MATH 6910 - Stochastic Calculus in Finance

WINTER 2009

This course will introduce the basic ideas and methods of stochastic calculus and will apply these methods to financial models, particularly the pricing and hedging of derivative securities. We will start by introducing the concepts of arbitrage, hedging and risk-neutral pricing in a discrete-time setting, and will then move to more sophisticated continuous-time models. Along the way we will cover the following mathematical topics: Brownian Motion, Stochastic Integrals, Ito's Formula, Martingales, and Girsanov transformations.

Prerequisites: A solid knowledge of calculus and basic probability theory (random variables, expectation, variance, etc.) will be assumed. The necessary concepts from measure theory and stochastic processes will be introduced as they are needed.

Text: Steven E. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models, Springer.

Instructor: Hanna Jankowski
E-mail: hkj[at]mathstat.yorku.ca
Please include "[6910]" in the subject of your e-mail. Also, plain text messages only, no html.
Office: N621B Ross

[Syllabus]

FINAL: Thursday, May 14th, 7pm-10pm, VH3009.

Office Hours:
Monday, May 11th, 2-3 pm, in my office.
Wednesday, May 13th, 5-7pm, in Ross N638.
Thursday, May 14th, 3-5pm, in my office.

N.B. Please note that I am away starting May 18th.

Announcements:

• [May 13] I have marked the third assignment. For those of you with a mailbox in Ross, I've put your there. I will bring the remaining to the office hours today. All left over will be placed in an envelope for you to pick up outside my door.
• [May 13] I have just posted partial solutions to the third assignment. Please take a look at these!
• [May 12] I get a lot of questions about this one: my current version of the exam is 9 questions, of which you have to do 8. I say "current version" because I'm still fiddling with it.
• [May 11] Here (new) and here (midterm) are the complete aid sheets for the final exam (thanks to Pavan and Cartland for their contributions). Typos on the midterm version have been corrected, but not on the scanned version above. Let me know if you see any typos in the new addition. FYI - I didn't put any Ito formula stuff on there, because I think it's too important to not be committed to memory. Also, I didn't put on any formulas for dX_t, d(D_tX_t), d(D_tS_t), ... I think that you should know dX_t from memory, and all the others can be derived, so you don't need them. If I remember, I will try to re-scan these on the good scanner when I have some spare time, so that you have a version that prints better.
• [April 28] In the L9 notes, pages 6 and 7 should be switched.
• [April 26] Here is your aid sheet from the midterm. I will provide this with your final as well. Suggestions (please, more detailed than last time) for the final are due no later than 3pm on Monday, May 11th.
• [March 31] A conditional probability problem.
• [March 22] In the next problem set I ask you to do some simulations (they help), using R. If you're not very familiar with computers, or would just like some more info on what R can do, here is a step-by-step guide.

Discussion Forum:

To access the forum you will need an AML account. Instructions should be there (in the link) on how to get one. If you have trouble, let me know.

This is the first time that I'm using MOODLE at York, so there will probably be a short period of time before I am fully familiar with the system and how best to use it for our needs. I expect that there will be an adjustment period for us all. For now, if you have questions about how to use the forum, you can send me an e-mail and I'll try to answer it.

Assignments:

• [Assignment 1] [partial solutions] Due Monday, April 6th at 9am.
  • Q6(d): the second distribution mu_2 should have mean theta_2 (and not theta_1).
  • Q5: the sigma algebra in the math display should be called G, and not F.
  • Q8(b): F_n is the sigma algebra generated by X_1 through X_n, that is F_n=sigma(X_1, ... , X_n). The sigma is missing in the type up.
• [Assignment 2] Due Monday, April 27th at 9am.
• [Assignment 3] [partial solutions]Due Monday, May 11th by noon.