**Mathematics 16:643:621 Mathematics Finance I**

#### Schedule

The course is offered during the Fall semester.- Class meeting dates: Please visit the University's academic calendar.
- Schedule and Instructor: Please visit the University's schedule of classes for the instructor, time, and room.
- Instructor and Teaching Assistant Office Hours: Please visit the Mathematical Finance program's office hour schedule.

#### Course Abstract

This course is an introduction to the mathematical theory of derivative security (or option) pricing. Fundamental concepts are briefly introduced first using the discrete-time binomial model: financial markets, derivative securities, arbitrage, hedging and replicating portfolios, risk-neutral probabilities, risk-neutral pricing formula, and market completeness. Basic ideas of probability and stochastic processes are reviewed for finite probability spaces and discrete-time processes: conditional expectation, martingales, and Markov processes. After this introduction to finance using discrete-time models, the emphasis shifts to continuous-time models and the main part of the course. Topics covered include a summary of probability measure theory and conditional expectation, Brownian motion and quadratic variation, martingales, Ito integral, stochastic calculus, replicating portfolios and hedging, Black-Scholes-Merton formulae for a European-style call option price, change of measure and Girsanov's Theorem, risk-neutral pricing pricing theory, no-arbitrage and existence of risk-neutral measure, market completeness and uniqueness of risk-neutral measure, Markov property, Feyman-Kac theorem and the connection between stochastic calculus and partial differential equations, and local volatility and stochastic volatility models.#### Pre-requisites and Co-requisites

Ordinary differential equations (01:640:244 or 01:640:252), multivariable calculus (01:640:251), linear algebra (01:640:250), and undegraduate probability theory with calculus (01:640:477 or 01:960:381). An undergraduate course on analysis (01:640:311-312 or 01:640:411-412) or engineering mathematics (01:640:421) or partial differential equations (01:640:423) is recommended but not required.Please visit the prerequisites page for descriptions of Rutgers undergraduate course prerequisites. A solid understanding of undergraduate probability at the level of the textbook by Sheldon Ross, A First Course in Probability, is especially important. Given this background, the course should be accessible to Mathematical Finance master's degree students and graduate students in Computer Science, Economics, Finance, Engineering, Mathematics, Physics, Operations Research, and Statistics.

#### Required Textbooks

Stephen E. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models, Springer Verlag, 2004, ISBN 0-387-40101-8. (Text errata available from author's web site.)Supplementary Textbooks: Stephen E. Shreve, Stochastic Calculus for Finance I: The Binomial Asset Pricing Model, Springer Verlag, 2004; John C. Hull,

*Options, Futures, and other Derivatives*, 7th Edition, Prentice Hall, 2008.

#### Sakai

All course content – lecture notes, homework assignments and solutions, exam solutions, supplementary articles, and computer programs – are posted on Sakai and available to registered students.#### Grading

Class attendance 5%, homework 10%, two midterm exams at 20% each, and final exam 45%. Exams are in-class.#### Class Policies

Please see the MSMF common class policies.#### Weekly Lecturing Agenda and Readings

The lecture schedule below is a sample; actual content may vary depending on the instructor. Please see the Sakai Wiki for the the latest lecture schedule.

Week | Topics | Reading |
---|---|---|

1 | Financial markets and derivative securities; No-arbitrage condition; One bond, one-stock model; Forward contracts No arbitrage pricing; No arbitrage price of an option for the binomial model |
Hull, § 1, 2, & 5 Hull, § 11; Shreve-I, § 1; Pliska, § 1.1, 1.2 |

2 | First Fundamental Theorem of Asset Pricing for a one period, finite state model; State-price vector State-price vectors and risk-neutral measure; Risk neutral pricing formula. Examples. |
Shreve-I, § 1; Hull, § 11; Pliska, § 1.3, 1.4, 1.5 Optional: Duffie, § 1; |

3 | Binomial trees (continued); Probability theory and discrete-time stochastic processes Binomial trees (continued); Risk-neutral measure and option pricing |
Shreve-I, § 2 & 3; |

