Stochastic Calculus in Finance

- Second Assignment (corrected). Solutions
- Notes on American options
- Notes on Asset allocation
- Notes on Girsanov transformations
- Practice problems (Ito's lemma) and solutions.
- First Assignment and solutions
- Course outline

- 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: Continuous-Time Models*

- 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

- 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 stochastic calculus.
- 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)

- 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)

- 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.

- 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.

- 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

- I continued with the discussion of Ito's lemma (section 4.4). In particular, I stated and compared several versions of this result - for Brownian motion, for semimartingales, and then multidimensional versions (section 4.6).
- 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.

- 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 is complete.
- 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

- I derived the BSM partial differential equation.

- 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.

- 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 problem.

- I finished discussing portfolio optimization (see the posted notes).
- I talked about American option pricing (see the posted notes)