这本《论波动率模型(英文版)》由易聪著，基于笔者于伦敦帝国理工学院和三菱UFJ证券国际(伦敦)的联合项目中所完成的金融数学博士论文。该联合项目始于2005年年底，旨在探讨随机波动率在股票、外汇、利率等金融资产的建模中的应用以及基于此模型之上对金融衍生品的定价。随机(非常量)的波动率模型是近些年来热门的金融数学研究方向，特别是在2008年金融危机动荡的市场中更是受到了学术界和金融业界的重视。《论波动率模型(英文版)》最大的贡献在于提供了当前最全面的随机金融模型架构，包括随机波动率、局部波动率、随机利率以及跳跃过程对外汇走势的建模，以及对金融衍生品(欧式期权)定价的半解析解。其他几个章节涉及了对另外的波动率模型的提出和讨论，以及随机波动率模型在金融业界中的实际应用和衍生品定价的范例。

笔者利用跨学术和金融业界的优势，为大家展现了国际金融工程学术研究和金融衍生品发展的最前沿画卷。

List of Figures

List of Tables

Abstract

Acknowledgements

1.General Introduction, Changing Volatility Models and European Options Pricing

  1.1 General Introduction

  1.2 Introduction to Changing Volatility Models

  1.3 Model Completeness and European Option Pricing

  1.4 Single Period Volatility Changing Problems

    1.4.1 Fixed Volatility Changing Time with Barrier B

    1.4.2 Random Volatility Changing Time with a Hitting Barrier B

  1.5 Multi-Period Volatility Changing Problems

  1.6 Extension to Incomplete Market

    1.6.1 A Simple Random Volatility Changing Model- Extension to Stochastic Volatility Model

    1.6.2 Future Research

  1.7 Appendix: Proof

    1.7.1 Proof of Proposition 1.1

    1.7.2 Proof of Proposition 1.2

    1.7.3 Proof of Proposition 1.3

    1.7.4 Proof of Proposition 1.4

    1.7.5 Proof of Proposition 1.5

    1.7.6 Proof of Proposition 1.6

    1.7.7 Theorem 2.2 of [132]: Uniqueness of the Equivalent Martingale Measure

2.Introduction to Stochastic Volatility and Local Stochastic Volatility Models

  2.1 Stochastic Volatility Models-A General Set-Up

    2.1.1 Model Set-Up

    2.1.2 Change of Measure and Model Incompleteness

  2.2 Making the Stochastic Volatility Economy Complete

  2.3 European Option Price

  2.4 Local Stochastic Volatility Models: An Introduction

  2.5 Adjustment to the Calculation of Greeks in a Non- Constant Implied Volatility Model

3.Foreign Exchange Options with Local Stochastic Volatility and Stochastic Interest Rates

  3.1 Introduction

  3.2 The FX-IR Hybrid Model

  3.3 Asymptotic Expansion

    3.3.1 A Brief Introduction

    3.3.2 European Option Pricing and Implied Volatility

 3.4 Model Implementation and Numerical Results

 3.5 FX Option Pricing via Fourier Transform under Stochastic Interest Rates, Stochastic Volatility and the Jump Process

    3.5.1 The Multi-Factor Model

    3.5.2 Change of Measure and Option Pricing

    3.5.3 Model Implementation

    3.5.4 Calibration Results for the USD/JPY Market

 3.6 Perfect Hedging with Stochastic Interest Rates and Local Stochastic Volatility

    3.6.1 Hedging with Options

    3.6.2 Hedging with Options and Bonds

 3.7 Partial Hedging with Hedging Error Analysis

    3.7.1 Hedging with One Option for the Volatility Risk

    3.7.2 Hedging with One Option for the Interest Rate Risk

 3.8 Model Mis-specification and Hedging Error Analysis

    3.8.1  Delta Hedging Difference between the CEV and CEV-SV Models

    3.8.2 Model Mis-speeifieation: The Importance of Stochastic Interest Rates

 3.9 Application to Power-Reverse-Dual-Currency Notes

    3.9.1 PRDC-TARN: The Structured Product

    3.9.2 Smile Impact on PRDC-TARN Product Valuation

 3.10 Conclusion and Future Research

 3.11 Appendix: Proof

    3.11.1 Derivation of the European Option Formula

    3.11.2 Conditional Expectations of the Multiple Weiner- Ito Integral

    3.11.3 Watanabe Theorem

    3.11.4 The European Option Formula from the Fourier Transform Method

4.Non-Biased Monte Carlo Simulation for a Heston-Type Stochastic Volatility Model

  4.1 Introduction

  4.2 Properties of the Square Root Process

  4.3 Simulation of Vt: Application of the Saddle Point Method

  4.4 Simulation of ∫t+Δ t Vsds given Vt and Vt+Δ: Moment Matching Technique

  4.5 Simulation of Ft+Δt, given Ft

  4.6 Conclusion and Future Research

  4.7 Appendix : Proof

    4.7.1 Characteristic Function and Moments of the Square Root Process

    4.7.2 The Lugannani and Rice Formula for the Cumulative Distribution Function

5.The LIBOR Market Model with Stochastic Volatility and Jump Processes

  5.1 Introduction

  5.2 The LIBOR Forward Rate Model

    5.2.1 Risk-Neutral Measure

    5.2.2 Change of Measure

  5.3 The LIBOR Swap Rate Model

    5.3.1 The Swap Market

    5.3.2 Change of Measure

  5.4 Caplet and Swaption Pricing Via Fourier Transform

    5.4.1 Caplet Pricing

    5.4.2 Swaption Pricing

  5.5 Conclusion and Future Research

  5.6 Appendix: Proof

    5.6.1 Proof of Lemma5.1

    5.6.2 Proof of Proposition 5.2

    5.6.3 Proof of Proposition 5.3

    5.6.4 Proof of Proposition 5.4

    5.6.5 The Marked Point Process

    5.6.6 The Girsanov Theorem and ho's Lemma on Jump Processes The Girsanov Theorem for Jump Processes

    5.6.7 The Derivation of Characteristic Functions Caplet

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