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Lecture: Mon Wed Fri 1:10 PM - 2:40 PM in CW 107
General Course Info / Math 454 Web Site:
This syllabus is only a part of an entire course website for Math 454.
Here is the
link to the home page
of that Math 454 web site.
The red boxes that come next have been written according to a provided Harpur template.
they are followed by additional information that is not part of the template.
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Bulletin Course Description
(A detailed course Description can be found further down):
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Interest rate models
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principle of no arbitrage
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fundamental theorem of asset pricing
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evaluation of derivatives
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put-call parity
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European put and call options
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binomial models
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Black-Scholes option-pricing model
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American options
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option Greeks
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exotic options
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lognormal distribution
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diffusion process
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Ito's lemma
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simulation and delta-hedging.
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The materials will partially cover the mathematical foundation of
actuarial Exam IFM (formerly MFE).
It is not a preparation for the exam.
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Student Learning Outcomes:
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By the end of this course, students should have acquired basic knowledge
of pricing contingent claims in both discrete time and continuous time markets.
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Contact Hours:
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This is a 4-credit course, which means that,
in addition to the scheduled meeting times,
students are expected to do at least 9.5 hours of course-related work
outside of class each week during the semester.
This includes time spent completing assigned readings,
studying for tests and examinations, preparing written assignments,
and other course-related tasks.
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Prerequisites:
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C or better in both MATH 346 and MATH 454, or consent of instructor.
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Course Schedule:
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See the
Schedule
page of the Math 454 website.
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Course materials:
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- This instructor's Math 454 - Lecture Notes.
We abbreviate this as MF or the MF doc.
(Required)
A downloadable PDF of this document and additional background material
can be found on the
Course material
page of the Math 454 website.
- Stochastic Calculus for Finance II - Continuous Time Models,
by Steven Shreve
We abbreviate this as the "SCF2 text", also just "SCF2".
(Required!)
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Assignments:
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Exams:
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There will be three midterm exams and one final exam.
No calculators, notes, books, cell phones, or laptops are allowed for tests.
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Exam dates can be found on the course home page and schedule page.
Make all arrangements necessary to take the tests at those dates
as it is extremely unlikely that they will be changed.
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Each midterm will last 60 minutes and counts 150 points.
Make an effort to show up 10 minutes early for those exams
so they can start on time.
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The final exam counts 300 points and it will last two hours.
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Date of the Final Exam:
TBD
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Final Exam makeup:
TBD
Makeup Exams:
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You can request a makeup final if you have another final at the same time
(direct conflict) or you have three final exams scheduled within 24 hours.
To request a makeup final,
please contact me by email no later than Monday, Apr 22.
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Quizzes:
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There will be approximately 10 quizzes.
Most of them will not be announced.
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The sum of points of the quizzes will be scaled to 250.
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No notes, books, cell phones, or laptops are allowed.
for the major exams.
By default, neither are calculators.
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Should I allow calculators for a specific quiz (very unlikely),
then note the following:
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Calculators can have statistical functions,
but they cannot be graphing calculators,
they cannot be programmable and cannot have internet access.
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The number of quizzes depends on how well the class understands how to aaply
the definitions, main propositions and theorems.
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Additional quizzes will be given if the class needs to do better.
Quizzes will often not be announced.
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Homework:
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Homework usually will be assigned weekly,
but it will not be graded.
Rather, homework assignments or just some fragments will become part of the quizzes
and you very likely will run out of time if you did not familiarize yourself
with the homework.
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How the Assignments Determine Your Grade:
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You can earn a total of 1,000 points in this course:
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| Final Exam |
300 pts |
300 pts |
| 3 Midterm Exams |
150 pts per exam |
450 pts |
| About n=10 Quizzes |
25 points per quiz.
The total will be scaled to 250 points. |
250 pts |
| Total: |
1,000 pts |
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TENTATIVE
Grading Scale
(unlikely to be adjusted):
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850 - 899 pts → A- |
900 - 1000 pts → A |
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700 - 749 pts → B- |
750 - 799 pts → B |
800 - 849 pts → B+ |
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550 - 599 pts → C- |
600 - 649 pts → C |
650 - 699 pts → C+ |
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500 - 549 pts → D |
Below 500 pts → F |
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Detailed Course Description:
As mentioned on the home page of the course website,
this course could be more aptly described as
"Introduction to Stochastic Calculus With Applications in Quantitative Finance".
We will spend a significant amount of time on the basics of continuous time
stochastic processes which can be represented as the sum of an ordinary
Riemann integral and a "stochastic integral" with respect to a Wiener process.
