1
Stat G6501 Section 002 Stochastic Processes with Applications
Lars Tyge Nielsen
Detailed Reading Assignments Fall 2015 v 2
Lecture 1
Chapter 1, up to and including Section 1.1, Length, Area, Volume, and Probability
Lecture 2
Chapter 1, up to and includi

Question 1 (Shreve 4.1, Vol. II) This proof is fully analogous to the one of Theorem 4.2.]. We want to show
that for 0 S s S t S T lE[I(t) |.F(s)] : 1(5). Assume again. that the s E [thtHﬂ and t E [tk,tk+1) for l S k. We
start by splitting up the sum into

Question 2 (Shreve 4.8, Vol. H)
(i) Let ﬁt: I) = emz. We have
a—f : eat :
I31“. ‘ 632
Using the Ito formula.J we can compute the differential of d[e"3IR(t) to be
{ﬂeﬂ‘PﬁflJ = rim. RUN
: ,seﬂ‘mtmt - eﬂ‘dmt)
: semmrmt —— eﬁ‘adt — ﬁemBﬂMt + e-Stgdwu)
= emoti

5.11.
Proof. We ﬁrst make an analysis which leads to the hint. then we give a formal proof.
(Analysis) If we want to construct a portfolio X that exactly replicates the cash ﬂow. we must ﬁnd a
solution to the backward SDE
dXt 2 Aids}; + Rtht * AtStMt * Ct

G6505 Stochastic Methods in Finance: Lecture 2
Yuchong Zhang
Monday January 25, 2016
Yuchong Zhang
G6505 Stochastic Methods in Finance: Lecture 2
Today
I
The binomial model
Yuchong Zhang
G6505 Stochastic Methods in Finance: Lecture 2
Model dependent/indep

G6505 Stochastic Methods in Finance: Lecture 1
Yuchong Zhang
Wednesday January 20, 2016
Acknowledgement: I would like to thank Mattias Jonsson (University of Michigan)
and Hongzhong Zhang (Columbia University) for sharing their lecture notes with me.
Yuch

G6505 Stochastic Methods in Finance: Lec 10
Yuchong Zhang
Monday February 22, 2016
Yuchong Zhang
G6505 Stochastic Methods in Finance: Lec 10
1 / 22
Plan
I
Quiz
I
Basics of continuous-time stochastic processes
I
Brownian motion
I
Symmetries of BM
I
Reflect

G6505 Stochastic Methods in Finance: Lec 16
Yuchong Zhang
Monday March 21, 2016
Yuchong Zhang
G6505 Stochastic Methods in Finance: Lec 16
1 / 19
Plan
I
Stochastic integral
I
It
os formula
I
Multi-dimensional It
os formula
I
It
o product rule
I
Decomposing

Chapter 8
The BlackScholes Model
This chapter investigates the BlackScholes model in detail. What we call
the BlackScholes model is not the formula for the value of a standard call
option, but rather the economy consisting of a money market account with
a

Index
adapted process, 151
almost everywhere identical processes,
111
almost surely, 12
Arnold, L., 177, 180, 182, 185
associativity of the product measure,
19
augmented ltration
generated
by a process, 154
augmented sigma-algebra, 152
generated
by a clas

Chapter 10
Gaussian Term Structure
Models
This chapter studies the Vasicek model of interest rates, the concept of instantaneous forward rates, and the Heath-Jarrow-Morton analysis of forward
rates.
In the Vasicek model, the interest rate follows an Ornst

Chapter 7
It Calculus
o
This chapter denes It processes and integration with respect to It proo
o
cesses, explains Its lemma, and develops the associated It calculus. We
o
o
work in detail through a number of examples of how to use Its formula,
o
such as

G6505 Stochastic Methods in Finance
Homework 1 Solutions
Question 1 (Shreve 1.2) Considering the cases of a head and of a tail on the rst toss, and utilizing the numbers
given in Example 1.1.1, we can write
5
X1 (H) = 80 + 30 (40 + 1.200 )
4
5
X1 (T ) = 2

G6505 Stochastic Methods in Finance: Lecture 8
Yuchong Zhang
Monday February 15, 2016
Yuchong Zhang
G6505 Stochastic Methods in Finance: Lecture 8
1 / 12
Plan
I
Summarize pricing and hedging in binomial model.
I
One-period trinomial model.
I
No-arbitrage

