# MVE095/MMG810, Options and mathematics, 2018/19

## Latest news

Welcome to the course!

The schedule for the course can be found in TimeEdit.

10/25: Lecture notes added, see Course literature

10/25: MATLAB project added (see here)

11/08: OBS! The location for the next Friday 11/16 class is not Pascal, but EA. Don't get lost!

12/04: Modified Matlab codes added (see here)

## Teachers

Course coordinator/Lecturer/Teaching assistant: Alexandr Usachev (usachev@chalmers.se) I reply to e-mails from Monday to Friday  9 am - 5 pm.

Course representatives:

TKMAS               brauns@student.chalmers.se              Simon Braun

TKIEK                 mattc@student.chalmers.se                Mattias Carlsson

TKIEK                 zingmark@student.chalmers.se          William Zingmark

TKIEK                 zingmark@student.chalmers.se          William Zingmark

## Course literature

Lecture notes: Introduction to options pricing theory (
pdf)

## Program

#### Lectures

Day
Sections Contents
6 Nov
1.1
Basic financial concepts. Long and short position. Portfolio.
7 Nov
1.1
Historical volatility.  Options. European/American financial derivatives.
8 Nov
1.1
Money market. Frictionless markets.
9 Nov 1.2 Qualitative properties of option prices. Put-call parity. Optimal exercise of American put options.

13 Nov 2.1, 2.2 Binomial markets. Self-financing portfolio
14 Nov
2.3 Arbitrage portfolio. Arbitrage-free binomial markets.
15 Nov
lektion

Proof of Th. 1.1 and Th. 1.2. Exercises 1.8, 1.9
16 Nov 3.1 Binomial price of European derivatives.

20 Nov 3.2 Hedging portfolio of European derivatives on binomial markets.
21 Nov
4.1, 4.2, 4.3 Binomial price of American derivatives. Optimal exercise time of American put options.
22 Nov
lektion

Proof of Th. 3.1. Exercises 3.2, 3.3
23 Nov 4.4 Hedging portfolio of American derivatives. Cash flow.

27 Nov 2.4, 3.3, 4.5 Computation of the binomial price of European/American derivatives with Matlab.
28 Nov
5.1, 5.2 Finite probability spaces. Random variables. Independence.
29 Nov
lektion

Exercise 3.7. Proof of Th. 4.1. Exercise 4.4
30 Nov 5.2, 5.3 Expectation and conditional expectation.  Stochastic processes. Martingales.

4 Dec 5.4 Applications of probability theory to the binomial model
5 Dec
5.5 General probability spaces. Central limit theorem. Brownian motion.
6 Dec
lektion

Exercises 5.9, 5.11, 5.12,  5.17
7 Dec 6.1 Black-Scholes markets.

11 Dec 6.2, 6.3 Black-Scholes price of European derivatives. Hedging portfolio. Black-Scholes price and hedging portfolio of European call and put options
12 Dec
6.4, 6.5, 6.6 Black-Scholes price of binary options. Implied volatility. Standard European derivatives on a dividend paying stock
13 Dec
lektion

Proof of Theorem 5.7(ii), Exercises 5.19, 5.20, 5.27
14 Dec 6.7 Optimal exercise time of American calls on a dividend paying stock.

8 Jan
Review
9 Jan
lektion

Review old exams
10 Jan
lektion

Review old exams
11 Jan
lektion

Review old exams

#### Recommended exercises

The recommended exercises are those marked with the symbol (●) or (☆) in the lecture notes, plus the following from

Appendix D:

Chapter 1: 1, 3, 5, 6

Chapter 3: 1, 2, 3, 4, 6

Chapter 4: 1, 2, 3

Chapter 5: 1, 2

Chapter 6: 1, 2, 3, 4, 5, 6

## Computer labs

#### Reference literature:

Learning MATLAB, Tobin A. Driscoll ISBN: 978-0-898716-83-2 (The book is published by SIAM).

