S17 Stochastic Methods in Finance: Homework 4
Due Wednesday April 5
1. Let B and W be independent Brownian motions. Consider stochastic processes X, Y with the following
dynamics
dXt = dt + dBt
dYt = Yt dt + Yt dWt
Compute the following stochastic dierent
G6501-001, Fall 2015: Homework set 2
Due Monday October 12
1. Page 119, Exercise 3.7.
2. Assume that X is a random variable on (, F, P). Let Q P be another probability measure on (, F)
with Radon-Nikodym derivative Z = dQ
dP . Assume X is integrable w.r.t
G6501 Stochastic Processes: Lecture 1
Yuchong Zhang
Wednesday September 9, 2015
Acknowledgement: I would like to thank Mattias Jonsson, Sergey Nadtochiy, Philip
Protter and Hongzhong Zhang for sharing their lecture notes with me.
Yuchong Zhang
G6501 Stoch
Chapter 9
It Calculus
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2015 Lars Tyge Nielsen All Rights Reserved Do not post or circulate
This chapter denes It processes and integration with respect to It proo
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cesses, explains Its lemma, and develops the associated It calculus. We
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Chapter 8
Stochastic Integrals
c
2015 Lars Tyge Nielsen All Rights Reserved Do not post or circulate
8.1
Stochastic Integrals Preliminaries
In this section, we dene and study the stochastic integral with respect to
a standard Wiener process.
The purpose
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FINAL EXAM
STOCHASTIC PROCESSES AND APPLICATIONS
Unless otherwise stated B = (Bt , Ft )t0 is one-dimensional standard
Brownian motion starting at zero
3
1. Prove that Yt = Bt 3tBt is a martingale
2. Use Itos lemma to compute:
t
(a) 0 uBu du,
t
(b) 0 eBu d
STOCHASTIC PROCESSES
(1) A certain town never has two sunny days in a row. Each day is classied
as being either sunny, cloudy (but dry), or rainy. If it is sunny one day,
then it is equally likely to be either cloudy or rainy the next day. If it is
rainy
S17 Stochastic Methods in Finance: Homework 2
Due Wednesday February 15
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 )
S17 Stochastic Methods in Finance: Homework 3
Due Wednesday March 1
Submit your solution as a hardcopy but the spreadsheet must be submitted on Canvas
1. For this problem you need to find a source for financial data. Please write down which one you use.
M
S17 Stochastic Methods in Finance: Homework 1
Due Wednesday February 1
1. Consider a European derivative security with the following payo structure V = V (S) as a function of
the underlying asset S:
8
0
S 60
>
>
>
<S 60 60 S 70
V (S) =
>
10
70 S 90
>
>
: