Introduction to Stochastic Finance with Market Examples, Second Edition presents an introduction to pricing and hedging in discrete and continuous-time financial models, emphasizing both analytical and probabilistic methods. It demonstrates both the power and limitations of mathematical models in finance, covering the basics of stochastic calculus for finance, and details the techniques required to model the time evolution of risky assets. The book discusses a wide range of classical topics including Black–Scholes pricing, American options, derivatives, term structure modeling, and change of numéraire. It also builds up to special topics, such as exotic options, stochastic volatility, and jump processes.
New to this Edition
- New chapters on Barrier Options, Lookback Options, Asian Options, Optimal Stopping Theorem, and Stochastic Volatility
- Contains over 235 exercises and 16 problems with complete solutions available online from the instructor resources
- Added over 150 graphs and figures, for more than 250 in total, to optimize presentation
57 R coding examples now integrated into the book for implementation of the methods
Substantially class-tested, so ideal for course use or self-study
With abundant exercises, problems with complete solutions, graphs and figures, and R coding examples, the book is primarily aimed at advanced undergraduate and graduate students in applied mathematics, financial engineering, and economics. It could be used as a course text or for self-study and would also be a comprehensive and accessible reference for researchers and practitioners in the field.
Author(s): Nicolas Privault
Series: Chapman and Hall/CRC Financial Mathematics Series
Edition: 2
Publisher: CRC Press/Chapman & Hall
Year: 2022
Language: English
Pages: 652
City: Boca Raton
Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Preface
Introduction
1. Assets, Portfolios, and Arbitrage
1.1. Portfolio Allocation and Short Selling
1.2. Arbitrage
1.3. Risk-Neutral Probability Measures
1.4. Hedging of Contingent Claims
1.5. Market Completeness
1.6. Example: Binary Market
Exercises
2. Discrete-Time Market Model
2.1. Discrete-Time Compounding
2.2. Arbitrage and Self-Financing Portfolios
2.3. Contingent Claims
2.4. Martingales and Conditional Expectations
2.5. Market Completeness and Risk-Neutral Measures
2.6. The Cox–Ross–Rubinstein (CRR) Market Model
Exercises
3. Pricing and Hedging in Discrete Time
3.1. Pricing Contingent Claims
3.2. Pricing Vanilla Options in the CRR Model
3.3. Hedging Contingent Claims
3.4. Hedging Vanilla Options
3.5. Hedging Exotic Options
3.6. Convergence of the CRR Model
Exercises
4. Brownian Motion and Stochastic Calculus
4.1. Brownian Motion
4.2. Three Constructions of Brownian Motion
4.3. Wiener Stochastic Integral
4.4. Itô Stochastic Integral
4.5. Stochastic Calculus
Exercises
5. Continuous-Time Market Model
5.1. Asset Price Modeling
5.2. Arbitrage and Risk-Neutral Measures
5.3. Self-Financing Portfolio Strategies
5.4. Two-Asset Portfolio Model
5.5. Geometric Brownian Motion
Exercises
6. Black–Scholes Pricing and Hedging
6.1. The Black–Scholes PDE
6.2. European Call Options
6.3. European Put Options
6.4. Market Terms and Data
6.5. The Heat Equation
6.6. Solution of the Black–Scholes PDE
Exercises
7. Martingale Approach to Pricing and Hedging
7.1. Martingale Property of the Itô Integral
7.2. Risk-Neutral Probability Measures
7.3. Change of Measure and the Girsanov Theorem
7.4. Pricing by the Martingale Method
7.5. Hedging by the Martingale Method
Exercises
8. Stochastic Volatility
8.1. Stochastic Volatility Models
8.2. Realized Variance Swaps
8.3. Realized Variance Options
8.4. European Options – PDE Method
8.5. Perturbation Analysis
Exercises
9. Volatility Estimation
9.1. Historical Volatility
9.2. Implied Volatility
9.3. Local Volatility
9.4. The VIX® Index
Exercises
10. Maximum of Brownian Motion
10.1. Running Maximum of Brownian Motion
10.2. The Reflection Principle
10.3. Density of the Maximum of Brownian Motion
10.4. Average of Geometric Brownian Extrema
Exercises
11. Barrier Options
11.1. Options on Extrema
11.2. Knock-Out Barrier
11.3. Knock-In Barrier
11.4. PDE Method
11.5. Hedging Barrier Options
Exercises
12. Lookback Options
12.1. The Lookback Put Option
12.2. PDE Method
12.3. The Lookback Call Option
12.4. Delta Hedging for Lookback Options
Exercises
13. Asian Options
13.1. Bounds on Asian Option Prices
13.2. Hartman–Watson Distribution
13.3. Laplace Transform Method
13.4. Moment Matching Approximations
13.5. PDE Method
Exercises
14. Optimal Stopping Theorem
14.1. Filtrations and Information Flow
14.2. Submartingales and Supermartingales
14.3. Optimal Stopping Theorem
14.4. Drifted Brownian Motion
Exercises
15. American Options
15.1. Perpetual American Put Options
15.2. PDE Method for Perpetual Put Options
15.3. Perpetual American Call Options
15.4. Finite Expiration American Options
15.5. PDE Method with Finite Expiration
Exercises
16. Change of Numéraire and Forward Measures
16.1. Notion of Numéraire
16.2. Change of Numéraire
16.3. Foreign Exchange
16.4. Pricing Exchange Options
16.5. Hedging by Change of Numéraire
Exercises
17. Short Rates and Bond Pricing
17.1. Vasicek Model
17.2. Affine Short Rate Models
17.3. Zero-Coupon and Coupon Bonds
17.4. Bond Pricing PDE
Exercises
18. Forward Rates
18.1. Construction of Forward Rates
18.2. LIBOR/SOFR Swap Rates
18.3. The HJM Model
18.4. Yield Curve Modeling
18.5. Two-Factor Model
18.6. The BGM Model
Exercises
19. Pricing of Interest Rate Derivatives
19.1. Forward Measures and Tenor Structure
19.2. Bond Options
19.3. Caplet Pricing
19.4. Forward Swap Measures
19.5. Swaption Pricing
Exercises
20. Stochastic Calculus for Jump Processes
20.1. The Poisson Process
20.2. Compound Poisson Process
20.3. Stochastic Integrals and Itô Formula with Jumps
20.4. Stochastic Differential Equations with Jumps
20.5. Girsanov Theorem for Jump Processes
Exercises
21. Pricing and Hedging in Jump Models
21.1. Fitting the Distribution of Market Returns
21.2. Risk-Neutral Probability Measures
21.3. Pricing in Jump Models
21.4. Exponential Lévy Models
21.5. Black–Scholes PDE with Jumps
21.6. Mean-Variance Hedging with Jumps
Exercises
22. Basic Numerical Methods
22.1. Discretized Heat Equation
22.2. Discretized Black–Scholes PDE
22.3. Euler Discretization
22.4. Milshtein Discretization
Exercises
Bibliography
Index
