Solution to Homework 1
1. What is the distribution of Bs + Bt , s t?
Solution. E (Bs +Bt ) = 0, var(Bs +Bt ) = var(Bs )+var(Bt )+2E (Bs Bt ) = s+t+2s =
3s + t. So Bs + Bt N (0, 3s + t).
2. Compute E [
MAFS5250 Computational Methods for Pricing Structured Products
Computer Assignment One
Instructor: Prof Y.K. Kwok
* Work as a group of two.
_
_
Pricing Behavior of an Equity-Linked Structured Product
0.1
0.2
Solution to Exercises
1. Which of the following are Brownian motions?
(a) cfw_ tB1 , t 0;
(b) cfw_B2t Bt , t 0.
Solution.
(a) Write Xt =
tB1 . Clearly, for s < t
E (Xt )
Cov (Xs , Xt )
=0
2
=
MAFS 501: Stochastic Calculus
Some of the questions in previous nal examinations
=
The following facts might be useful to you.
t
t
Hs dXs ,
t
Ks dYs =
0
Hs K s d X , Y
0
s.
0
Levys Characterization o
0.1
Exercises
1. A function, f , is said to be Lipschitz-continuous on [0, T ] if there exists a constant
C > 0 such that for any t, t [0, T ],
|f (t) f (t )| < C |t t |.
Show that a Lipschitz-continu
cfw_
&%
'1
o p p i p
h p h r j h
d 7c td 7c e|5au~ya i raxr5E VuFhq'd 7c md c ta'hrsxrn5sha5VWV
d c
x t' c md d d 7c c 9 d d 7c c d c
q r p h w g p u & w v s p w p t h d
gxrxrVaVshyphs5a
Stochastic Calculus and Financial Applications Mid-Term Take Home Exam (Fall 2006) THE SOLUTIONS
Problem 1. Show that for square integrable martingales cfw_Xn and cfw_Yn with Y0 = 0 one has for all
MATH 543: Advanced Mathematical Statistics I
Academic Year 20102011, Fall Semester
Instructor: Dr. Bing-Yi JING.
Email: [email protected]
Room: 3489.
Phone: 2358 7458.
Main References:
1. Mathematical St
Chapter 10
Applications of Stochastic
Calculus in Finance
Since Black and Scholes (1973) and Merton (1973), the idea of using stochastic
calculus for modelling prices of risky assets (e.g., share pric
Chapter 3
Unbiased Estimation
3.1
Criteria of estimation
Suppose that we have a random sample X = cfw_X1 , , Xn from a family cfw_F :
of probability distributions ( could be a vector.) Let T (X) =
Chapter 3
Conditional expectation
Given a probability space (, F , P ), let A, B be events, X, Y be r.v.s, and
A be a sub- -algebra. We are interested in introducing E (X |A), one of the
most importan
Chapter 1
Probability Models
Broadly speaking, there are two types of models.
1. Parametric models: A family of distributions F = cfw_F :
is said to be a parametric family i Rd for some xed positive
0.1
Exercises
2
1. Let V ar(X |F ) = E (X 2 |F ) (E (X |F ) . Show that
V ar(X ) = E (V ar(X |F ) + V ar (E (X |F ) .
Proof.
RHS
=
=
=
=
=
E (V ar(X |F ) + V ar (E (X |F )
E E (X 2 |F ) (E (X |F )
EE
MAFS5250 Computational Methods for Pricing Structured Products
Course objective
This course introduces the computational techniques in pricing of structured products.
First we discuss the three common
MAFS5250 - Computational Methods for Pricing Structured Products
Computer Assignment Two
* Work as a group of two.
Course instructor: Prof. Y.K. Kwok
Pricing of Accumulators
Product nature
The accumul