随机波动金融市场衍生品 [Derivatives in Financial Markets with Stochasic Volatility] pdf epub mobi txt 电子书 下载 2026

内容简介

This book addresses problems in financial mathematics of pricing and hedging derivative securities in an environment of uncertain and changing market volatility. These problems are important to investors ranging from large trading institutions to pension funds. The authors present mathematical and statistical tools that exploit the "bursty" nature of market volatility. The mathematics is introduced through examples and illustrated with simulations, and the approach described is validated and tested on market data.
The material is suitable for a one-semester course for graduate students who have been exposed to methods of stochastic modeling and arbitrage pricing theory in finance. It is easily accessible to derivatives practitioners in the inancial engineering industry.

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目录

Introduction
1 The Black-Scholes Theory of Derivative Pricing
1.1 Market Model
1.1.1 Brownian Motion
1.1.2 Stochastic Integrals
1.1.3 Risky Asset Price Model
1.1.4 Itos Formula
1.1.5 Lognormal Risky Asset Price
1.2 Derivative Contracts
1.2.1 European Call and Put Options
1.2.2 American Options
1.2.3 Other Exotic Options
1.3 Replicating Strategies
1.3.1 Replicating Self-Financing Portfolios
1.3.2 The Black-Scholes Partial Differential Equation
1.3.3 Pricing to Hedge
1.3.4 The Black-Scholes Formula
1.4 Risk-Neutral Pricing
1.4.1 Equivalent Martingale Measure
1.4.2 Self-Financing Portfolios
1.4.3 Risk-Neutral Valuation
1.4.4 Using the Markov Property
1.5 Risk-Neutral Expectations and Partial Differential Equations
1.5.1 Infinitesimal Generators and Associated Martingales
1.5.2 Conditional Expectations and Parabolic Partial Differential Equations
1.5.3 Application to the Black-Scholes Partial Differential Equation
1.5.4 American Options and Free Boundary Problems
1.5.5 Path-Dependent Derivatives
1.6 Complete Market

2 Introduction to Stochastic Volatility Models
2.1 Implied Volatility and the Smile Curve
2.1.1 Interpretation of the Smile Curve
2.1.2 What Data to Use
2.2 Implied Deterministic Volatility
2.2.1 Time-Dependent Volatility
2.2.2 Level-Dependent Volatility
2.2.3 Short-Time Tight Fit versus Long-Time Rough Fit
2.3 Stochastic Volatility Models
2.3.1 Mean-Reverting Stochastic Volatility Models
2.3.2 Stock-Price Distribution under Stochastic Volatility
2.4 Derivative Pricing
2.5 Pricing with Equivalent Martingale Measures
2.6 Implied Volatility as a Function of Moneyness
2.7 Market Price of Volatility Risk and Data
2.8 Special Case： Uncorrelated Volatility
2.8.1 Hull-White Formula
2.8.2 Stochastic Volatility Implies Smile
2.8.3 Remark on Correlated Volatility
2.9 Summary and Conclusions

3 Scales in Mean-Reverting Stochastic Volatility
3.1 Scaling in Simple Models
3.2 Models of Clustering
3.2.1 Example： Markov Chain
3.2.2 Example： Another Jump Process
3.2.3 Example： Ornstein-Uhlenbeck Process
3.2.4 Summary
3.3 Convergence to Black-Scholes under Fast Mean-Reverting Volatility
3.4 Scales in the Returns Process
3.4.1 The Returns Process
3.4.2 Returns Process with Jump Volatility
3.4.3 Returns Process with OU Volatility
3.4.4 S&P; 500 Returns Process

4 Tools for Estimating the Rate of Mean Reversion
4.1 Model and Data
4.1.1 Mean-Reverting Stochastic Volatility
4.1.2 Discrete Data
4.2 Variogram Analysis
4.2.1 Computation of the Variogram
4.2.2 Comparison and Sensitivity Analysis with Simulated Data
4.2.3 The Day Effect
4.3 Spectral Analysis

5 Asymptotics for Pricing European Derivatives
5.1 Preliminaries
5.1.l The Rescaled Stochastic Volatility Model
5.1.2 The Rescaled Pricing Equation
5.1.3 The Operator Notation
5.2 The Formal Expansion
5.2.1 The Diverging Terms
5.2.2 Poisson Equations
5.2.3 The Zero-Order Term
5.2.4 The First Correction
5.2.5 Universal Market Group Parameters
5.2.6 Probabilistic Interpretation of the Source Term
5.2.7 Put-Call Parity
5.2.8 The Skew Effect
5.3 Implied Volatilities and Calibration
5.4 Accuracy of the Approximation
5.5 Region of Validity

