Module 1
Section 1: Accumulation Functions
The accumulation function, denoted a(t), gives the value at time t for an initial
time 0 investment of 1. So a(0) = 1. The amount function, denoted A(t), gives the
value at time t for an initial time 0 investment
EXAM 1
MAP 2302
Spring 2017
Name:
Read all of the following information before starting the exam:
Show all work, clearly and in order, if you want to get full credit. I reserve the right to
take off points if I cannot see how you arrived at your answer (
Set 6: Interpolation Part 2
Kyle A. Gallivan
Department of Mathematics
Florida State University
Introduction to Computational Finance
Spring 2014
1
Piecewise Polynomial Interpolation
Use local interpolants of lower order rather than one global polynomial
Set 1: Basics
David Kopriva and Kyle A. Gallivan
Department of Mathematics
Florida State University
Introduction to Computational Finance
Spring 2014
1
Finite Precision
All discussions so far have assumed we can compute with elements
of R and by extensi
Spring 2014
Introduction to Computational Finance
MAP 5611
Details
Time and Place : 11:15 12:05 , 201 Love Building
Instructor: K. A. Gallivan (5-0306, 318 Love Building, [email protected])
Homepage http:/www.math.fsu.edu/~gallivan
Oce Hours: 8:00
Set 17: Options and PDE Methods
David Kopriva and Kyle A. Gallivan
Department of Mathematics
Florida State University
Introduction to Computational Finance
Spring 2014
1
Scaling
Recall the generic advection diffusion equation:
ut + aux = uxx
and consider
Set 14: Ordinary Differential Equations:
Stiffness
Kyle A. Gallivan
Department of Mathematics
Florida State University
Introduction to Computational Finance
Spring 2014
1
Stiffness: Stability vs. Accuracy
A-stable methods are neither always effective no
Set 16: Numerical PDE Basics
David Kopriva and Kyle A. Gallivan
Department of Mathematics
Florida State University
Introduction to Computational Finance
Spring 2014
1
A Simple Example
The diffusion equation
ut = uxx , x (, ),
u(0, x) = g(x)
is an initial
Set 13: Ordinary Differential Equations Part
2
Kyle A. Gallivan
Department of Mathematics
Florida State University
Introduction to Computational Finance
Spring 2014
1
Quadrature Approach Backward Euler
Using right rectangle rule quadrature, i.e., a const
Set 8: Newton-Cotes Quadrature
Kyle A. Gallivan
Department of Mathematics
Florida State University
Introduction to Computational Finance
Spring 2014
1
Numerical Quadrature
Let f (x) C[a, b]. Numerical quadrature approximates the denite
integral
b
In (f )
Set 11: Monte Carlo Quadrature
David Kopriva and Kyle A. Gallivan
Department of Mathematics
Florida State University
Introduction to Computational Finance
Spring 2014
1
Probability
A random variable, X, is a real number assigned to random event E.
The
Set 12: Ordinary Differential Equations Part
1
David Kopriva and Kyle A. Gallivan
Department of Mathematics
Florida State University
Introduction to Computational Finance
Spring 2014
1
Sources and References
U. Ascher and L. Petzold, Computer Methods fo
Set 9: Composite and Adaptive Quadrature
Kyle A. Gallivan
Department of Mathematics
Florida State University
Introduction to Computational Finance
Spring 2014
1
Newton-Cotes Composite Formulas
When b a is large or n is too large to trust a Newton-Cotes
Set 15: Partial Differential Equations
David Kopriva and Kyle A. Gallivan
Department of Mathematics
Florida State University
Introduction to Computational Finance
Spring 2014
1
Partial Differential Equations
An unknown function u is specied in terms of
Set 10: Differentiation
Kyle A. Gallivan
Department of Mathematics
Florida State University
Introduction to Computational Finance
Spring 2014
1
Derivatives
Given samples of an unknown function y(x)
(x0 , y0 ), (x1 , y1 ), . . . , (xn , yn )
estimate y (
Set 7: Least Squares
Kyle A. Gallivan
Department of Mathematics
Florida State University
Introduction to Computational Finance
Spring 2014
1
Discrete Interpolation and Approximation
Given discrete values of f (x) and possibly some of its derivatives
ex
Set 3: Solving Nonlinear Equations Part 1
David Kopriva and Kyle A. Gallivan
Department of Mathematics
Florida State University
Introduction to Computational Finance
Spring 2014
1
Value of an Account with Simple Interest
Dene a time interval to be T tim
Set 5: Interpolation Part 1
David Kopriva and Kyle A. Gallivan
Department of Mathematics
Florida State University
Introduction to Computational Finance
Spring 2014
1
Motivation
Volatility, , is a derived quantity.
It is inferred from stock prices.
van
Set 4: Solving Nonlinear Equations Part 2
David Kopriva and Kyle A. Gallivan
Department of Mathematics
Florida State University
Introduction to Computational Finance
Spring 2014
1
Order of a Method
We would like to do much better than the bisection metho
Set 2: Stability
David Kopriva and Kyle A. Gallivan
Department of Mathematics
Florida State University
Introduction to Computational Finance
Spring 2014
1
Stability
Conditioning is a property of a problem that is independent of the
algorithm used to sol
Solutions for Homework 4 Optimization
Spring 2012
Problem 4.1
Problem 26 on page 260 of the textbook.
Solution:
For Steepest Descent we have the result
e(k+1)
A
1 (k)
e
+1
A
1
+1
k+1
e(0)
A.
6 places accuracy implies relative error of 106 if the size of
Solutions for Homework 3 Optimization
Spring 2012
Problem 3.1
Problem 15 on page 259 of the textbook.
Solution: We have for > 0
xk+1 = xk + k rk , (1 )k k (1 + )k
k =
T
rk rk
T
rk Ark
First note that if = 1 then = 0 and there is no progress toward x = A1
Solutions for Homework 2 Optimization
Spring 2012
Problem 2.1
Let A Rnn be a symmetric positive denite matrix, C Rnn be a symmetric nonsingular
matrix, and b Rn be a vector. The matrix M = C 2 is therefore symmetric positive denite.
Also, let A = C 1 AC 1
Homework 5 Optimization
Spring 2012
The solutions will be posted on Monday, 3/26/12
Problem 5.1
Problem 1 on page 28 of the textbook.
Solution: Using slack and surplus variables yields the conversion
min x + 2y + 3z
subject to
x+y 2
x+y 3
x+z 4
x+z 5
x0
y
Homework 1 Optimization
Spring 2012
Problem 1.1
Let f (x) = x3 3x + 1. This polynomial has three distinct roots.
(1.1.a) Consider using the iteration function
1
1 (x) = (x3 + 1)
3
Which, if any, of the three roots can you compute with 1 (x) and how would
Solutions Homework 5 Introduction to Computational
Finance Spring 2013
Answers to the homework problems should be sent via email to [email protected]
before 11:59 PM on the due date. For problems with written solutions you may scan handwritten solutio
Homework 4 Introduction to Computational Finance Spring
2013
Solutions due Friday, 4/5/13
Solutions to the programming tasks should be sent via email to [email protected]
before 11:59 PM on the due date. For problems with written solutions you may sca