7.1.3. Pricing Standard European Options -- 7.1.4. The Carr-Madan Method for Pricing Vanilla Options -- 7.2. Stochastic Volatility Diffusion Models -- 7.2.1. Local Volatility Models -- 7.2.2. Constant Elasticity of Variance Model -- 7.3. The Heston model -- 7.3.1. Solution to the Ricatti Equation -- 7.3.2. Implied Volatility for the Heston Model -- 7.4. Models with Jumps -- 7.4.1. The Poisson Process -- 7.4.2. Jump Diffusion Models with a Compound Poisson Component -- 7.4.3. The Merton Jump Diffusion Model -- 7.4.4. Characteristic Function for a Jump Diffusion Process -- 7.4.5. Change of Measure for Jump Diffusion Processes -- 7.4.6. The Variance Gamma Model -- 7.5. Exercises -- A. Essentials of General Probability Theory -- A.1. Random Variables and Lebesgue Integration -- A.2. Multidimensional Lebesgue Integration -- A.3. Multiple Random Variables and Joint Distributions -- A.4. Conditioning -- A.5. Changing Probability Measures -- B. Some Useful Integral (Expectation) Identities and Symmetry Properties of Normal Random Variables -- C. Answers and Hints to Exercises -- C.1. Chapter 1 -- C.2. Chapter 2 -- C.3. Chapter 3 -- C.4. Chapter 4 -- C.5. Chapter 5</subfield></datafield><datafield tag="505" ind1="8" ind2=" "><subfield code="a">C.6. Chapter 6 -- C.7. Chapter 7 -- D. Glossary of Symbols and Abbreviations -- References -- Index</subfield></datafield><datafield tag="653" ind1=" " ind2="6"><subfield code="a">Electronic books</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Makarov, Roman N.</subfield><subfield code="e">Sonstige</subfield><subfield code="4">oth</subfield></datafield><datafield tag="776" ind1="0" ind2="8"><subfield code="i">Erscheint auch als</subfield><subfield code="n">Druck-Ausgabe</subfield><subfield code="a">Campolieti, Giuseppe</subfield><subfield code="t">Financial Mathematics</subfield><subfield code="d">Milton : CRC Press LLC,c2022</subfield><subfield code="z">9781138603639</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">ZDB-30-PQE</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-034007798</subfield></datafield><datafield tag="966" ind1="e" ind2=" "><subfield code="u">https://ebookcentral.proquest.com/lib/hwr/detail.action?docID=7144681</subfield><subfield code="l">HWR01</subfield><subfield code="p">ZDB-30-PQE</subfield><subfield code="q">HWR_PDA_PQE</subfield><subfield code="x">Aggregator</subfield><subfield code="3">Volltext</subfield></datafield></record></collection>
id	DE-604.BV048632780
illustrated	Not Illustrated
index_date	2024-07-03T21:16:06Z
indexdate	2024-07-10T09:44:33Z
institution	BVB
isbn	9780429889097 9781138603639
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-034007798
oclc_num	1351752074
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owner	DE-2070s
owner_facet	DE-2070s
physical	1 Online-Ressource (511 Seiten)
psigel	ZDB-30-PQE ZDB-30-PQE HWR_PDA_PQE
publishDate	2022
publishDateSearch	2022
publishDateSort	2022
publisher	CRC Press LLC
record_format	marc
series2	Textbooks in Mathematics Ser
