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E-book
Author Nicolay, David, author

Title Asymptotic chaos expansions in finance : theory and practice / David Nicolay
Published London : Springer, [2014]
©2014

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Description 1 online resource
Series Springer finance
Springer finance.
Contents 880-01 Introduction -- Volatility dynamics for a single underlying: foundations -- Volatility dynamics for a single underlying: advanced methods -- Practical applications and testing -- Volatility dynamics in a term structure -- Implied Dynamics in the SV-HJM framework -- Implied Dynamics in the SV-LMM framework -- Conclusion
880-01/(S Machine generated contents note: 1. Introduction -- 1.1. Motivation -- 1.2. State of the Art -- 1.2.1. Stochastic Volatility Market Models -- 1.2.2. Asymptotic Methods -- 1.3. Objectives -- 1.4. Asymptotic Chaos Expansion -- 1.5. Outline and Main Results -- 1.6. General Spirit and Edited Material -- References -- pt. I Single Underlying -- 2. Volatility Dynamics for a Single Underlying: Foundations -- 2.1. Framework and Objectives -- 2.1.1. Market and Underlyings -- 2.1.2. Vanilla Options Market and Sliding Implied Volatility -- 2.1.3. Two Stochastic Volatility Model Frameworks -- 2.1.4. Objectives -- 2.2. Derivation of the Zero-Drift Conditions -- 2.2.1. Main Zero-Drift Condition -- 2.2.2. Immediate Zero Drift Conditions -- 2.2.3. IATM Identity -- 2.2.4. Synthesis and Overture -- 2.3. Recovering the Instantaneous Volatility: The First Layer -- 2.3.1. Computing the Dynamics of σt -- 2.3.2. Interpretation and Comments -- 2.4. Generating the Implied Volatility: The First Layer -- 2.4.1. Computing the Immediate ATM Differentials -- 2.4.2. Interpretation and Comments -- 2.5. Illustrations and Applications -- 2.5.1. Overview of Possible Applications -- 2.5.2. Illustration: Qualitative Analysis of a Classical SV Model Class -- 2.5.3. Second Illustration: Smile-Specification of SInsV Models -- 2.6. Conclusion and Overture -- References -- 3. Volatility Dynamics for a Single Underlying: Advanced Methods -- 3.1. Higher-Order Expansions: Methodology and Automation -- 3.1.1. Tools and Roadmap -- 3.1.2. Computing the First Column of the Differentiation Matrix -- 3.1.3. Computing Subsequent Columns of the Differentiation Matrix -- 3.2. Higher-Order Expansions: Illustration and Interpretation -- 3.2.1. Justification and Outline -- 3.2.2. Interpretation of the Results -- 3.2.3. Illustration of the Maturity Effect -- 3.3. Framework Extensions and Generalisation -- 3.3.1. Building Blocks and Available Extensions -- 3.3.2. Important Example: The Normal Baseline via Its ZDC -- 3.3.3. Generic Baseline Transfer -- 3.4. Multi-dimensional Extensions, or the Limitations of Recovery -- 3.4.1. Framework -- 3.4.2. Derivation of the Zero-Drift Conditions -- 3.4.3. Recovering the Instantaneous Volatility: The First Layer -- 3.4.4. Generating the Implied Volatility: The First Layer -- 3.5. Illustration of the Vectorial Framework: The Basket Case -- 3.5.1. Motivation -- 3.5.2. Framework and Objectives -- 3.5.3. Coefficient Basket -- 3.5.4. Asset Basket in the General Case -- 3.5.5. Asset Basket Specialised to Fixed Weights -- 3.5.6. Interpretation and Applications -- References -- 4. Practical Applications and Testing -- 4.1. General Considerations on Practical Applications -- 4.2. Application to the Generic SABR Class -- 4.2.1. Presentation of the Model -- 4.2.2. Coefficients of the Chaos Dynamics -- 4.2.3. Mapping the Model and the Smile -- 4.3. Application to the CEV-SABR Model -- 4.3.1. Presentation of the Model -- 4.3.2. Coefficients of the Chaos Dynamics -- 4.3.3. Mapping the Model and the Smile Shape -- 4.3.4. Compatibility with Hagan et al -- 4.4. Application to the FL-SV Class (Exercise) -- 4.4.1. Presentation of the Model -- 4.4.2. Derivation Exercise -- 4.5. Numerical Implementation and Testing -- 4.5.1. Testing Environment and Rationale -- 4.5.2. Tests Data and Results -- 4.5.3. Conclusions -- References -- pt. II Term Structures -- 5. Volatility Dynamics in a Term Structure -- 5.1. Framework and Objectives -- 5.1.1. Numeraires, Underlyings and Options -- 5.1.2. Absolute and Sliding Implied Volatilities -- 5.1.3. Two Stochastic Volatility Models -- 5.1.4. Objectives -- 5.2. Derivation of the Zero-Drift Conditions -- 5.2.1. Main Zero-Drift Condition -- 5.2.2. Immediate Zero Drift Condition -- 5.2.3. IATM Identity -- 5.3. Recovering the Instantaneous Volatility -- 5.3.1. Establishing the Main Result -- 5.3.2. Interpretation and Comments -- 5.4. Generating the SIV Surface: The First Layer -- 5.4.1. Computing the Differentials -- 