Stochastic processes

Spring 2014

MATH 672 is intended to be a more mathematical variant of Finance 651 which I taught in Fall 2012.

While not a prerequisite, this course is a natural continuation of Math 671 taught by Prof. Ross in Fall 2013. However, we may focus more on (a) discrete, finite models, (b) applications (to finance), and less on (c) measure theory, although it probably cannot be completely avoided.

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## Textbooks

Shreve, “Stochastic calculus for finance I: The binomial asset pricing model”, and “II: Continuous time models”.

Recommended: Bernt Øksendal, Stochastic Differential Equations.

## Grading scheme

Homework will be collected and corrected, but grades not recorded.

Best of two midterms counts 50%, and Final exam or alternatively a class presentation counts 50%.

The first midterm is February 19 and covers Shreve volume I, chapters 1-5.

The second midterm is April 2 and covers Shreve volume I, chapter 6, and volume II, chapters 1-3.

The final exam is May 12 (12-2pm) and covers Shreve volume II, chapter 4-6.

Syllabus for final exam on May 12, 2014:

Volume I

Chapters

1

2

4

Volume II

Chapters

1

2

3.1-5

4.1-6

5.1-4

**Week #1, Jan. 13-17: Shreve I-1**

**1.1** (binomial pricing model): pricing a derivative (in the case where the interest is 0).

**1.2**: Calculation of the no-arbitrage price of a derivative asset by replicating using a stock and the money market.

**Week #2, Jan. 20-24: Shreve I-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: **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.

**Week #3, Jan. 27-31: Shreve I-3**

3.1 (change of measure), 3.2 (Radon-Nikodym)

**Week #4, Feb. 3-7: Shreve I-4**

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).

**Week #5, Feb. 10-14: Shreve I-5**

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 $\tau_2$.

**Week #6, Feb. 17-21: Shreve I-6**

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

**Week #7, Feb. 24-28: Shreve II-1**

Volume II

Chapter II-1 (probability and measure theory)

**Week #8, Mar. 3-7: Shreve II-2**

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

**Week #9, Mar. 10-14: Shreve II-3**

Chapter II-3 (Brownian motion) 3.1-3.5

**Week #10, Mar. 17-21: Shreve II-3**

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

**Week #11, March 31 – April 4: Shreve II-4**

All Vol. I homework returned; ready to study for the Midterm!

Chapter II-4 (Black-Scholes PDE and formula)

**Week #13, April 14-18: Shreve II-5**

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.

**Week #14, April 21-25: Shreve II-5**

More Black-Scholes and review.

**Week #15, April 28-May 2: Shreve II-6**

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]

**Week #16, May 5-7: Review**

Final exam covering Volume II, Chapters 1-4

651 Fall 2012 Final Solutions

**Volume I homework problems**

HW1: Ch. 1 #**1,4,6**

HW2: Ch. 2 #**1,2,8**

HW3: Ch. 3 #**1,2,3**

Ch. 4 #**1**

HW4: Ch. 4 #**3,4**

Ch. 5 #**1,5**

HW5: Ch. 5 #6, Ch. 6 #1,3,**5**

**Vol. II homework problems**

HW6: Ch. 1 #2,6,7 and Ch. 2 #5

HW7: Ch. 2 #7, 8 and Ch. 3 #2, **5**

HW8: Ch. 3 #6 (GBM and BM with drift) and Ch. 4 # 4 (Stratonovich), 5 (GBM), 8 (Vasicek)