G6505 Stochastic Methods in Finance: Lec 21
Yuchong Zhang
Wednesday April 6, 2016
Yuchong Zhang
G6505 Stochastic Methods in Finance: Lec 21
1 / 14
G6505 Stochastic Methods in Finance: Lec 21
2 / 14
Plan
I
Implied volatility
I
Local volatility model
I
Dupi

G6505 Stochastic Methods in Finance: Lec 26
Yuchong Zhang
Monday April 25, 2016
Yuchong Zhang
G6505 Stochastic Methods in Finance: Lec 26
1 / 15
Plan
I
Short rate models
I
Forward and forward measure
I
Option pricing with stochastic interest rate
Yuchong

G6505 Stochastic Methods in Finance: Lecture 4
Yuchong Zhang
Monday February 1, 2016
Yuchong Zhang
G6505 Stochastic Methods in Finance: Lecture 4
1 / 10
Plan
I
Risk-neutral pricing of general cash flow
I
Risk-neutral pricing with dividends.
I
Utility maxi

G6505 Stochastic Methods in Finance: Lec 24
Yuchong Zhang
Monday April 18, 2016
Yuchong Zhang
G6505 Stochastic Methods in Finance: Lec 24
1 / 19
Plan
I
Examples of change of numeraire
I
Domestic and foreign risk-neutral measures
I
Currency derivative
Yuch

G6505 Stochastic Methods in Finance: Lecture 3
Yuchong Zhang
Wednesday January 27, 2016
Yuchong Zhang
G6505 Stochastic Methods in Finance: Lecture 3
1 / 22
Today
I
Review basic probability and martingale theory
I
Risk-neutral pricing
Yuchong Zhang
G6505 S

G6505 Stochastic Methods in Finance: Lec 11
Yuchong Zhang
Wednesday February 24, 2016
Yuchong Zhang
G6505 Stochastic Methods in Finance: Lec 11
1 / 19
Plan
I
Brownian motion
I
Symmetries of BM
I
Reflection principle and barrier options
Yuchong Zhang
G6505

G6505 Stochastic Methods in Finance: Lec 12
Yuchong Zhang
Monday February 29, 2016
Yuchong Zhang
G6505 Stochastic Methods in Finance: Lec 12
1 / 24
Plan
I
Black-Scholes model (or Black-Scholes-Merton model)
I
Stochastic integral
I
It
os formula
Yuchong Zh

G6505 Stochastic Methods in Finance: Lecture 2
Yuchong Zhang
Monday January 25, 2016
Yuchong Zhang
G6505 Stochastic Methods in Finance: Lecture 2
Today
I
The binomial model
Yuchong Zhang
G6505 Stochastic Methods in Finance: Lecture 2
Model dependent/indep

G6505 Stochastic Methods in Finance: Lec 18
Yuchong Zhang
Monday March 28, 2016
Yuchong Zhang
G6505 Stochastic Methods in Finance: Lec 18
1 / 20
Plan
I
Girsanov theorem
I
Arbitrage
I
Hedging in BS model
I
Risk-neutral pricing
I
Market completness
I
Fundam

G6505 Stochastic Methods in Finance: Lec 19
Yuchong Zhang
Wednesday March 30, 2016
Yuchong Zhang
G6505 Stochastic Methods in Finance: Lec 19
1 / 18
Plan
I
Arbitrage
I
Hedging in BS model
I
Risk-neutral pricing
I
Market completness
I
Fundamental Theorem of

G6505 Stochastic Methods in Finance: Lec 27
Yuchong Zhang
Wednesday April 27, 2016
Yuchong Zhang
G6505 Stochastic Methods in Finance: Lec 27
1 / 17
Yuchong Zhang
G6505 Stochastic Methods in Finance: Lec 27
2 / 17
Plan
I
Forward rate
I
The HJM model
I
*Cal

G6505 Stochastic Methods in Finance: Lec 22
Yuchong Zhang
Monday April 11, 2016
Yuchong Zhang
G6505 Stochastic Methods in Finance: Lec 22
1 / 22
Plan
I
American option
I
Variational inequality
I
Free boundary problem and smooth fit principle
I
Perpetual A

G6505 Stochastic Methods in Finance: Lecture 9
Yuchong Zhang
Wednesday February 17, 2016
Yuchong Zhang
G6505 Stochastic Methods in Finance: Lecture 9
1/1
Plan
I
Complete and incomplete markets.
I
Binomial option pricing with Excel
I
Basics of continuous-t