6 Implementation and Stability
6.1 Step-by-Step Procedure
6.2 Comments about the Method
6.3 Dividends
6.4 The Second Correction

7 Hedging Strategies
7.1 Black-Scholes Delta Hedging
7.1.1 The Strategy and Its Cost
7.1.2 Averaging Effect
7.2 Mean Self-Financing Hedging Strategy
7.3 Staying Close to the Price

8 Application to Exotic Derivatives
8. l European Binary Options
8.2 Barrier Options
8.3 Asian Options

9 Application to American Derivatives
9.1 American Problem under Stochastic Volatility
9.2 Stochastic Volatility Correction for an American Put
9.2.1 Expansions
9.2.2 First Approximation
9.2.3 The Stochastic Volatility Correction
9.2.4 Uncorrelated Volatility
9.2.5 Probabilistic Representation
9.3 Numerical Computation
9.3.1 Solution of the Black-Scholes Problem
9.3.2 Computation of the Correction

10 Generalizations
10.1 Portfolio Optimization under Stochastic Volatility
10.1.1 Constant Volatility Merton Problem
10.1.2 Stochastic Volatility Merton Problem
10.1.3 A Practical Solution
10.2 Periodic Day Effect
10.3 Other Markovian Volatility Models
10.3.1 Markovian Jump Volatility Models
10.3.2 Pricing and Asymptotics
10.4 Martingale Approach
10.4.1 Main Argument
10.4.2 Decomposition Result
10.4.3 Comparison with the PDE Approach
10.5 Non-Markovian Models of Volatility
10.5.1 Setting： An Example
10.5.2 Asymptotics in the Non-Markovian Case
10.6 Multidimensional Models

11 Applications to Interest-Rate Models
11.1 Bond Pricing in the Vasicek Model
11.1.1 Review of the Constant Volatility Vasicek Model
11.1.2 Stochastic Volatility Vasicek Models
11.2 Bond Option Pricing
11.2.1 The Constant Volatility Case
11.2.2 Correction for Stochastic Volatility
11.2.3 Implications
11.3 Asymptotics around the CIR Model
11.4 Illustration from Data
11.4.1 Variogram Analysis
11.4.2 Yield Curve Fitting
Bibliography
Index

精彩书摘

The Black-Scholes model rests upon a number of assumptions that are，to some extent.“counterfactual.”Among these are continuity ofthe stock. price process it does not iump），the ability to hedge continuously without transaction costs，inde-pendent Gaussian returns. and constant volatility.We shall focus here on relaxing the last assumption by allowing volatility to vary randomly，for the following rea-son：a well. known discrepancy between Black-Scholes-predicted European op-tion prices and market-traded options prices，the smile curve，can be accounted for by stochastic volatility models. That iS.this modification of the Black-Scholes theory has a posteriori success in one area where the classical model fails.In fact.modeling volatility as a stochastic process iS motivated a priori by em-pirical studies of stock.price returns in which estimated volatility iS observed to exhibit“random”characteristics.Additionally，the effects of transaction costs show up. under many models，as uncertainty in the volatility；fat-tailed returns distributions can be simulated by stochastic volatility；and market‘jump”phe-nomena are often best modeled as volatility iump processes.Stochastic volatility modeling therefore iS not iust a simple fix to one particular Biacl（Scholes as-sumption but rather a powerful modification that describes a much more complex market.We cite literature that explores possible causes of stochastic volatility in the notes at the end Of this chapter. In Chapter 1，we introduced the notation and tools for pricing and hedging deriv-ative securities ander a constant volatility lognormal model（1.2）.This iS the sim-plest example of pricing in a complete market.However，pricing in a market with stochastic volatility is an incomplete markets problem.a distinction that（as we shall explain）has far-reaching consequences-particularly for the hedging prob-lem and the problem of parameter estimation. It iS the latter inverse problem that iS the biggest mathematical and practical challenge introduced by such models，and also perhaps the one that benefits most from the asymptotic methods of Chapter 5.

