Lecture 1 - Sp 2012
Probability review
M.C.Spruill
January 10, 2012
Denition 1 A function P mapping events A,
subsets of an outcome space A, to the real
numb ers and satisfying the following three properties is called a probability.
1. P (A) 0
2. P (A) =
Lecture 16 - Sp 2012
Pricing derivatives
M.C.Spruill
March 12, 2012
It follows that a = S, b = S so that
1
dC = (Ct + SCs + 2S 2)dt + Cs SdW.
2
Below we will use this information to help determine the function C.
127
First we correct some misstatements ma
Lecture 17 - Sp 2012
Pricing derivatives
M.C.Spruill
March 15, 2012
The heat equation is ut = uxx subject to u(x, 0) =
f (x), f smo oth. The strategy is to change variables in the original equation (?) and transform it into the heat equation whose solutio
Lecture 19 - Sp 2012
Pricing derivatives
M.C.Spruill
March 28, 2012
Let V b e the market price of the b ond. Then
V > 0 since it is riskless and there is no arbitrage. To see that there is a value y solving the
equation, note rst that the rhs of equation
Lecture 20 - Sp 2012
Pricing derivatives
Intro to Portfolio selection
M.C.Spruill
April 2, 2012
The following result from advanced calculus
nds frequent use in statistics.
Theorem 1 Let X = (X1, . . . , Xn) have a probability density f (x) with resp ect t
Lecture 21 - Sp 2012
Portfolio Theory
M.C.Spruill
April 4, 2012
Example 1 Supp ose there are two securities S
and C and the parameters are S = 0.08, S =
0.03, C = 0.14, C = 0.06, S C = 0.7. Plot the
lower b oundary
cfw_(c, (c) : c (, )
of the opp ortunity
Lecture 22 - Sp 2012
Portfolio Theory
M.C.Spruill
April 10, 2012
If we now add the existence of a risk-free asset then the calculations we have made above
hold no longer; for the addition of the risk-free
results in a covariance matrix which is singular.
Lecture 23 - Sp 2012
Regression
M.C.Spruill
April 11, 2012
We see that when f > r an investor will choose
a point along the ecient frontier representing
his particular preferred risk-return combination
by combining the riskless asset with the portfolio x(
Lecture 24 - Sp 2012
Regression
M.C.Spruill
April 22, 2012
In this basis one has
n
Y=
Xj bj ,
j =1
or, denoting by B the matrix of real numbers
whose j th column consists of the coordinates
of bj relative to the standard basis in Rn (see
Appendix ?, takin
Lecture 25 - Sp 2012
Review - Odds-n-Ends
M.C.Spruill
April 28, 2012
Principal Components Suppose we have random variables X1, . . . , Xk and their covariance
matrix is S. We shall show that there is an
orthogonal matrix P and a diagonal matrix L
whose en
Lecture 15 - Sp 2012
Mo deling sto ck prices
M.C.Spruill
March 9, 2012
Denition 1 If
lim
n
n
X
j =1
g (tn,i1)(W (tn,i) W (tn,i1)
exists then it is dened as
Zb
a
g (s)dW (s)
Theorem 1 If g (s, w) is a non-anticipating function on [a, b] ( means g (s, .)isF
Lecture 14 - Sp 2012
Mo deling sto ck prices
M.C.Spruill
March 6, 2012
IndTst.lst
Printed: Wednesday, February 22, 2012 9:13:34 AM
Page 1 of 1
Independence of Ln Returns
08:11 Wednesday, February 2
The FREQ Procedure
Table of Pre by post
Pre
post
Frequenc
Lecture 13 - Sp 2012
Mo deling sto ck prices
M.C.Spruill
February 22, 2012
RWI
Taking
rj = ln(Pj /Pj 1) = pj pj 1
the RW1 hypothesis with drift is that rj are iid
(, 2); then
pj pj 1 = ej
ej iid (, 2). To test the hyp othesis RW1 following Cowles and Jone
Lecture 1 - Sp 2012
Probability review
M.C.Spruill
January 11, 2012
Example 1 The prop ortion of sto cks which
will increase in value by a factor of approximately 4 in a year is around 0.001. Denote this
prop erty by + and the opp osite by -. An excellent
Lecture 1 - Sp 2012
Probability review
M.C.Spruill
January 12, 2012
It follows that the functions, for > 0,
x1ex/
f (x) =
, x>0
()
are probability density functions. If X has this
distribution then we say that X is gamma distributed with parameters , and
Lecture 4 - Sp 2012
Probability and Statistics review
M.C.Spruill
January 23, 2012
A certain technical analyst has a probability
p of correctly predicting an up movement
in the price of a sto ck over a months time
of any randomly selected sto ck. Then th
Lecture 5 - Sp 2012
Probability and Statistics review
M.C.Spruill
January 26, 2012
The analysis of some real data - IBM and SP
500 nding mean and variance of ln rets and
the correlation b etween them - output.
32
The R-program that generates this output i
Lecture 7 - Sp 2012
Probability and Statistics review
M.C.Spruill
February 1, 2012
Comp ounded continuously , since
one has
ba n
lim (1 + r
) = er(ba)
n
n
Pt = ertP0.
Supp ose we lo ok at the price of an equity at
time t = 1. We have, breaking up the inte
Lecture 8 - Sp 2012
Probability and Statistics review
M.C.Spruill
February 7, 2012
Back to the subject under consideration b efore the discussion of the solutions to problem
set 2, namely asymptotic normality. We b egin
with a simple example.
L
Supp ose n
Lecture 9 - Sp 2012
Probability and Statistics review
M.C.Spruill
February 8, 2012
R
1
2
3
4
5
6
0.05
0.05
0.05
1
0
0.01
0.741
0.088
0.149
0
1
0.249
67
More generally A test is a function (x) :
[0, 1], where
(x) = P ( reject H0|x).
If (x) cfw_0, 1 for all
Lecture 10 - Sp 2012
Probability and Statistics review
M.C.Spruill
February 13, 2012
Now let be any test of size no more than
for H1 and let 1 be arbitrary (that is; let
1 > 1 b e arbitrary.). Then by continuity of
the p ower function it follows that s s
Lecture 11 - Sp 2012
Mo deling sto ck prices
M.C.Spruill
February 15, 2012
Often the likeliho o d ratio technique do es not
yield a closed form. In large sample cases we
have the following theorem by Wilks.
Theorem 1 Under appropriate regularity condition
Lecture 12 - Sp 2012
Mo deling sto ck prices
M.C.Spruill
February 20, 2012
This mo del is imp ortant to pricing and hedging of derivatives, as we shall see, but one key
ingredient in its successful application to real
data is the empirical question of the
Alex Slette
Questions:
1. Discuss the rationale for expecting an efficient capital market. What factor would you look for to
differentiate the market efficiency for two alternative stocks?
You would expect a market to be efficient because nowadays, there