Fall 2021 - STAT 490 D100

Selected Topics in Probability and Statistics (3)

Stochastic Processes

Class Number: 5146

Delivery Method: In Person

Overview

  • Course Times + Location:

    Tu 12:30 PM – 2:20 PM
    AQ 5015, Burnaby

    Th 12:30 PM – 1:20 PM
    AQ 5015, Burnaby

  • Exam Times + Location:

    Dec 20, 2021
    7:00 PM – 10:00 PM
    AQ 5046, Burnaby

  • Prerequisites:

    Dependent on the topic covered.

Description

CALENDAR DESCRIPTION:

Topics in areas of probability and statistics not covered in the regular undergraduate curriculum of the department.

COURSE DETAILS:

Prerequisties: None. Students should have some knowledge of option pricing and undergraduate nonmeasure
theoretic probability.

Cross-listing: This course is cross-listed with ACMA 830.

Student Learning Objectives:
As a result of taking STAT 490, students should be able to:

1. Understand the probabilistic foundations needed for stochastic calculus (e.g., sample space, probability
measure, sigma-algebra, measurable space).
2. Understand what are stochastic processes as well as the notion of filtration.
3. Compute expectations and conditional expectations (as well as other relevant moments).
4. Understand the notion of independence.
5. Describe what are martingales.
6. Describe and construct the Brownian motion.
7. Apply stochastic integration.
8. Define stochastic differential equation.
9. Apply Ito’s lemma.
10. Understand how to construct jump processes.
11. Understand the stylized facts of asset returns.
12. Define and construct the main models used in financial econometrics.
13. Understand the main parameter estimation strategies for asset models.
14. Describe the main features of economic scenario generators.
15. Describe the risk-neutral pricing.
16. Understand the binomial option pricing model.
17. Design advanced discrete-time market model.
18. Apply the Radon-Nikodym theorem.
19. Apply Girsanov’s theorem.
20. Explain and employ replication.
21. Understand the martingale representation theorem.
22. Apply option pricing to realistic scenarios.

Course Outline:
This course is divided into fifteen chapters.

Part 1: Stochastic Processes
  • Chapter 1, Probabilistic Foundations: Sample space, Random variable, Probability measure, Distribution, Sigma-algebra, Measurable space, Probability triple.
  • Chapter 2, Stochastic Processes: Stochastic process, Filtration.
  • Chapter 3, Expectations: Independence, Conditional probability, Expectation, Moments, Conditional expectation.
  • Chapter 4, Martingales: Definition, Examples.
  • Chapter 5, Brownian Motion: Scaled random walks, Construction of the Brownian motion.
  • Chapter 6, Stochastic Integral: Riemann integration, Ito integration.
  • Chapter 7, Stochastic Di fferential Equations and Ito’s Lemma: Ordinary differential equations, Ito’s lemma, Product rule, Multidimensional Ito’s lemma.
  • Chapter 8, Jump Processes: Poisson process, Compound Poisson process, Jump processes and their integrals, Stochastic calculus for jump processes.
Part 2: Financial Econometrics Models
  • Chapter 9, Asset Models: Stylized facts of returns, Continuous-time models (the Black-Scholes-Merton model, the Merton model, the Heston model, the Bates model, the Duffie, Pan, and Singleton framework), Discretization, Discrete-time models (Regime-switching models, Autoregressive conditional heteroskedasticity, generalized ARCH, the stochastic volatility model).
  • Chapter 10, Parameter Estimation: Moment-based methods, Likelihoodism and likelihood-based methods, Direct likelihood calculations, Presence of latent variables, Bayesianism and Bayesian-based methods, Markov chain Monte Carlo, Issues and challenges.
Part 3: Actuarial Applications
  • Chapter 11, Economic Scenario Generators: Overview of ESGs, Need for ESGs, Roles of physical and risk-neutral scenarios, Inflation models, Interest rate models, Corporate bond models, Equity index models, International considerations, Integrated models (e.g., Wilkie, AAA model).
Part 4: Advanced Option Pricing
  • Chapter 12, Discrete-Time Market Models: Review of the binomial option pricing model, Price systems and martingale measures, Self-financing strategy and arbitrage, Arbitrage and martingale measures, Attainable claims and price uniqueness, Admissible strategy and martingales, Relationship between replication and pricing, Risk-neutral and martingale measures, Market completeness.
  • Chapter 13, Girsanov’s Theorem and Fundamental Theorems of Asset Pricing: Radon-Nikodym theorem, Girsanov’s theorem, Multidimensional Girsanov’s theorem.
  • Chapter 14, Replication Strategies and Martingale Representation Theorem: Self-financing strategies, Martingale representation theorem, Asset replication.
  • Chapter 15, Option Pricing in Practice: Common option pricing models (Black-Scholes-Merton, Merton, Heston, Heston-Nandi), The Dueffie, Pan and Singleton (2000) framework, Monte Carlo simulation, Fourier inversion.
Course Format:

