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 [Bt1 Bt2 Bt3 ] for t1 < t2 < t3 .
Solution. Let with 0 =
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
= Cov ( sB1 , tB1 ) = stEB1 = st = s.
Thus, cfw_ tB1 , t
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
You are invited to explore the pricing behavior of an e
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 of Brownian motion:
Mt is a BM i Mt is a continuous mart
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-continuous function has bounded variation and zero 2-variation
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
gxrxrVaVshyphs5aug5VnCd c ~asu#vuy1
p hrt w g
d 7c r 'aaxrp'xvwa'hra
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 n = 0, 1, 2. that
2 E(Xn+1 )
(E(Xn+1 Yn+1 )2 (E(Xn Yn
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 Statistics by J. Shao.
2. Theory of Point Estimation by E
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 prices of stock, stock
indices such as Dow Jones, foreign e
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) = T (X1 , , Xn )
be such an estimator for or g (). The pu
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 important (and perhaps one of most dicult) concepts in probabil
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
integer d, and each F is known when is known. The set i
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 (X 2 |F ) (EE (X |F )
E (X 2 ) (EX )2
LHS
2
2
2
+ E (E
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 classes of numerical methods: lattice tree methods,
fi
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 accumulator or accumulative forward is a daily accumulated and