spelling	Campolieti, Giuseppe Verfasser aut Financial Mathematics A Comprehensive Treatment in Continuous Time Volume II. Milton CRC Press LLC 2022 ©2023 1 Online-Ressource (511 Seiten) txt rdacontent c rdamedia cr rdacarrier Textbooks in Mathematics Ser Cover -- Half Title -- Series Page -- Title Page -- Copyright Page -- Dedication -- Contents -- List of Figures -- Preface -- Authors -- I. Stochastic Calculus with Brownian Motion -- 1. One-Dimensional Brownian Motion and Related Processes -- 1.1. Multivariate Normal Distributions -- 1.1.1. Multivariate Normal Distribution -- 1.1.2. Conditional Normal Distributions -- 1.2. Standard Brownian Motion -- 1.2.1. One-Dimensional Symmetric Random Walk -- 1.2.2. Formal Definition and Basic Properties of Brownian Motion -- 1.2.3. Multivariate Distribution of Brownian Motion -- 1.2.4. The Markov Property and the Transition PDF -- 1.2.5. Quadratic Variation and Nondifferentiability of Paths -- 1.3. Some Processes Derived from Brownian Motion -- 1.3.1. Drifted Brownian Motion -- 1.3.2. Geometric Brownian Motion -- 1.3.3. Processes related by a monotonic mapping -- 1.3.4. Brownian Bridge -- 1.3.5. Gaussian Processes -- 1.4. First Hitting Times and Maximum and Minimum of Brownian Motion -- 1.4.1. The Reflection Principle: Standard Brownian Motion -- 1.4.2. Translated and Scaled Driftless Brownian Motion -- 1.4.3. Brownian Motion with Drift -- 1.5. Exercises -- 2. Introduction to Continuous-Time Stochastic Calculus -- 2.1. The Riemann Integral of Brownian Motion -- 2.1.1. The Riemann Integral -- 2.1.2. The Integral of a Brownian Path -- 2.2. The Riemann-Stieltjes Integral of Brownian Motion -- 2.2.1. The Riemann-Stieltjes Integral -- 2.2.2. Integrals w.r.t. Brownian Motion: Preliminary Discussion -- 2.3. The Itô Integral and Its Basic Properties -- 2.3.1. The Itô Integral for Simple Processes -- 2.3.2. The Itô Integral for General Processes -- 2.4. Itô Processes and Their Properties -- 2.4.1. Gaussian Processes Generated by Itô Integrals -- 2.4.2. Itô Processes -- 2.4.3. Quadratic and Co-Variation of Itô Processes 2.5. Itô's Formula for Functions of BM and Itô Processes -- 2.5.1. Itô's Formula for Functions of BM -- 2.5.2. An "Antiderivative" Formula for Evaluating Itô Integrals -- 2.5.3. Itô's Formula for Itô Processes -- 2.6. Stochastic Differential Equations -- 2.6.1. Solutions to Linear SDEs -- 2.6.2. Existence and Uniqueness of a Strong Solution to an SDE -- 2.7. The Markov Property, Martingales, Feynman-Kac Formulae, and Transition CDFs and PDFs -- 2.7.1. Forward Kolmogorov PDE -- 2.7.2. Transition CDF/PDF for Time-Homogeneous Diffusions -- 2.8. Radon-Nikodym Derivative Process and Girsanov's Theorem -- 2.8.1. Some Applications of Girsanov's Theorem -- 2.9. Brownian Martingale Representation Theorem -- 2.10. Stochastic Calculus for Multidimensional BM -- 2.10.1. The Itô Integral and Itô's Formula for Multiple Processes on Multidimensional BM -- 2.10.2. Multidimensional SDEs, Feynman-Kac Formulae, and Transition CDFs and PDFs -- 2.10.3. Girsanov's Theorem for Multidimensional BM -- 2.10.4. Martingale Representation Theorem for Multidimensional BM -- 2.11. Exercises -- II. Continuous-Time Modelling -- 3. Risk-Neutral Pricing in the (B, S) Economy: One Underlying Stock -- 3.1. From the CRR Model to the BSM Model -- 3.1.1. Portfolio Strategies in the Binomial Tree Model -- 3.1.2. The Cox-Ross-Rubinstein Model and its Continuous-Time Limit -- 3.2. Replication (Hedging) and Derivative Pricing in the Simplest Black-Scholes Economy -- 