前言/序言

《金融市场随机波动下的投资策略与风险管理》 在瞬息万变的现代金融世界中，市场价格的波动是恒定的主题，而波动率本身并非静止不变，它也随着时间推移而呈现出复杂的变化规律。这种“波动的波动”现象，即随机波动，构成了金融市场中最深层的动力之一，对各类资产的定价、投资组合的构建以及风险的度量都产生了至关重要的影响。本书旨在深入探讨在随机波动背景下，投资者如何制定有效的投资策略，以及如何进行系统性的风险管理。 本书的篇幅将聚焦于以下几个核心领域，力求为读者提供一套全面而实用的分析框架和操作指南： 第一部分：随机波动模型理论与实证分析 我们将从随机波动模型的基本概念入手，介绍经典的随机波动模型（如 the GARCH family models）及其演变。重点将放在理解这些模型的数学结构，包括均值方程、方差方程以及条件分布的设定。理论讲解将与实际数据分析紧密结合，通过案例分析展示如何使用统计软件对真实市场数据进行模型拟合和参数估计，评估模型的拟合优劣，以及如何进行模型诊断和选择。此外，本书还将介绍一些非参数方法和机器学习方法在捕捉随机波动特性上的应用，为读者提供更广阔的视野。 第二部分：随机波动下的资产定价与投资组合优化 理解随机波动对于准确评估金融资产的内在价值至关重要。本书将深入研究随机波动模型如何影响资产定价，特别是在股票、债券、外汇以及商品等不同资产类别上的应用。我们将探讨随机波动对期权定价的影响，例如Black-Scholes模型在随机波动下的修正和扩展。 在投资组合优化方面，本书将重点阐述如何将随机波动纳入投资组合的构建过程中。传统的均值-方差优化模型在随机波动环境下会失效，因此，我们将介绍基于随机波动模型进行投资组合优化的新方法，包括如何衡量和管理组合的波动率风险，如何构建能够有效对冲波动率风险的投资组合，以及如何根据市场动态调整投资组合的资产配置。我们将重点关注风险预算、条件在险价值（CVaR）等风险度量指标，以及在随机波动下的优化应用。 第三部分：随机波动下的风险管理技术与工具 风险管理是金融实践的核心，而随机波动更是风险管理中不可回避的挑战。本书将详细介绍如何在随机波动的市场环境中识别、度量和管理各类风险。 市场风险管理： 我们将深入分析随机波动对市场风险度量的影响，如在险价值（VaR）和条件在险价值（CVaR）的计算。本书将探讨如何在存在随机波动时，更准确地估计这些风险指标，以及如何利用这些指标来指导交易和投资决策。 信用风险管理： 尽管本书侧重于市场波动，但我们也将触及随机波动对信用风险的间接影响，例如在经济下行或市场剧烈波动时，企业违约概率的增加。 流动性风险管理： 在市场波动加剧时，流动性往往会枯竭。本书将探讨随机波动如何影响市场流动性，以及如何在流动性风险显著的市场环境下进行有效的风险管理。 操作风险与合规风险： 虽然不是本书的重点，但我们将简要提及在市场剧烈波动期间，操作风险和合规风险也可能被放大，需要投资者保持高度警惕。 第四部分：实际应用案例与交易策略 理论与实践相结合是本书的一大特色。我们将通过多个精心挑选的实际应用案例，展示随机波动模型和风险管理工具在不同市场环境下的应用效果。这些案例将涵盖： 高频交易中的波动率套利策略： 分析如何在短期市场波动中捕捉交易机会。 宏观经济事件对波动率的影响与应对： 探讨突发事件（如地缘政治冲突、疫情）如何引发市场剧烈波动，以及投资者如何进行风险对冲。 量化对冲基金的波动率交易策略： 剖析一些利用波动率产品（如VIX期货期权）进行交易的策略。 养老金和保险公司在不确定市场下的资产配置： 讨论如何在长期投资中考虑随机波动对收益和风险的影响。 第五部分：前沿研究与未来展望 本书的最后部分将对当前金融学界在随机波动研究领域的最新进展进行梳理，例如高阶矩模型、非线性随机波动模型、以及与高频数据相结合的波动率建模方法。我们将讨论这些前沿研究的潜在应用价值，并展望随机波动研究的未来发展方向，例如机器学习与深度学习在波动率预测和风险管理中的深度融合，以及更精细化的微观结构模型对波动率形成机制的揭示。 本书适合于金融机构的风险管理人员、投资经理、量化分析师、对冲基金经理，以及对金融市场波动性有深入研究需求的学术界研究人员和研究生。通过阅读本书，您将能够更深刻地理解金融市场的内在运行机制，掌握在不确定性环境中进行投资决策和风险管理的先进方法，从而在复杂多变的金融市场中游刃有余，实现稳健的投资回报。