# Fall 2020 - ACMA 850 G100

## Actuarial Science: Selected Topics (4)

Stochastic Processes

## Overview

• #### Course Times + Location:

Sep 9 – Dec 8, 2020: Tue, 10:30 a.m.–12:20 p.m.
Burnaby

• #### Exam Times + Location:

Dec 18, 2020
Fri, 3:30–6:30 p.m.
Burnaby

## Description

#### COURSE DETAILS:

Course Title: Stochastic Processes for Insurance and Finance

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

Cross-listing: This course is cross-listed with STAT 490.

Student Learning Objectives:
As a result of taking ACMA 850, 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 di erential 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.
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 erential Equations and Ito’s Lemma: Ordinary di erential 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 Bayesianbased 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).
• 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 Due, 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-ofclass learning will consist of: readings, internet research, and problem solving.
• There will be four hours of virtual 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.
• Office 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

Mode of teaching:

• Lecture: mix of synchronous (not recorded) and asynchronous (recorded)
• Final exam: synchronous; date: TBA

#### 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.

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

#### 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.

The virtual exam will be proctored. Specifically, we will be using Zoom which in many ways is equivalent
to live proctoring for in-person exams. The session will not be recorded. Students should be muted to
minimize background noise (invigilators can randomly unmute at any time), be instructed to turn off virtual
backgrounds, and that they must not turn o ff cameras. We may ask students to show their work area before
starting the exam, and should remind students to set aside cell phones or other devices.

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)

## Materials

#### MATERIALS + SUPPLIES:

None

References:

• 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.

Important dates and deadlines for graduate students are found here: http://www.sfu.ca/dean-gradstudies/current/important_dates/guidelines.html. The deadline to drop a course with a 100% refund is the end of week 2. The deadline to drop with no notation on your transcript is the end of week 3.