3.2.1. Pricing Standard European Calls and Puts -- 3.2.2. Hedging Standard European Calls and Puts -- 3.2.3. Europeans with Piecewise Linear Payoffs -- 3.2.4. Power Options -- 3.2.5. Dividend Paying Stock -- 3.2.6. Option Pricing with the Stock Numéraire -- 3.3. Forward Starting, Chooser, and Compound Options -- 3.4. Some European-Style Path-Dependent Derivatives -- 3.4.1. Risk-Neutral Pricing under GBM. 3.4.2. Pricing Single Barrier Options -- 3.4.3. Pricing Lookback Options -- 3.5. Structural Credit Risk Models -- 3.5.1. The Merton Model -- 3.5.2. The Black-Cox Model -- 3.6. Exercises -- 4. Risk-Neutral Pricing in a Multi-Asset Economy -- 4.1. General Multi-Asset Market Model: Replication and Risk-Neutral Pricing -- 4.2. Equivalent Martingale Measures: Derivative Pricing with General Numéraire Assets -- 4.3. Black-Scholes PDE and Delta Hedging for Standard Multi-Asset Derivatives -- 4.3.1. Standard European Option Pricing for Multi-Stock GBM -- 4.3.2. Explicit Pricing Formulae for the GBM Model -- 4.3.3. Cross-Currency Option Valuation -- 4.3.4. Option Valuation with General Numéraire Assets -- 4.4. Exercises -- 5. American Options -- 5.1. Basic Properties of Early-Exercise Options -- 5.2. Arbitrage-Free Pricing of American Options -- 5.2.1. Optimal Stopping Formulation and Early-Exercise Boundary -- 5.2.2. The Smooth Pasting Condition -- 5.2.3. Put-Call Symmetry Relation -- 5.2.4. Dynamic Programming Approach for Bermudan Options -- 5.3. Perpetual American Options -- 5.3.1. Pricing a Perpetual Put Option -- 5.3.2. Pricing a Perpetual Call Option -- 5.4. Finite-Expiration American Options -- 5.4.1. The PDE Formulation -- 5.4.2. The Integral Equation Formulation -- 5.5. Exercises -- 6. Interest-Rate Modelling and Derivative Pricing -- 6.1. Basic Fixed Income Instruments -- 6.1.1. Bonds -- 6.1.2. Forward Rates -- 6.1.3. Arbitrage-Free Pricing -- 6.1.4. Fixed Income Derivatives -- 6.2. Single-Factor Models -- 6.2.1. Diffusion Models for the Short Rate Process -- 6.2.2. PDE for the Zero-Coupon Bond Value -- 6.2.3. Affine Term Structure Models -- 6.2.4. The Ho-Lee Model -- 6.2.5. The Vasiček Model -- 6.2.6. The Cox-Ingersoll-Ross Model -- 6.3. Heath-Jarrow-Morton Formulation -- 6.3.1. HJM under Risk-Neutral Measure 6.3.2. Relationship between HJM and Affine Yield Models -- 6.4. Multifactor Affine Term Structure Models -- 6.4.1. Gaussian Multifactor Models -- 6.4.2. Equivalent Classes of Affine Models -- 6.5. Pricing Derivatives under Forward Measures -- 6.5.1. Forward Measures -- 6.5.2. Pricing Stock Options under Stochastic Interest Rates -- 6.5.3. Pricing Options on Zero-Coupon Bonds -- 6.6. LIBOR Model -- 6.6.1. LIBOR Rates -- 6.6.2. The Brace-Gatarek-Musiela Model of LIBOR Rates -- 6.6.3. Pricing Caplets, Caps, and Swaps -- 6.7. Exercises -- 7. Alternative Models of Asset Price Dynamics -- 7.1. Characteristic Functions -- 7.1.1. Definition and Properties -- 7.1.2. Recovering the Distribution Function -- 7.1.3. Pricing Standard European Options -- 7.1.4. The Carr-Madan Method for Pricing Vanilla Options -- 7.2. Stochastic Volatility Diffusion Models -- 7.2.1. Local Volatility Models -- 7.2.2. Constant Elasticity of Variance Model -- 7.3. The Heston model -- 7.3.1. Solution to the Ricatti Equation -- 7.3.2. Implied Volatility