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MATH 6910 3.0MW (Winter 2007)
Stochastic Calculus in Finance
Announcements and assignments will be posted here as they
- Grades are posted outside my office. If you can't drop by, send
me an e-mail and I will send you your results. I'm told that FGS will
have them mounted on-line by May 25 at the latest. Your graded project reports
are available to be picked up. If nobody from a group can pick the
graded report up, I'll be happy to mail it out - just tell me a mailing
address. You are welcome to go over your exams in
my office, but you can't take them away.
- The final exam was held 7-10pm on April 17, in CB115 (ie in the
Chemistry Building). The exam
covered course material through Girsanov, and the fundamental theorem
of asset pricing, but not portfolio optimaztion or later topics.
The exam was NOT open book, but as on the midterm you could bring up
to 7 sheets of paper (2-sided) into the exam,
with whatever you like written on them.
- The course met Tuesdays, 7-10pm in S105 Ross (Note: the room
changed from what was originally in the lecture schedule)
- The text was Steven Shreve's Stochastic Calculus for Finance II:
Groups gave 20" class presentations, and submited reports to me
(roughly 10-15 pages). The report was due by the end of April.
Presentations were held 7-10pm on April 10 (in our
regular classroom), and 2-4pm on April 12 (in TEL 0015). A data
projector and an overhead projector were available, but you needed
to bring your own laptops. The schedule for the presentations
- April 10
- Calibration of Hull-White and Vasicek models
- Stochastic Volatility
- American Options
- Convertible bonds
- Barrier options
- April 12
- Pricing of swaps in a CIR model
- Credit VaR
- Copulas and CDOs
- Pricing using the variance Gamma model
On January 9
On January 16
- I explained the the approach the course will take,
and I described the role of continuous time models and complete markets in financial engineering
- I summarized the topics that will be covered, including the role of
- I described the binomial tree model, with emphasis on hedging, completeness,
the no-arbitrage nature of prices, and the role of risk-neutral expectations.
(This model isn't in our text, but can be found in Volume I of Shreve's book,
or in many of the other references. Eg Hull, Baxter-Rennie, Etheridge)
On January 23
- I discussed the basic mathematical model for probability
and random variables (sections 1.1 and 1.2 of the text)
- I explained how sigma-fields model information (section 2.1)
On January 30
- I discussed expectations (sections 1.3 to 1.5) and conditional
expectations (section 2.3), focussing on problems that can arise with
expectations, and on how to manipulate the definition of conditional
expectations. This was related to the construction of martingales, which
tie in naturally to the notion of risk-neutral valuation.
On Feb 6
- I introduced Brownian motion
(section 3.3) and its quadratic variation (section 3.4).
- I mentioned the
reflection principle and the Markov property, but didn't go into many
details about those (see sections 3.5-3.7).
- I introduced the notion
of a stochastic integral for simple integrands, interpreted as the
gains from trading (section 4.2.1). I also gave out the first assignment.
There was no class on Feb 13 (York reading week) or Feb 20 (Schulich
I constructed the stochastic integral (sections 4.2 and 4.3), deriving
the mean, variance, martingale property, and quadratic variation in the
case of simple integrands.
- I sketched (using the Ito isometry) how to
take limits and get the integral for general non-anticipating integrands.
- I worked through an example showing how crazy behaviour (eg infinite
gains from trading) can arise without the non-anticipating condition.
- I stated Ito's lemma, and gave a preview of how it would be used
to solve the SDE for GBM.
On Feb 27
On March 6
- I continued with the discussion of Ito's lemma (section 4.4).
In particular, I stated and compared several versions of this result -
motion, for semimartingales, and then multidimensional versions
- I sketched the proof of the Brownian version of Ito's lemma.
- I defined semimartingales and their quadratic variation.
- I defined the solution to a stochastic differential equation.
- I applied Ito's lemma to get the solutions to the SDE for GBM,
and to the SDE for the Vasicek term-structure model.
On March 13 we had the midterm test. Afterwards
- I did some examples of using Ito's lemma for semimartingales, and I
gave additional discussion of quadratic variation for semimartingales.
- I stated the martingale representation theorem and a preliminary
version of Girsanov's theorem.
- I defined self-financing portfolios and the risk neutral measure, and
I showed that discounted self-financing portfolios correspond to
risk-neutral martingales. Using that I showed that the Brownian market
- As a consequence I obtained the no-arbitrage prices of contingent
claims as discounted risk-neutral expectations. In the case of a Call
option this implies the BSM pricing formula, and more generally gives
non-path-dependent option prices as integrals against a state-price density.
Likewise it shows that prices can be computed by Monte Carlo techniques.
See sections 5.2.2 to 5.2.5
On March 20
- I derived the BSM partial
On March 27
- I derived a formula for the Delta hedge.
- I discussed parameter estimation, ie why the volatility can be
computed perfectly, but the growth rate can't.
- I discussed volatility and implied volatility
- I sketched the proof of the martingale representation theorem.
- I began a discussion of Girsanov tranformations.
On April 3
- I finished discussing Girsanov transformations
(see the posted notes).
- I briefly discussed incomplete markets, and
the fundamental theorem of asset pricing, finishes the
material you are responsible for the final exam.
- I started presenting Merton's solution to the portfolio optimization
On April 10 and 12 there were project
presentations. The exam was on April 17.
- I finished discussing portfolio optimization (see the posted
- I talked about American option pricing (see the posted notes)