Shreve, “Stochastic calculus for finance I: The binomial asset pricing model”, and “II: Continuous time models”.
Recommended: Neftci, “An introduction to the mathematics of financial derivatives” and Hull, “Options, futures, and other derivatives”.

Midterm 30%
Homework 20%
Final exam or alternatively presentation: 50%.

There are 8 homework sets which count 2% of the course each (the two best count twice). They each consist of 3-4 problems of which only some will be graded.

12-12:40 Lecture 1 (Math refresher, chapters 1-2: number theory and algebra)
12:45-1:25 Problem session 1
1:30-2:10 Lecture 2 (Math refresher, chapter 3: calculus (we skip chapter 4: statistics)
2:15-2:45 Problem session 2

Lecture 1: 2.1 and 2.2: Sample spaces, probability experiments, outcomes and events, random variables, probability distribution of a RV, expectation of a RV.
Lecture 2: 1.1 (binomial pricing model): pricing a derivative (in the case where the interest is 0).
Session 3: Interactive “examples game”
Session 4: Discussion of a student’s example of a derivative asset in a 3-element sample space.

Lecture 1: 1.2: Calculation of the no-arbitrage price of a derivative asset by replicating using a stock and the money market.
Session 2: Continuation.

Lecture 2: 2.3 (conditional expectation $\mathbb E_n S_{N}$), 2.4 (martingales $E_n S_{n+1} = S_n$), 2.5 (Markov processes $E_n f(X_{N}) = g(X_n)$). See also post about discrete Ito’s lemma.

Session 4: Calculated the price of the derivative $V_n=(S_n-5)^+$ for $n=1$, $n=2$, $n=3$, and obtained 1.2, 1.76, 2.304.

3.1 (change of measure), 3.2 (Radon-Nikodym), 4 (American derivatives).
The Radon-Nikodym derivative of $\tilde{\mathbb{P}}$ w.r.t. $\mathbb P$ is
$$
Z(\omega)=\frac{\tilde{\mathbb P}(\omega)}{\mathbb P(\omega)}
$$
The state price of $\omega$ is
$$
\frac{\tilde{\mathbb P}(\omega)}{(1+r)^N}
$$
since this is the discounted risk-neutral expected value of $1_{\omega}$. See also post on State prices and post on American derivatives.
These things will be revisited in II-5.4 if we get that far. We skip 3.3 (utility functions).

5.2, 5.3 (Random walk), 5.4 (perpetual put).
Went over perpetual put, random walk, principle of reflection. Got as far as the recursive formula for the Catalan numbers and the distribution of $\tau_1$; Class #6 is to start with the simplified formula for the distribution of $\tau_2$.

6.2, 6.3; did not get to 6.4, 6.5 (futures and marking to market) (Interest rates)

Volume II
Chapter II-1 (probability and measure theory)

Chapter II-2 ($\sigma$-algebras)

Chapter II-3 (Brownian motion) 3.1-3.5

Chapter II-4, 4.1-4.6 (Ito integral, Ito-Doeblin formula; and HW about forward measures and about the Reflection Principle)

All Vol. I homework returned; ready to study for the Midterm!
Chapter II-4 (Black-Scholes PDE and formula)

Midterm covering Volume I.

651 Fall 2012 Midterm

651 Fall 2012 Midterm Solutions
Showed upper and lower probabilities for a trinomial asset pricing problem.
Derived the Black-Scholes-Merton formula for option prices.
Verified a solution as in problem II-3.5, and then went over the numerical example from my post on doing Black-Scholes by hand.
Partly derived the solution to the Vasicek SDE.

More Black-Scholes and review.

Early Final exam covering Volume II, Chapters 1-4
The final exam will focus on sections:
3.2 Scaled random walks [which in the limit become Brownian motion]
3.3 Brownian motion [which is a basis for modeling continuous-time trading]
3.4 Quadratic variation [which explains what exactly is meant by $(dW)^2=dt$ and $dWdt=0$]
4.2 Ito integral for simple integrands [which explains what is meant by $\Delta_t dW_t$]
4.3 Ito integral for general integrands
4.4 Ito-Doeblin formula [which calculates $df(t,W_t)$ and allows for solving SDEs]
4.5 Black-Scholes-Merton equation [which must be satisfied by the price process of a derivative]

Final exam covering Volume II, Chapters 1-4
651 Fall 2012 Final Solutions

HW1: Ch. 1 #1,4,6 due Sep. 28

HW2: Ch. 2 #1,2,8 due Oct. 5

HW3: Ch. 3 #1,2,3 due Oct. 12
Ch. 4 #1 due Oct. 12

HW4: Ch. 4 #3,4 due Oct. 26
Ch. 5 #1,5 due Oct. 26

HW5: Ch. 5 #6, Ch. 6 #1,3,5 due Oct. 26

HW6: Ch. 1 #2,6,7 and Ch. 2 #5: due Nov. 2
HW7: Ch. 2 #7, 8 and Ch. 3 #2, 5: due Nov. 16
HW8: Ch. 3 #6 (GBM and BM with drift) and Ch. 4 # 4 (Stratonovich), 5 (GBM), 8 (Vasicek) due Nov. 30