for the Heston Model -- 7.4. Models with Jumps -- 7.4.1. The Poisson Process -- 7.4.2. Jump Diffusion Models with a Compound Poisson Component -- 7.4.3. The Merton Jump Diffusion Model -- 7.4.4. Characteristic Function for a Jump Diffusion Process -- 7.4.5. Change of Measure for Jump Diffusion Processes -- 7.4.6. The Variance Gamma Model -- 7.5. Exercises -- A. Essentials of General Probability Theory -- A.1. Random Variables and Lebesgue Integration -- A.2. Multidimensional Lebesgue Integration -- A.3. Multiple Random Variables and Joint Distributions -- A.4. Conditioning -- A.5. Changing Probability Measures -- B. Some Useful Integral (Expectation) Identities and Symmetry Properties of Normal Random Variables -- C. Answers and Hints to Exercises -- C.1. Chapter 1 -- C.2. Chapter 2 -- C.3. Chapter 3 -- C.4. Chapter 4 -- C.5. Chapter 5 C.6. Chapter 6 -- C.7. Chapter 7 -- D. Glossary of Symbols and Abbreviations -- References -- Index Electronic books Makarov, Roman N. Sonstige oth Erscheint auch als Druck-Ausgabe Campolieti, Giuseppe Financial Mathematics Milton : CRC Press LLC,c2022 9781138603639
spellingShingle	Campolieti, Giuseppe Financial Mathematics A Comprehensive Treatment in Continuous Time Volume II. Cover -- Half Title -- Series Page -- Title Page -- Copyright Page -- Dedication -- Contents -- List of Figures -- Preface -- Authors -- I. Stochastic Calculus with Brownian Motion -- 1. One-Dimensional Brownian Motion and Related Processes -- 1.1. Multivariate Normal Distributions -- 1.1.1. Multivariate Normal Distribution -- 1.1.2. Conditional Normal Distributions -- 1.2. Standard Brownian Motion -- 1.2.1. One-Dimensional Symmetric Random Walk -- 1.2.2. Formal Definition and Basic Properties of Brownian Motion -- 1.2.3. Multivariate Distribution of Brownian Motion -- 1.2.4. The Markov Property and the Transition PDF -- 1.2.5. Quadratic Variation and Nondifferentiability of Paths -- 1.3. Some Processes Derived from Brownian Motion -- 1.3.1. Drifted Brownian Motion -- 1.3.2. Geometric Brownian Motion -- 1.3.3. Processes related by a monotonic mapping -- 1.3.4. Brownian Bridge -- 1.3.5. Gaussian Processes -- 1.4. First Hitting Times and Maximum and Minimum of Brownian Motion -- 1.4.1. The Reflection Principle: Standard Brownian Motion -- 1.4.2. Translated and Scaled Driftless Brownian Motion -- 1.4.3. Brownian Motion with Drift -- 1.5. Exercises -- 2. Introduction to Continuous-Time Stochastic Calculus -- 2.1. The Riemann Integral of Brownian Motion -- 2.1.1. The Riemann Integral -- 2.1.2. The Integral of a Brownian Path -- 2.2. The Riemann-Stieltjes Integral of Brownian Motion -- 2.2.1. The Riemann-Stieltjes Integral -- 2.2.2. Integrals w.r.t. Brownian Motion: Preliminary Discussion -- 2.3. The Itô Integral and Its Basic Properties -- 2.3.1. The Itô Integral for Simple Processes -- 2.3.2. The Itô Integral for General Processes -- 2.4. Itô Processes and Their Properties -- 2.4.1. Gaussian Processes Generated by Itô Integrals -- 2.4.2. Itô Processes -- 2.4.3. Quadratic and Co-Variation of Itô Processes 2.5. Itô's Formula for Functions of BM and Itô Processes -- 2.5.1. Itô's Formula for Functions of BM -- 2.5.2. An "Antiderivative" Formula for Evaluating Itô Integrals -- 2.5.3. Itô's Formula for Itô Processes -- 2.6. Stochastic Differential Equations -- 2.6.1. Solutions to Linear SDEs -- 2.6.2. Existence and Uniqueness of a Strong Solution to an SDE -- 2.7. The Markov Property, Martingales, Feynman-Kac Formulae, and Transition CDFs and PDFs -- 2.7.1. Forward Kolmogorov PDE -- 2.7.2. Transition CDF/PDF for Time-Homogeneous Diffusions -- 2.8. Radon-Nikodym Derivative Process and Girsanov's Theorem -- 2.8.1. Some Applications of Girsanov's Theorem -- 2.9. Brownian Martingale Representation Theorem -- 2.10. Stochastic Calculus for Multidimensional BM -- 2.10.1. The Itô Integral and Itô's Formula for Multiple Processes on Multidimensional BM -- 2.10.2. Multidimensional SDEs, Feynman-Kac Formulae, and Transition CDFs and PDFs -- 2.10.3. Girsanov's Theorem for Multidimensional BM -- 2.10.4. Martingale Representation Theorem for Multidimensional BM -- 2.11. Exercises -- II. Continuous-Time Modelling -- 3. Risk-Neutral Pricing in the (B, S) Economy: One Underlying Stock -- 3.1. From the CRR Model to the BSM Model -- 3.1.1. Portfolio Strategies in the Binomial Tree Model -- 3.1.2. The Cox-Ross-Rubinstein Model and its Continuous-Time Limit -- 3.2. Replication (Hedging) and Derivative Pricing in the Simplest Black-Scholes Economy -- 3.2.1. Pricing Standard European Calls and Puts -- 3.2.2. Hedging Standard European Calls and Puts -- 3.2.3. Europeans with Piecewise Linear Payoffs -- 3.2.4. Power Options -- 3.2.5. Dividend Paying Stock -- 3.2.6. Option Pricing with the Stock Numéraire -- 3.3. Forward Starting, Chooser, and Compound Options -- 3.4. Some European-Style Path-Dependent Derivatives -- 3.4.1. Risk-Neutral Pricing under GBM. 3.4.2. Pricing Single Barrier Options -- 3.4.3. Pricing Lookback Options -- 3.5. Structural Credit Risk Models -- 3.5.1. The Merton Model -- 3.5.2. The Black-Cox Model -- 3.6. Exercises -- 4. Risk-Neutral Pricing in a Multi-Asset Economy -- 4.1. General Multi-Asset Market Model: Replication and Risk-Neutral Pricing -- 4.2. Equivalent Martingale Measures: Derivative Pricing with General Numéraire Assets -- 4.3. Black-Scholes PDE and Delta Hedging for Standard Multi-Asset Derivatives -- 4.3.1. Standard European Option Pricing for Multi-Stock GBM -- 4.3.2. Explicit Pricing Formulae for the GBM Model -- 4.3.3. Cross-Currency Option Valuation -- 4.3.4. Option Valuation with General Numéraire Assets -- 4.4. Exercises -- 5. American Options -- 5.1. Basic Properties of Early-Exercise Options -- 5.2. Arbitrage-Free Pricing of American Options -- 5.2.1. Optimal Stopping Formulation and Early-Exercise Boundary -- 5.2.2. The Smooth Pasting Condition -- 5.2.3. Put-Call Symmetry Relation -- 5.2.4. Dynamic Programming Approach for Bermudan Options -- 5.3. Perpetual American Options -- 5.3.1. Pricing a Perpetual Put Option -- 5.3.2. Pricing a Perpetual Call Option -- 5.4. Finite-Expiration American Options -- 5.4.1. The PDE Formulation -- 5.4.2. The Integral Equation Formulation -- 5.5. Exercises -- 6. Interest-Rate Modelling and Derivative Pricing -- 6.1. Basic Fixed Income Instruments -- 6.1.1. Bonds -- 6.1.2. Forward Rates -- 6.1.3. Arbitrage-Free Pricing -- 6.1.4. Fixed Income Derivatives -- 6.2. Single-Factor Models -- 6.2.1. Diffusion Models for the Short Rate Process -- 6.2.2. PDE for the Zero-Coupon Bond Value -- 