AR Model Building
Building an AR Model
1. Order specification
1. Partial ACF: PACF cuts off at lag p for an AR(p) model.
2. Akaike information criterion.
AIC(p) = 2 (log likelihood) + 2 (# of estimated parameters).
Note: First term of the AIC measures goo
Review
September 16, 2016
Random Walk: Suppose that pt = lnPt are the log prices and rt are the corresponding log returns from an
asset, then we have
pt p0 = rt + rt1 + + r1 .
In other words,
Pt = P0 exp (rt + rt1 + + r1 ) .
Assume the log returns rt are
Review and Homework Hints
September 9, 2016
Question from class: How to prove the asymptotic results that given a simple random sample of size n (n
large) from a normal distribution, then
S(X)
N (0, 6/n)
and
K(X)
N (0, 24/n).
Sorry I could not find out
Autoregressive Model at Lag 1
October 21, 2016
AR(1) Model
1. Form:
rt = 0 + 1 rt1 + at
(1)
where 0 and 1 are real numbers, which are referred to as parameters (to be estimated from the data
in an application).
2. Stationarity: necessary and sufficient co
Model Comparison
November 2, 2016
Model Comparison and Averaging
In applications, there is no true model for a given time series. All statistical models are approximations
used to describe the dynamic dependence of the data. It is then common to see that
Autocorrelation Function
October 19, 2016
Definition:
Lag-k autocovariance:
k = Cov(rt , rtk ) = E[(rt )(rtk )]
Serial (or auto-) correlations: for a weakly stationary return series rt the correlation coefficient between
rt and rtk is called the lag-k a
Moving Average Model at Lag q
November 9, 2016
MA(q) Model
1. Form:
rt = + at 1 at1 2 at2 q atq , t = 1, , T
where , 1 , 2 , , q are parameters and at W N (0, a2 ).
2. Compact form:
rt = + (1 1 B 2 B 2 q B q )at
3. Stationarity: always stationary.
4. Mean
Geometric Brownian Motion Simulation
In this file, we show how to simulate a geometric brownian motion process with drift and volatility
2
St = S0 exp (
)t + Wt
2
on the interval [0, T ]. Based on Itos Lemma, a geometric brownian motion process satisfie
Geometric Brownian Motion
1
Binomial tree:
Pro: simple
Con: unrealistic
2
Solution: increase the number of steps in a binomial model
so that we could divide the life of the option into many steps
with small time length between adjacent steps.
3
Result: bi
STATS 461
Fall 2016
Black-Scholes Formula
1. Assumptions.
Stock price obeys geometric brownian motion with constant drift and volatility during [0, T ].
No dividend payment on the stock.
There is no arbitrage opportunity.
It is possible to borrow and
Autoregressive Model at Lag P
October 28, 2016
AR(p) Model
1. Form:
rt = 0 + 1 rt1 + 2 rt2 + + p rtp + at
where 0 , 1 , 2 , p are parameters and at
2. Mean:
= E(rt ) =
(1)
W N (0, a2 ).
0
.
1 1 2 p
3. Alternative representation: plugging 0 = (1 1 2 ) in
Moving Average Model at Lag 1
November 4, 2016
MA(1) Model
1. Form:
rt = + at 1 at1 , t = 1, , T
where and 1 are parameters and at W N (0, a2 ).
2. Mean:
E(rt ) =
3. Variance:
Var(rt ) = a2 + 12 a2 = (1 + 12 )a2
4. Compact form:
rt = + (1 1 B)at
5. Stati
One-step Binomial Tree
General situations:
Non-dividend-paying stock.
Current price: S0 .
Price at expiration could move up to S0 u or move down to
S0 d.
Risk-free interest rate: r.
Consider a European call option with maturity T and strike price
K. Furth
Multi-step Binomial Tree
Example: consider a stock with $50 as currect price and two
time steps with six months in length for each time step. At each
two time steps, the stock price may move up by 10% or down by
10%. The risk-free interest rate is 6%. Con
CS 525 Class Project
Breast Cancer Diagnosis via Quadratic
Programming
Fall, 2015
Due 15 December 2015, 5:00pm
In this project, we apply quadratic programming to breast cancer diagnosis.
We use the Wisconsin Diagnosis Breast Cancer Database (WDBC) made
pu
CS368-2 Fall 2015
Homework Assignment 3
Table of Contents
Problem 1: Plotting Polynomials . 1
Problem 2: Approximation . 4
Problem 3: Heat Plate . 5
Name: Yingwei Zhu
Due Date: Monday, November 9 by 11:59 pm
Problem 1: Plotting Polynomials
Plot each of
STATS 461
Fall 2015
Final Project
1. Project Overview.
For this project, you need to collect one (or two) financial series and analyze using the tools
we discussed in class. Your final goal is to show that you have achieved a high level of skill
and know
% CS368-2 Fall 2015 Homework Assignment 3
% * Name: Yingwei Zhu
% * Due Date: Monday, November 9 by 11:59 pm
% Problem 1: Plotting Polynomials
%
% Plot each of these polynomials on a separate plot
%
% $p(x) = x^3 - 3x^2 + 5$ in green over the range x=-2.1
%
%#1
%a)
mu = 0.001;
[data test]=wdbcData('wdbc.data.txt',30,0.1,0);
%call QP function for determining misclassified points in training set
[w g cvx_optval num_mis]=QP(data,mu);
disp('w=');
disp(w);
disp(['gamma=',num2str(g),',Optimal Value=',num2str(cvx