AN INTRODUCTION TO FINANCIAL OPTION VALUATION Mathematics Stochastics and Computation 1st edition by Desmond Higham ISBN 0521547571 978-0521547574

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ISBN 10: 0521547571
ISBN 13: 978-0521547574
Author: Desmond Higham 

This is a lively textbook providing a solid introduction to financial option valuation for undergraduate students armed with a working knowledge of a first year calculus. Written in a series of short chapters, its self-contained treatment gives equal weight to applied mathematics, stochastics and computational algorithms. No prior background in probability, statistics or numerical analysis is required. Detailed derivations of both the basic asset price model and the Black–Scholes equation are provided along with a presentation of appropriate computational techniques including binomial, finite differences and in particular, variance reduction techniques for the Monte Carlo method. Each chapter comes complete with accompanying stand-alone MATLAB code listing to illustrate a key idea. Furthermore, the author has made heavy use of figures and examples, and has included computations based on real stock market data.

AN INTRODUCTION TO FINANCIAL OPTION VALUATION Mathematics Stochastics and Computation 1st Table of contents:

1. Options

1.1 What are options?

1.2 Why do we study options?

1.3 How are options traded?

1.4 Typical option prices

1.5 Other financial derivatives

1.6 Notes and references

1.7 Program of Chapter 1 and walkthrough

2. Option valuation preliminaries

2.1 Motivation

2.2 Interest rates

2.3 Short selling

2.4 Arbitrage

2.5 Put–call parity

2.6 Upper and lower bounds on option values

2.7 Notes and references

2.8 Program of Chapter 2 and walkthrough

3. Random variables

3.1 Motivation

3.2 Random variables, probability and mean

3.3 Independence

3.4 Variance

3.5 Normal distribution

3.6 Central Limit Theorem

3.7 Notes and references

3.8 Program of Chapter 3 and walkthrough

4. Computer simulation

4.1 Motivation

4.2 Pseudo-random numbers

4.3 Statistical tests

4.4 Notes and references

4.5 Program of Chapter 4 and walkthrough

5. Asset price movement

5.1 Motivation

5.2 Efficient market hypothesis

5.3 Asset price data

5.4 Assumptions

5.5 Notes and references

5.6 Program of Chapter 5 and walkthrough

6. Asset price model: Part I

6.1 Motivation

6.2 Discrete asset model

6.3 Continuous asset model

6.4 Lognormal distribution

6.5 Features of the asset model

6.6 Notes and references

6.7 Program of Chapter 6 and walkthrough

7. Asset price model: Part II

7.1 Computing asset paths

7.2 Timescale invariance

7.3 Sum-of-square returns

7.4 Notes and references

7.5 Program of Chapter 7 and walkthrough

8. Black–Scholes PDE and formulas

8.1 Motivation

8.2 Sum-of-square increments for asset price

8.3 Hedging

8.4 Black–Scholes PDE

8.5 Black–Scholes formulas

8.6 Notes and references

8.7 Program of Chapter 8 and walkthrough

9. More on hedging

9.1 Motivation

9.2 Discrete hedging

9.3 Delta at expiry

9.4 Large-scale test

9.5 Long-Term Capital Management

9.6 Notes

9.7 Program of Chapter 9 and walkthrough

10. The Greeks

10.1 Motivation

10.2 The Greeks

10.3 Interpreting the Greeks

10.4 Black–Scholes PDE solution

10.5 Notes and references

10.6 Program of Chapter 10 and walkthrough

11. More on the Black–Scholes formulas

11.1 Motivation

11.2 Where is μ?

11.3 Time dependency

11.4 The big picture

11.5 Change of variables

11.6 Notes and references

11.7 Program of Chapter 11 and walkthrough

12. Risk neutrality

12.1 Motivation

12.2 Expected payoff

12.3 Risk neutrality

12.4 Notes and references

12.5 Program of Chapter 12 and walkthrough

13. Solving a nonlinear equation

13.1 Motivation

13.2 General problem

13.3 Bisection

13.4 Newton

13.5 Further practical issues

13.6 Notes and references

13.7 Program of Chapter 13 and walkthrough

14. Implied volatility

14.1 Motivation

14.2 Implied volatility

14.3 Option value as a function of volatility

14.4 Bisection and Newton

14.5 Implied volatility with real data

14.6 Notes and references

14.7 Program of Chapter 14 and walkthrough

15. Monte Carlo method

15.1 Motivation

15.2 Monte Carlo

15.3 Monte Carlo for option valuation

15.4 Monte Carlo for Greeks

15.5 Notes and references

15.6 Program of Chapter 15 and walkthrough

16. Binomial method

16.1 Motivation

16.2 Method

16.3 Deriving the parameters

16.4 Binomial method in practice

16.5 Notes and references

16.6 Program of Chapter 16 and walkthrough

17. Cash-or-nothing options

17.1 Motivation

17.2 Cash-or-nothing options

17.3 Black–Scholes for cash-or-nothing options

17.4 Delta behaviour

17.5 Risk neutrality for cash-or-nothing options

17.6 Notes and references

17.7 Program of Chapter 17 and walkthrough

18. American options

18.1 Motivation

18.2 American call and put

18.3 Black–Scholes for American options

18.4 Binomial method for an American put

18.5 Optimal exercise boundary

18.6 Monte Carlo for an American put

18.7 Notes and references

18.8 Program of Chapter 18 and walkthrough

19. Exotic options

19.1 Motivation

19.2 Barrier options

19.3 Lookback options

19.4 Asian options

19.5 Bermudan and shout options

19.6 Monte Carlo and binomial for exotics

19.7 Notes and references

19.8 Program of Chapter 19 and walkthrough

20. Historical volatility

20.1 Motivation

20.2 Monte Carlo type estimates

20.4 Maximum likelihood estimate

20.5 Other volatility estimates

20.6 Example with real data

20.7 Notes and references

20.8 Program of Chapter 20 and walkthrough

21. Monte Carlo Part II: variance reduction by antithetic variates

21.1 Motivation

21.2 The big picture

21.3 Dependence

21.4 Antithetic variates: uniform example

21.5 Analysis of the uniform case

21.6 Normal case

21.7 Multivariate case

21.8 Antithetic variates in option valuation

21.9 Notes and references

21.10 Program of Chapter 21 and walkthrough

22. Monte Carlo Part III: variance reduction by control variates

22.1 Motivation

22.2 Control variates

22.3 Control variates in option valuation

22.4 Notes and references

22.5 Program of Chapter 22 and walkthrough

23. Finite difference methods

23.1 Motivation

23.2 Finite difference operators

23.3 Heat equation

23.4 Discretization

23.5 FTCS and BTCS

23.6 Local accuracy

23.7 Von Neumann stability and convergence

23.8 Crank–Nicolson

23.9 Notes and references

23.10 Program of Chapter 23 and walkthrough

24. Finite difference methods for the Black–Scholes PDE

24.1 Motivation

24.2 FTCS, BTCS and Crank–Nicolson for Black–Scholes

24.3 Down-and-out call example

24.4 Binomial method as finite differences

24.5 Notes and references

24.6 Program of Chapter 24 and walkthrough

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