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
Summarize pricing and hedging in binomial model.
One-period trinomial model.
No-arbitrage and ri

G6505-001-S16: Homework set 3
Due Wednesday March 2
Submit your solutions as a hardcopy but the spreadsheet must be submitted on CourseWorks
1. For this problem you need to find a source for financial data. Please write down which one you use.
Make sure t

Stochastic Methods in Finance
September 8, 2015
Lecture 1
Introduction, binomial pricing model
Vol I. Chapter 1
Lecture 2
Lecture 3
Binomial pricing model (contd.), options
Discrete cond. expectations and martingales
Vol I. Chapter 1
Vol I. Chapters 2.1-2

G6505 Stochastic Methods in Finance
Homework 3 Solutions
Question 1 (Shreve 3.2, Vol. II) We have
E[W 2 (t) t | F(s)] = E[W (t) W (s)2 + 2W (t)W (s) W 2 (s) t | F(s)]
= E[(W (t) W (s)2 ] + 2W (s)E[W (t) | F(s)] W 2 (s) t
= Var(W (t) W (s) + 2W 2 (s) W 2 (

G6505 Stochastic Methods in Finance
Homework 5 Solutions
Question 1 (Shreve 4.7, Vol. II)
(i) Let f (t, x) = x4 . We have
f
= 0,
t
f
= 4x3 ,
x
2f
= 12x2 .
x2
Using the Ito formula to compute the differential of d(W 4 (t) yields
d(W 4 (t) = 4W 3 (t)dW (t)

G6505 Stochastic Methods in Finance
Homework 4 Solutions
Question 1 (Shreve 4.1, Vol. II) This proof is fully analogous to the one of Theorem 4.2.1. We want to show
that for 0 s t T E[I(t) | F(s)] = I(s). Assume again, that the s [tl , tl+1 ) and t [tk ,

G6505 Stochastic Methods in Finance
Homework 6 Solutions
Question 1 (Shreve 5.4, Vol. II)
(i) Let f (t, x) = ln(x). We have
f
= 0,
t
f
1
= ,
x
x
1
2f
= 2
x2
x
and
(dS(t)2 = 2 (t)S 2 (t)dt.
Applying Itos lemma, the differential of the log stock price d ln

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: 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: Lecture 9
Yuchong Zhang
Wednesday February 17, 2016
Yuchong Zhang
G6505 Stochastic Methods in Finance: Lecture 9
1 / 18
Plan
I
Complete and incomplete markets.
I
Binomial option pricing with Excel
I
Basics of continuou

G6505-001-S16: Homework set 1
Due Wednesday February 3
1. Consider a European derivative security with the
the underlying asset S:
0
S 60
V (S) =
10
S 80
following payoff structure V = V (S) as a function of
S 60
60 S 70
70 S 90
S 90
The option expires on

G6505-001-S16: Homework set 2
Due Wednesday February 17
1. The price S per share of Fish stock is modeled using a two-period binomial model. We have S0 = 48,
S1 (H) = 54, S1 (T ) = 45, S2 (HH) = 57, S2 (HT ) = 51, S2 (T H) = 48, S2 (T T ) = 42. The intere