Stochastic Calculus in Finance

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.

- Course outline
- Assignment 1 ( Solutions)
- Practice problems (not to be handed in), and solutions. Note that an earlier version of the solutions had two misprints (an 8 instead of a 4 at one point in A, and a 4 for a 2 at one point in C). I also added a few lines of explanation at points where people had trouble following the argument.
- Assignment 2 ( Solutions)
- Notes on Girsanov transformations

- 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, and 5.4).
- 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 measure
- 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.