6.2.3. Affine Term Structure Models -- 6.2.4. The Ho-Lee Model -- 6.2.5. The Vasiček Model -- 6.2.6. The Cox-Ingersoll-Ross Model -- 6.3. Heath-Jarrow-Morton Formulation -- 6.3.1. HJM under Risk-Neutral Measure 6.3.2. Relationship between HJM and Affine Yield Models -- 6.4. Multifactor Affine Term Structure Models -- 6.4.1. Gaussian Multifactor Models -- 6.4.2. Equivalent Classes of Affine Models -- 6.5. Pricing Derivatives under Forward Measures -- 6.5.1. Forward Measures -- 6.5.2. Pricing Stock Options under Stochastic Interest Rates -- 6.5.3. Pricing Options on Zero-Coupon Bonds -- 6.6. LIBOR Model -- 6.6.1. LIBOR Rates -- 6.6.2. The Brace-Gatarek-Musiela Model of LIBOR Rates -- 6.6.3. Pricing Caplets, Caps, and Swaps -- 6.7. Exercises -- 7. Alternative Models of Asset Price Dynamics -- 7.1. Characteristic Functions -- 7.1.1. Definition and Properties -- 7.1.2. Recovering the Distribution Function -- 7.1.3. Pricing Standard European Options -- 7.1.4. The Carr-Madan Method for Pricing Vanilla Options -- 7.2. Stochastic Volatility Diffusion Models -- 7.2.1. Local Volatility Models -- 7.2.2. Constant Elasticity of Variance Model -- 7.3. The Heston model -- 7.3.1. Solution to the Ricatti Equation -- 7.3.2. Implied Volatility for the Heston Model -- 7.4. Models with Jumps -- 7.4.1. The Poisson Process -- 7.4.2. Jump Diffusion Models with a Compound Poisson Component -- 7.4.3. The Merton Jump Diffusion Model -- 7.4.4. Characteristic Function for a Jump Diffusion Process -- 7.4.5. Change of Measure for Jump Diffusion Processes -- 7.4.6. The Variance Gamma Model -- 7.5. Exercises -- A. Essentials of General Probability Theory -- A.1. Random Variables and Lebesgue Integration -- A.2. Multidimensional Lebesgue Integration -- A.3. Multiple Random Variables and Joint Distributions -- A.4. Conditioning -- A.5. Changing Probability Measures -- B. Some Useful Integral (Expectation) Identities and Symmetry Properties of Normal Random Variables -- C. Answers and Hints to Exercises -- C.1. Chapter 1 -- C.2. Chapter 2 -- C.3. Chapter 3 -- C.4. Chapter 4 -- C.5. Chapter 5 C.6. Chapter 6 -- C.7. Chapter 7 -- D. Glossary of Symbols and Abbreviations -- References -- Index
title	Financial Mathematics A Comprehensive Treatment in Continuous Time Volume II.
title_auth	Financial Mathematics A Comprehensive Treatment in Continuous Time Volume II.
title_exact_search	Financial Mathematics A Comprehensive Treatment in Continuous Time Volume II.
title_exact_search_txtP	Financial Mathematics A Comprehensive Treatment in Continuous Time Volume II.
title_full	Financial Mathematics A Comprehensive Treatment in Continuous Time Volume II.
title_fullStr	Financial Mathematics A Comprehensive Treatment in Continuous Time Volume II.
title_full_unstemmed	Financial Mathematics A Comprehensive Treatment in Continuous Time Volume II.
title_short	Financial Mathematics
title_sort	financial mathematics a comprehensive treatment in continuous time volume ii
title_sub	A Comprehensive Treatment in Continuous Time Volume II.
work_keys_str_mv	AT campolietigiuseppe financialmathematicsacomprehensivetreatmentincontinuoustimevolumeii AT makarovromann financialmathematicsacomprehensivetreatmentincontinuoustimevolumeii

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