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MATH6910.03, Fall 2002
Stochastic Calculus in Finance
Announcements and assignments will be posted here as they
Final grades are posted outside my office. Anyone who can't
come in to check grades can e-mail me to ask what they got. People
can drop by my office to pick up project reports. If people want
to look over their final exams, they should make an appointment to
do so in my office.
Project presentation days/times/rooms are listed at the bottom
of this page on the daily schedule. The writeups are due anytime
before December 20. There will be a data projector and laptop
available in the room for presentations, as well as an overhead
projector. Presentations should be 15-20 minutes, and are meant
to give the motivation and main ideas of the topic. Details should
be left to the reports, which should be 10-20 pages in length.
Note that one group will be presenting before
class on December 3, starting at 11:00, so please come early that
day if you can.
Tuesdays, 11:30-2:30, in 129 CCB, with the normal 30 minutes of breaks.
Note that since several students have a class immediately beforehand,
we will normally start at 11:40, take a 10 minute break roughly half
way through, and then finish at 2:20.
Stochastic Calculus and Financial Applications, by J.M. Steele (Springer Verlag)
The first midterm will cover the material we have done in class, up
to the end of Ito's lemma. Since stochastic integrals and Ito's lemma
weren't on the first assignment, you should try the practice problems
(see below) on those topics. Solutions will be available on this webpage
in due course.
Date of First Class
Note that graduate courses in the Department of Mathematics and Statistics
normally start the second week of term. But as I will cancel class during the
Schulich reading week, I in fact held the first lecture on September 10.
Students who missed the first class can make up the material later on, as
the material from the first class won't be applied till later in the course.
Students unable to make up the material should contact me.
Assignments or Documents
Topics and Lecture Schedule
On September 10:
- I surveyed the main ideas of the course.
- I covered the binomial tree model, including hedging, completeness
no-arbitrage pricing, and the role of risk-neutral valuation.
On September 17:
- I described the basic mathematical model for probability (section 4.1).
- I introduced sigma-fields, and discussed how they model information.
On September 24:
- I discussed expectations and the dominated convergence
theorem (p. 277-278).
- I defined conditional expectations (section 4.2)
- I derived the basic properties of conditional expectations.
On October 1:
- I defined Brownian motion (sections 3.0-3.1).
- I sketched several constructions of Brownian motion (sections 3.2-3.5,
- I showed that Brownian motion is not differentiable, and has
infinite variation, so that ordinary calculus fails to apply to
Brownian paths (sections 5.1-5.2).
On October 8:
- I gave out the first assignment
- I proved that Brownian motion has quadratic variation t (top of p. 87).
- I defined elementary stochastic integrals, and explained why
the theory breaks down for anitipating integrands.
- I stated the Ito isometry and used it to construct the general
stochastic integral (section 6.1).
On October 15:
- I proved the Ito isometry (remainder of section 6.1).
- I introduced stochastic integrals as processes (section
6.2), and as martingales.
On October 22:
- I proved the basic version of Ito's lemma.
- I defined solutions to SDE's, and stated the basic existence theorem
- I solved the SDE for geometric Brownian motion
On October 29 there was no class (Schulich reading week)
On November 5:
- I reviewed the course material
- I defined semimartingales and their quadratic variation
- People wrote the midterm exam (1 hour)
On November 12:
- I went over the exam
- I defined joint quadratic variation, and the box calculus
- I stated the n-dimensional version of Ito's lemma, and applied
it to the integration-by-parts formula
- I stated the martingale representation theorem, and a preliminary
version of Girsanov's theorem
On November 19:
- I defined a self-financing portfolio, and the risk-neutral probability
- I proved that the Brownian market is complete
- I showed how to price general options using risk-neutral expectations
- I derived the BSM pricing formula, and the BSM PDE
- I derived the formula for delta-hedging
On November 26:
- I discussed parameter estimation, numerical pricing
algorithms, and completeness.
- I solved the Ornstein-Uhlenbeck SDE.
I held class on December 3. If anybody missed it because of course
conflicts, they should come to see me, in order to go over the
material covered. There was a project presentation on lookback options.
- I discussed Girsanov's theorem, using a prepared handout (see above)
- In particular, I explained how the market price of risk is used
if one needs to price using the real-world measure.
- I described the basic set up of term structure models.
- I explained how to price bonds in the Vasicek model, using
either the SDE or PDE approach.
December 4, 11:30-2:30 in S170 Ross: project presentations
(GARCH, Barrier options)
December 6, 11:30-2:30 in S170 Ross: project presentations
(American options, stochastic volatility, Black-Derman-Toy model,
investment timing, asset allocation, term structure models)
December 13, 11:30-2:30, in S203 Ross: Final Exam.