W4635 (Spring 2015)
Lecture 9
Hongzhong Zhang
Department of Statistics
Columbia University
February 25, 2015
1 / 16
Review of continuous random variables
Let X be a random variable that takes value on R. How to
characterize the distribution of X ?
P(X = c
Stochastic Calculus and Option Pricing
Week 1: From the Random walk to the Wiener
process, passing to the continuous-time limit,
modeling the flow of information by using
filtrations, application to the modeling of asset
prices.
Agn`es Tourin
January 16,
FRE 6233 Stochastic Calculus and Option Pricing
The martingale approach
Lecture week 4
Agn`es Tourin
October 4, 2016
In these slides, I draw heavily from a number of books, in particular
the textbook by S. Shreve, Stochastic Calculus for Finance II:
conti
Stochastic Calculus and Option Pricing
Week 9 : Finite Difference methods for the
American Option
Agn`es Tourin
April 7, 2016
Introduction
I
In practice, pure Finite Difference schemes are only useful in
1,2 or at most 3 spatial dimensions. They may be th
Stochastic Calculus and option pricing
Week 6: The Asian Option
Agn`es Tourin
March 29, 2016
Acknowledgement
In part I of these slides, I draw heavily from the textbook by S.
Shreve entitled Stochastic Calculus for Finance, II, chapter 7. Any
mistake is t
Stochastic Calculus and Option pricing
Week 13: Option pricing in a jump-diffusion
model
Agn`es Tourin
April 26, 2016
Most of the material in this Lecture is from the Textbook by S.
Shreve entitled Stochastic Calculus for Finance, Part 2, Springer,
chapte
Stochastic Calculus and Option pricing
Week 12: Stochastic Calculus for jump-diffusion
processes
Agn`es Tourin
April 26, 2016
Most of the material in this Lecture is from the Textbook by S.
Shreve entitled Stochastic Calculus for Finance, Part 2, Springer
Stochastic Calculus and Option Pricing, week 8 :
The American Option
Agn`es Tourin
November 11, 2016
I
I am going to cover the theory for the American Option today
and the Finite Difference method for the American Option
next week.
I
I will draw heavily f
Quantitative methods
Lecture 4-5
A basic introduction to stochastic processes
October 3, 2016
1
Introduction
Stochastic processes are used for modeling the time evolution of financial
assets. In the course of the past 3 lectures, we have already encounter
Quantitative methods
Lecture 6
The arithmetic random walk and the
geometric random walk
October 3, 2016
1
Introduction
In this lecture, we draw heavily from the textbook in Stochastic Calculus by
S. Shreve, Stochastic Calculus for Finance. In particular y
Quantitative Methods
Week 2
Convergence concepts, Law of large numbers,
Central Limit Theorem, Markov sequences,
the Martingale property
September 16, 2016
For the concepts of convergence, the Law of large numbers and the central
limit Theorem, we draw he
FRE 6083 Final Review
Problem 1
Compute the characteristic function E[eiW (t) ] for any R, and W (t) a Wiener process.
1
2
Problem 2
A stock is currently $100. Over the next two six-month periods is expected to go up by
10% or down 10%. The risk free rate
Quantitative methods, Homework Assignment for Lecture 1
Where appropriate, show the space or universe, the field of subsets of the
space, and the probability measure for outcomes. Clearly state the events
for which probabilities are to be calculated.
Prob
Quantitative Methods
Lecture 3
Markov chains
September 19, 2016
1
1.1
Markov chains
Introduction
This week, we study Markov chains. This study is motivated by the numerous applications to Insurance and Finance, in particular to credit risk
but also to the
Tandon School of Engineering
FRE 6083
Final Review
(1) We construct a Binomial tree with 3 periods for the parameters
r = 1/4; d = 1/2; u = 2
and for the current stock price S0 = 8, for computing the price C0 of a call option
at time 0 with payoff
C3 = S3
Quantitative methods, Homework Assignment 6
Due December 10th
In the following exercises B(t), t 0 is a standard Brownian motion process and Ta denotes the time it takes this process to hit a.
1. What is the distribution of B(s) + B(t), s t?
2. Compute th
Quantitative methods, Homework Assignment 3
Problem 3.1
Let X1 , X2 , ., X10 be a sample of size 10 from the standardized normal distri = X1 +X2 +.+X10 .
bution N (0, 1). Determine probability P (X 1) where X
10
Problem 3.2
Let the random variables X and
Quantitative methods,
Homework Assignment 4
Problem 4.1
Consider the process X(t), t 0 defined by
1, t Y ;
X(t) =
0, t > Y .
where Y is a uniformly distributed random variable on the interval (0, 1).
1. Compute, for t (0, 1), the first-order probability m
FRE 6083 Homework 2
Problem 2.1
We let the random variables Xn be a binomial process,
Xn =
n
X
Bi ,
i=1
where each Bi is independent and is distributed according to
(
1 with probability p
Bi =
0 with probability 1 p
1. Calculate the probability P (X4 > k)
FRE6303, Midterm Examination
Monday March 30, 2015
Notes:
1. This examination contains 4 pages, including this one.
2. For this examination, you may only use a 2-page cheat sheet. No other notes, books or electronic
devices may be used. Cell phones may no
FRE6303, Midterm Examination
Monday October 27 2014, 6:00pm-8:30pm
Notes:
1. This examination contains 3 pages, including this one.
2. For this examination, you may only use a 2-page cheat sheet. No other notes, books or electronic
devices may be used. Ce
FRE6233, Assignment 3, week 3
Question 1 (30 points)
Consider the Black-Scholes framework with the standard notations and a claim of the form
(S(T ) at time T > 0. We assume that the function satisfies the property
(t s) = t (s), t > 0.
1. (5 points) Writ
Stochastic Calculus and Option pricing,
assignment week 5
Agn`es Tourin
October 9, 2015
Problem: the Heston stochastic volatility model (drawn from the
textbook by Shreve, Stochastic Calculus)
Consider the Heston stochastic volatility model under a risk-n
FRE6233, assignment week 1
Agn`es Tourin
September 11, 2015
In the textbook by Bjork,
1. In chapter 4, exercises 4.2,4.3,4.4,4.8.
2. In chapter 5, exercises 5.5,5.6,5.7,5.8
1
FRE 6233 Stochastic Calculus and Option pricing
Week 2: The Ito integral and Itos lemma in
several dimensions
Agn`es Tourin
January 2, 2016
This Lecture draws heavily on the Textbook by S. Shreve entitled
Stochastic Calculus for Finance, Part 2, Springer,
Stochastic Calculus and option pricing
Week 5: The Partial Differential Equations
approach to option pricing
Agn`es Tourin
August 7, 2015
Partial Differential Equations
Most of the material for this Lecture is adapted from the textbook
by S. Shreve entitl
FRE 6233 Stochastic Calculus and option pricing
Application of Stochastic Calculus to the
Black-Scholes model
Lecture week 3
Agn`es Tourin
January 2, 2016
In these slides, I draw heavily from a number of books, in particular
1. Tomas Bjork, Arbitrage Theo
FRE 6233 Stochastic Calculus and Option Pricing
The martingale approach
Lecture week 4
Agn`es Tourin
August 7, 2015
In these slides, I draw heavily from a number of books, in
particular the textbook by S. Shreve, Stochastic Calculus for
Finance II: contin
Stochastic Calculus and Option Pricing
Week 1: From the Random walk to the Wiener
process, passing to the continuous-time limit,
modeling the flow of information by using
filtrations, application to the modeling of asset
prices.
Agn`es Tourin
September 14