  • Students will learn in-class through: formal lectures, presentations and problem solving. Out-of-class learning will consist of: readings, internet research, and problem solving.
  • There will be four hours of lecture per week. Some lectures might be rescheduled if necessary.
  • Course material, references, links and messages will be posted on the course website (Canvas). Students are expected to read the material and to attempt as many problems as possible before attending the lectures.
  • Regular assignments will be given throughout the semester.
  • Office hours may be used for further explanations and relevant discussions.
  • You are encouraged to discuss problems in small groups. However, you must work alone when writing up the solutions to the problems.
  • Every attempt will be made for the exam questions to verify the objectives given above. Therefore, it is recommended that students regularly consult these objectives when working on problems during the semester and when studying for the exams

COURSE-LEVEL EDUCATIONAL GOALS:

This course aims to develop a thorough understanding of stochastic calculus and the standard probabilistic tools used in the financial and the insurance practice.

Grading

  • Assignments 20%
  • Projects and Presentations 50%
  • Final Exam 30%

NOTES:

The pass mark is 50%. The final grade will be allocated according to the student’s achievement in the course. Under no circumstances will late assignments be accepted.

Missing an exam will result in a mark of 0 unless the student was prevented from taking it due to medical reasons with convincing evidence. Students should use the “Health Care Provider Statement” form available at http://www.sfu.ca/content/dam/sfu/students/pdf/healthcare-statement-general.pdf. Should you miss an exam, you must let the instructor know as soon as possible. Under no circumstances will make up exams be given. A student present at the start of an exam will have his/her exam marked even if he/she leaves early for any reason.

Academic Dishonesty:
Students are expected to take responsibility for their learning. To assist students in understanding scholarly expectations, the following actions are examples of violations of the Student Academic Integrity Policy. Please note that this list is not exhaustive.

  • Plagiarism
  • Submitting shared work as individual work (i.e, collusion)
  • Submitting an assignment or paper more than once (i.e, cheating)
  • Cheating during an exam: Using any device/mobile phone to receive or share information during the exam
  • Submitting a purchased assignment or essay (i.e, contract cheating)
  • Falsifying documents (e.g., Doctor’s note) to gain an advantage Accessing exam information from an unauthorized source (i.e., pay to pass)
For further details, visit https://www.sfu.ca/students/academicintegrity/what-is-it.html.

Materials

REQUIRED READING:

None

RECOMMENDED READING:

  • Hull, J. C. (2006). Options, Futures, and Other Derivatives. Pearson.
  • Lyaso, A. (2017). Stochastic Methods in Asset Pricing. The MIT Press.
  • McDonald, R. L. (2006). Derivatives Markets. Pearson.
  • Shreve, S. (2012). Stochastic Calculus for Finance I: The Binomial Asset Pricing Model. Springer Science & Business Media.
  • Shreve, S. (2004). Stochastic Calculus for Finance II: Continuous-Time Models. Springer Science & Business Media.
Many other references will be provided throughout the semester.

Department Undergraduate Notes:

Students with Disabilities:
Students requiring accommodations as a result of disability must contact the Centre for Accessible Learning 778-782-3112 or csdo@sfu.ca


Tutor Requests:
Students looking for a tutor should visit hhttps://www.sfu.ca/stat-actsci/all-students/other-resources/tutoring.html. We accept no responsibility for the consequences of any actions taken related to tutors.

Registrar Notes:

ACADEMIC INTEGRITY: YOUR WORK, YOUR SUCCESS

SFU’s Academic Integrity web site http://www.sfu.ca/students/academicintegrity.html is filled with information on what is meant by academic dishonesty, where you can find resources to help with your studies and the consequences of cheating.  Check out the site for more information and videos that help explain the issues in plain English.

Each student is responsible for his or her conduct as it affects the University community.  Academic dishonesty, in whatever form, is ultimately destructive of the values of the University. Furthermore, it is unfair and discouraging to the majority of students who pursue their studies honestly. Scholarly integrity is required of all members of the University. http://www.sfu.ca/policies/gazette/student/s10-01.html

TEACHING AT SFU IN FALL 2021

Teaching at SFU in fall 2021 will involve primarily in-person instruction, with approximately 70 to 80 per cent of classes in person/on campus, with safety plans in place.  Whether your course will be in-person or through remote methods will be clearly identified in the schedule of classes.  You will also know at enrollment whether remote course components will be “live” (synchronous) or at your own pace (asynchronous).

Enrolling in a course acknowledges that you are able to attend in whatever format is required.  You should not enroll in a course that is in-person if you are not able to return to campus, and should be aware that remote study may entail different modes of learning, interaction with your instructor, and ways of getting feedback on your work than may be the case for in-person classes.

Students with hidden or visible disabilities who may need class or exam accommodations, including in the context of remote learning, are advised to register with the SFU Centre for Accessible Learning (caladmin@sfu.ca or 778-782-3112) as early as possible in order to prepare for the fall 2021 term.