5.4.2. Interpretation and Comments -- 5.5. Extensions, Further Questions and Conclusion -- References -- 6. Implied Dynamics in the SV-HJM Framework -- 6.1. Definitions, Notations and Objectives -- 6.1.1. HJM Framework in a Chaos Context -- 6.1.2. Tenor Structures and Simplified Notations -- 6.1.3. Objectives and Assumptions -- 6.1.4. Relative Pertinence of the SV-HJM and SV-LMM Classes -- 6.2. Dynamics of Rebased Bonds -- 6.2.1. Dynamics of the Rebased Zero Coupons -- 6.2.2. Dynamics of a Fixed-Weights Rebased Zero Coupon Basket -- 6.3. Bond Options -- 6.3.1. Casting the Bond Options Into the Generic Framework -- 6.3.2. Dynamics of the Underlying Rebased Bond -- 6.3.3. Interpretation -- 6.4. Caplets -- 6.4.1. Casting the Caplets into the Generic Framework -- 6.4.2. Dynamics of the Underlying Libor Rate -- 6.4.3. Interpretation of the Libor Rate HJM Dynamics -- 6.5. Swaptions -- 6.5.1. Casting the Swaptions into the Generic Framework -- 6.5.2. Dynamics of the Underlying Swap Rate -- 6.6. Indirect Approaches: Assets vs Rates -- 6.6.1. Applying the Asymptotic Approach to Caplets -- 6.6.2. Applying the Asymptotic Approach to Swaptions -- References -- 7. Implied Dynamics in the SV-LMM Framework -- 7.1. Definitions, Notations and Objectives -- 7.1.1. LMM Framework in a Chaos Context -- 7.1.2. Tenor Structures and Simplified Notations -- 7.1.3. Objectives and Assumptions -- 7.2. Chaos Dynamics of the Zeros in an LMM Framework -- 7.2.1. State Variables and Rationale for Rebasing -- 7.2.2. Computing the Chaos Dynamics -- 7.3. Bond Options -- 7.3.1. Casting the Bond Options into the Generic Framework -- 7.3.2. Dynamics of the Underlying Rebased Bond -- 7.4. Caplets -- 7.4.1. Casting the Caplets into the Generic Framework -- 7.4.2. IATM Differentials of the Caplet Smile -- 7.5. Swaptions -- 7.5.1. Casting the Swaptions into the Generic Framework -- 7.5.2. Dynamics of the Underlying Par Swap Rate -- 7.6. Approximating the Swap Rate Volatility -- 7.6.1. Basket Approximation for Swap Rates -- 7.6.2. Exact Swap Rate Dynamics in the Basket Representation -- 7.6.3. Impact of the Freezing Approximation in the General Case -- 7.6.4. Impact of the Freezing Approximation in a Simplified Case -- References -- 8. Conclusion -- 8.1. Summary of Achievements -- 8.2. Advantages of the Methodology -- 8.3. Limitations of the Methodology -- 8.4. Extensions and Further Work -- References
Summary Stochastic instantaneous volatility models such as Heston, SABR or SV-LMM have mostly been developed to control the shape and joint dynamics of the implied volatility surface. In principle, they are well suited for pricing and hedging vanilla and exotic options, for relative value strategies or for risk management. In practice however, most SV models lack a closed form valuation for European options. This book presents the recently developed Asymptotic Chaos Expansions methodology (ACE) which addresses that issue. Indeed its generic algorithm provides, for any regular SV model, the pure asymptotes at any order for both the static and dynamic maps of the implied volatility surface. Furthermore, ACE is programmable and can complement other approximation methods. Hence it allows a systematic approach to designing, parameterising, calibrating and exploiting SV models, typically for Vega hedging or American Monte-Carlo. Asymptotic Chaos Expansions in Finance illustrates the ACE approach for single underlyings (such as a stock price or FX rate), baskets (indexes, spreads) and term structure models (especially SV-HJM and SV-LMM). It also establishes fundamental links between the Wiener chaos of the instantaneous volatility and the small-time asymptotic structure of the stochastic implied volatility framework. It is addressed primarily to financial mathematics researchers and graduate students, interested in stochastic volatility, asymptotics or market models. Moreover, as it contains many self-contained approximation results, it will be useful to practitioners modelling the shape of the smile and its evolution
Analysis wiskunde
mathematics
partial differential equations
numerieke methoden
numerical methods
waarschijnlijkheidstheorie
probability theory
stochastische processen
stochastic processes
wiskundige modellen
mathematical models
toegepaste wiskunde
applied mathematics
finance
Mathematics (General)
Wiskunde (algemeen)
Bibliography Includes bibliographical references and index
Notes Online resource; title from PDF title page (EBSCO, viewed January 8, 2015)
Subject Finance -- Mathematical models.
BUSINESS & ECONOMICS -- Finance.
Finance -- Mathematical models.
Genre/Form Electronic books
Form Electronic book
ISBN 9781447165064
1447165063