To do this in a 100% exact fashion, with rigorous proofs,
requires as prerequisite a graduate level
course in a measure theoretically founded probability theory.
But this is an undergraduate level class, and we have to take a different approach.
I will proceed as follows:
- Develop the basics of abstract integration and probability with focus on
conditional expectations and continuous time stochastic processes.
Most proofs will only be outlined or even entirely omitted,
but many precise definitions
and formulations of important theorems and propositions will be given
and continually referenced later on.
- Study a discrete time version (the binomial) model of the pricing of an option,
in a market which rules out arbitrage,
to gain some intuition concerning the continuous time approach.
- Become proficient at working with the Itô formula.
Many financial math problems can be solved with help of that formula.
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A major example of the above is the pricing of financial derivatives
such as a European call option.
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No knowledge of ordinary and partial differential equations is assumed,
and yet we will learn how to associate with certain PDEs
(partial differential equations)
so called diffusion processes.
Those stochastic processes lead to stochastic representations of the solutions
of the underlying PDE.
We will study in detail how this works for the pricing of a European call,
and thus become acquanted with the famous Black Scholes model.
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A possible misconception:
I have heard from Porf.David Biddle, who is involved in undergraduate advising,
that many of you believe this course is specifically meant for students
who want an actuarial degree.
This is definitely not the case.
Rather, this course will prove most suitable to those of you who have a bend towards
the abstract side of mathematics and would like to have a background in mathematical
finance that allows them to study this subject on a graduate level, not necessarily
as a Math major and/or to work as a "quant" in the finance industry.
Helpful , but not required:
Basics of linear algebra:
I will treat a sample as an n-dimensional vector.
See the bottom of the
Course material
page of the Math 454 website for more.
Lectures during campus wide class cancellations:
BU admin leaves it at the discretion of the instructor whether or not
lectures will be held in case of a cancellation due to inclement weather or other
circumstances that transportation to/from campus is suspended.
My policy for this situation, should it occur, is as follows.
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I will hold lectures via Zoom.
I will create a separate session for each such meeting
and publish the link for the meeting via email.
This may happen on short notice, so check your email for an announcement
in case there is a campus wide cancellation of classes.
- I plan to record those sessions and publish the links on the course materials page.
- If a midterm or quiz was scheduled for such a date, then it will be administered
during the next in-class lecture date.
MyCourses (Brightspace):
I plan to use Brightspace as little as possible.
I will only use this software to keep track of your grades.
In particular, all written announcements will be made
on the announcement page of this course site and/or by email.
Students With Disabilities:
Students requesting disability-related accommodations should register with
the Services for Students with Disabilities office (SSD).
They are the appropriate entity on campus to determine and authorize
disability-related accommodations.
For more info please
click here.
Once you are registered with SSD you should approach me during office hours or after lecture
so that you and I can discuss the implementation of your accommodations.
Success:
See the
Advice
page of the Math 454 website.
Attendance Policy and Make-up Policy:
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Attendance will not be taken, but you are advised not to skip lecture:
I will follow the Shreve text and my lecture notes closely whenever possible,
but I will teach some of the material from other sources.
If you cannot spare the time to go to lecture then you should
consider dropping/withdrawing from the course before
your GPA is messed up.
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For all exams and quizzes you miss:
Make-ups will only be given in response to an excused absence.
Excused absences include illness, religious holidays, a major tragedy in the family,
and participation in official BU athletic events.
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I usually count the next quiz double rather than issuing a make-up quiz.
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To be excused, absences must be properly documented.
Such a document should cover the day of the test.
The makeup will usually be scheduled within 1 to 3 days after the missed exam.
You must request a make-up in writing by sending an email.
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When you receive a grade, whether on MyCourses or in class,
you will have one week to discuss that grade before it becomes FINAL.
ChatGPT and other Generative AI tools:
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Generative AI is a moving target,
so the following may be subject to change on short notice!
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You may use generative AI in this course.
I do not consider it an issue for the following reasons.
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All exams and quizzes are taken in class without access to the internet
and they make up 100% of the grade.
Academic Honesty:
Incidents of academic dishonesty will be dealt with severely.
There is precedent for giving an "F" for the course to a student who attempts to advance
his/her grade illegally.
Dishonesty includes, but is not limited to:
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Copying another student's work
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Letting someone copy your work
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lying to or intentionally misleading an instructor
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Signing someone else's name to a document
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Homework assignments only:
only having your name on the PDF but not understanding the proofs.
To eliminate suspicion, only writing/erasing utensils will be permitted
on desks during an in-class exam.
Best wishes for a successful semester!
Michael Fochler
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