This textbook provides the necessary techniques from financial mathematics and stochastic analysis for the valuation of more complex financial products and strategies. The author discusses how to make use of mathematical methods to analyse volatilities in capital markets. Furthermore, he illustrates how to apply and extend the Black-Scholes theory to several fields in finance. In the final section of the book, the author introduces the readers to the fundamentals of stochastic analysis and presents examples of applications. This book builds on the previous volume of the author’s trilogy on quantitative finance. The aim of the second volume is to present and discuss more complex and advanced techniques of modern financial mathematics in a way that is intuitive and easy to follow. As in the previous volume, the author provides financial mathematicians with insights into practical requirements when applying financial mathematical techniques in the real world.
Author(s): Gerhard Larcher
Series: Springer Texts in Business and Economics
Publisher: Springer
Year: 2023
Language: English
Pages: 362
City: Cham
Contents
1 Volatilities
1.1 Volatility I: Historical Volatility
1.2 Volatility II: ARCH Models
1.3 Volatility III: How to Use and Forecast Volatility
1.4 Volatility IV: Magnitude of the Historical Volatility of the S&P500
1.5 Volatility V: Volatility in Derivatives Pricing
1.6 Derivatives Pricing with Time- (and Price-) Dependent Volatility: the Dupire Model
1.7 Implied Volatility
1.8 Implied Volatilities of Call Options and Put Options with Same Expiration and Strike
1.9 Volatility Skews, Volatility Smiles, and Volatility Surfaces
1.10 Inferences From Implied Volatilities About the Market-Anticipated Distribution of the Underlying Asset's Price
1.11 Volatility Indices
1.12 Basic Properties of the VIX
1.13 Relation and Correlations Between VIX and SPX
1.14 Influence of Price- and/or Time-Dependent Volatility on Delta, Gamma, and Theta
1.15 Combined Trading of SPX and VIX for Hedging Purposes
1.16 Relation and Correlations of VIX with Historical and Realized Volatility
1.17 The CBOE S&P500 Put Write Index
1.18 The VIX Calculation Methodology
1.19 The Volatility Weekend Effect
1.20 Derivatives on the VIX: VIX Futures
1.21 VIX Options
1.22 Payoff and Profit Functions of a Trading Strategy for Combinations of SPX and VIX Options
References
2 Extensions of the Black-Scholes Theory to Other Types of Options (Futures Options, Currency Options, American Options, Path-Dependent Options, Multi-asset Options)
2.1 Introduction and Discussions So Far
2.2 Currency Options
2.3 Futures Options
2.4 Valuation of American Options and of Bermudan Options Through Backwardation (the Algorithm)
2.5 Valuation Examples for American Options in the Binomial and in the Wiener Model
2.6 Hedging American Options
2.7 Path-Dependent (Exotic) Derivatives, Definition and Examples
2.8 Valuation of Path-Dependent Options, the Black-Scholes Formula for Path-Dependent Options
2.9 Numerical Valuation Example of a Path-Dependent Option in a Three-Step Binomial Model (European and American)
2.10 The Complexity of Pricing Path-Dependent Options in an N-Step Binomial Model in General and, For Example, for Lookback Options
2.11 Valuation of an American Lookback Option in a Four-Step Binomial Model (Numerical Example)
2.12 Explicit Formulas for European Path-Dependent Options, For Example, Barrier Options
2.13 Explicit Formulas for European Path-Dependent Options, For Example, Geometric Asian Options
2.14 Brief Comment on Hedging Path-Dependent Derivatives
2.15 Valuation of Derivatives Using Monte Carlo Methods, Basic Principle
2.16 Valuation of European Path-Dependent Derivatives with Monte Carlo Methods
2.17 Monte Carlo Valuation of Asian Options
2.18 Monte Carlo Valuation of Barrier Options
2.19 Barrier Options in Turbo and Bonus Certificates
2.20 Estimating Greeks (Especially Delta and Gamma) of Derivatives with Monte Carlo
2.21 Estimating Delta and Delta Hedging for Path-Dependent Derivatives (e.g. Geometric Asian Option)
2.22 Some Fundamental Remarks on Monte Carlo Methods and on the Convergence of Monte Carlo Methods
2.23 Some Remarks on Random Numbers
2.24 A Remark on Quasi-Monte Carlo Methods
2.25 An Example of Low-Discrepancy QMC Point Sets: The Hammersley Point Sets
2.26 Variance Reduction Methods for Monte Carlo
2.27 Using Monte Carlo with Control Variates to Value an Arithmetic Asian Option
2.28 Multi-asset Options
2.29 Modelling Correlated Financial Products in the Wiener Model, Cholesky Decomposition
2.30 Valuation of Multi-asset Options
2.31 Example of Pricing a Multi-asset Option with MCand with QMC
References
3 Fundamentals: Stochastic Analysis and Applications, Interest Rate Dynamics, and Basic Principles of Pricing Interest Rate Derivatives
3.1 Modelling of Interest Rate Dynamics
3.2 Differential Representation of Stochastic Processes: Heuristic Introduction
3.3 Simulation of Ito Processes and Basic Models
3.4 Excursus: The Ito Formula and Differential Notation of the GBM
3.5 Interest Rate Modelling Using Mean-Reverting Ornstein-Uhlenbeck
3.6 Examples of Interest Rate Derivatives and a Principal Methodology for Pricing such Derivatives
3.7 Basic Concepts of Frictionless Interest Rate Markets: Zero-Coupon Bonds and Interest Rates
3.8 Fixed and Floating Rate Coupon Bonds
3.9 Interest Rate Swaps
3.10 Valuation of Bond Prices and Interest Rate Derivatives in a Short-Rate Approach
3.11 The Mean-Reverting Vasicek Model and the Hull-White Model for the Short Rate
3.12 Affine Model Structures of Bond Prices
3.13 Bond Prices in the Vasicek Model and Calibration in the Vasicek Model
3.14 Bond Prices in the Hull-White Model and Calibration in the Hull-White Model
3.15 Valuation and Put-Call Parity of Call and Put Options on Bond Prices
3.16 Valuation of Caplets and Floorlets (as Well as Interest Rate Caps and Interest Rate Floors)
3.17 The Black-Scholes Differential Equation
3.18 The Stochastic Ito Integral: Heuristic Explanation and Basic Properties
3.19 Conditional Expectations and Martingales
3.20 The Feynman-Kac Formula
3.21 The Black-Scholes Formula
3.22 The Black-Scholes Model as a Complete Market and Hedging of Derivatives
3.23 The Multidimensional Black-Scholes Model and Its Completeness
3.24 Incomplete Markets (e.g. the Trinomial Model)
3.25 Incomplete Markets (e.g. Non-tradable Underlying Asset)
References