Modified Matlab codes:

European derivatives (see here)

American derivatives (see here)

How to use the function argument (see here)

## Course requirements

The learning goals of the course can be found in the course plan.

## Assignments

Matlab project (pdf)

The deadline for the submission of the report is December, 14th. The results will be announced the week thereafter.

Remarks:
(1) The Matlab project is NOT compulsary.
(2) The bonus points of the Matlab project count for the exam in January and the two following re-exams.
(3) Note that each option in the project has many different variants (e.g.,compound options exist as call on call, call on put, etc.). While you should describe all possible variants, for the numerical part you can focus on one example for each option.
(4) Attach only the most relevant Matlab codes (e.g., the Matlab functions to compute the price). In particular, DO NOT attach the Matlab scripts used to create the plots
(5) The main part of the project is the Matlab code and the pictures. As far as the theoretical evaluation you can simply list the different approaches used in the literature and give some reference to where details can be found. If there exists an exact formula, you can write it in your report. Moreover you could give an example with the binomial model for N=2 or 3... this is really up to you! As I ask you to write no more than 5 pages for each option (including the Matlab code and the figures), you really can't write much.

## Examination

The exam is on January 18th, 2019

The test comprises 15 points and to pass at least 6 points are required
- at GU a result greater than or equal to 11 points is graded VG;
- at Chalmers a result greater than or equal to 9 points and smaller than12 points is graded 4 and a result greater than or equal to 12 points is graded 5.

The Matlab project gives max 1 point

The test is divided in three parts, each one giving a maximum of 5 points.

One part will be of theoretical nature and will require to state and prove one or more of the following theorems in the lecture notes (max. 4 points) :

Theorem 1.2, Theorem 2.2, Theorem 2.3, Theorem 3.3, Theorem 4.3, Theorem 5.4, Theorem 5.5, Theorem 6.1, Theorem 6.2, Theorem 6.3, Theorem 6.5, Theorem 6.6, Theorem 6.9

and to provide and explain one of the following definitions in the lecture notes (max. 1 point):

Definition 1.1, Definition 2.3, Definition 2.4, Definition 3.1, Definition 3.2,Definition 3.3, Definition 4.1, Definition 4.2, Definition 4.3, Definition 5.15, Definition 6.1, Definition 6.2

Remarks:
(i) If in the exam it is asked to prove theorem X and the proof requires the result of theorem Y, you don't need to prove also Y.
(ii) When asked to prove one of the above theorems, the question does not necessarily contain the exact statement as it appears in the lecture notes. For instance, a question asking to prove theorem 6.5 could read like "Derive the Black-Scholes price of European call options".
(iii) The explanation of the definition need not be the same as in the lecture notes. You can use your own intuition.

## Examination procedures

In Chalmers Student Portal you can read about when exams are given and what rules apply on exams at Chalmers. In addition to that, there is a schedule when exams are given for courses at University of Gothenburg.

Before the exam, it is important that you sign up for the examination. If you study at Chalmers, you can do this from the Chalmers Student Portal, and if you study at University of Gothenburg, you sign up via GU's Student Portal.

At the exam, you should be able to show valid identification.

At the annual (regular) examination:
When it is practical, a separate review is arranged. The date of the review will be announced here on the course homepage. Anyone who can not participate in the review may thereafter retrieve and review their exam at the Mathematical Sciences Student office. Check that you have the right grades and score. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.

At re-examination:
Exams are reviewed and retrieved at the Mathematical Sciences Student office. Check that you have the right grades and score. Any complaints about the marking must be submitted in writing at the office, where there is a form to fill out.

## Old exams

January 2018 (pdf)   April 2018 (pdf)

January 2017 (pdf)   April 2017 (pdf)    August 2017 (pdf)

April 2016 (pdf)    June 2016 (pdf)   August 2016 (pdf)

April 2015 (pdf)     June 2015 (pdf)     August 2015 (pdf)

May 2014 (pdf)   August 2014 (pdf)

Some older exams

2012 (pdf1, pdf2, pdf3)

2013 (pdf1, pdf2, pdf3)