4 | Binomial trees (continued) Probability spaces |
Shreve-I, § 2 & 3 Shreve-I, § 2; Shreve-II, § 1.1, 1.2, 1.3 |

5 | Expectation, information, and σ-algebras Conditional expectation |
Shreve-I, § 2.2; Shreve-II, § 1.3, 1.5 Shreve-I, § 2.3, 2.4, 2.5; Shreve-II, § 2.1, 2.2, 2.3 pdf Rutgers Math 591 Notes (97 KB) , pdf Chicago Stat 313 Notes (159 KB) , pdf Lyons Notes (339 KB) , pdf Harvey Mudd Math 157 Notes (84 KB) (pdf) |

6 | Brownian motion: Random walks and the central limit theorem Brownian motion: Definition, martingale property, quadratic variation |
Shreve-II, § 3.2 Shreve-II, § 3.3 |

7 | Brownian motion: Markov property The Itô integral: Introduction |
Shreve-II, § 3.3 Shreve-II, § 4.2, 4.3, & 4.4 |

8 | The Itô formula The Black-Scholes-Merton PDE and its solution for European-style call and put option prices. |
Shreve-II, § 4.4 Shreve-II, § 4.5 |

9 | The Black-Scholes-Merton formula, geometry of hedging, put-call parity Multivariable stochastic calculus, Lévy's characterization of Brownian motion, Gaussian processes, Brownian bridge. |
Shreve-II, § 4.5 Shreve-II, § 4.6 & 4.7 |

10 | Change of measure, Radon-Nikodym derivative, Girsanov's theorem for single Brownian motion Discounted stock and portfolio processes as martingales |
Shreve-II, § 1.6, 5.1, & 5.2.1 Shreve-II, § 5.2.2, 5.2.3, & 5.2.4 |

11 | Pricing under risk-neutral measure, derivation of Black-Scholes-Merton formula Martingale representation theorem, Multi-dimensional market model |
Shreve-II, § 5.2.4, & 5.2.5 Shreve-II, § 5.3, 5.4.1 & 5.4.2 |

12 | Existence of risk-neutral measure, no arbitrage, and First fundamental theorem of asset pricing Uniqueness of risk-neutral measure, completeness, and Second fundamental theorem of asset pricing |
Shreve-II, § 5.4.3 Shreve-II, § 5.4.4 |

13 | Option pricing and PDEs | Shreve II, § 6.1, 6.2, 6.3, 6.4, & 6.6 |

14 | Risk-neutral, martingale measure pricing theory and explicit portfolio hedge ratios Overview of Dupire local volatility, Heston stochastic volatility, and jump models Course sequels Introduction |
Steele § 14.3, Shreve II chapters 5 & 6 Shreve II, chapters 6 and 11 Mathematical Finance II, Computational Finance |

#### Library Reserves

All textbooks referenced on this page should be on reserve in the Hill Center Mathematical Sciences Library (1st floor). Please contact the instructor if reserve copies are insufficient or unavailable. Please visit the Mathematical Finance Reference Text List blog for additional textbook suggestions.#### Additional Textbooks

Class lectures will draw on material from the following texts and current research articles. Please see the Rutgers Mathematical Finance Reference Texts blog for additional textbooks. K. Back,*A Course in Derivative Securities: Introduction to Theory and Computation*, Springer, 2005

M. Baxter and A. Rennie,

*Financial Calculus: An Introduction to Option Pricing*, Cambridge, 1996

T. Björk,

*Arbitrage Theory in Continuous Time*, Oxford, 2004

J. C. Hull, Options, Futures, and other Derivatives, 6th Edition, Prentice Hall, 2006

P. Hunt and J. Kennedy,

*Financial Derivatives in Theory and Practice*, Wiley, 2004

M. Jackson and M. Staunton,

*Advanced Modelling in Finance using Excel and VBA*, Wiley, 2001

R. Jarrow and S. Turnbull,

*Derivative Securities*, 2nd edition, South-Western College,1999

I. Karatzas and S. E. Shreve,

*Brownian Motion and Stochastic Calculus*, Springer, 1997