Financial Engineering
MRM 8610, Spring 2014
The Black-Scholes Option Pricing Model
We mentioned before that the Bachelier model, the simplest continuous-time mathematical
model for the stock market, assumes that the stock price follows a Brownian motion w

Financial Engineering
MRM 8610, Spring 2014
Assignment 1 -
due on January 30th, 7:15pm.
1. Suppose that the initial stock price is S0 = $80. The stock price increases or decreases
in each period of time with the same probability. Every time the stock pric

Lecture 2: Static Monto Carlo
Ajay Subramanian
January 24, 2013
Ajay Subramanian ()
Lecture 2: Static Monto Carlo
January 24, 2013
1 / 37
Acceptance-Rejection Methods
PX target distribution on D = Im(X )
Random sample Y drawn from a distribution PY with s

Financial Engineering
MRM 8610, Spring 2014
It integral and It calculus
o
o
Consider that now on, we will work on (, F, P, F), the probability space with ltration or
ltered space where F is the natural ltration.
1
A new type of integral
By using the multi

Financial Engineering
MRM 8610, Spring 2014
The Binomial (non-arbitrage pricing) model
In Finance, the Binomial model is an option pricing model that was proposed by Cox, Ross
and Rubinstein in 1979. It is discrete-time model that considers that the price

Financial Engineering
MRM 8610, Spring 2014
Generalization of the Black-Scholes Model
We have seen that the Black-Scholes model, and its results, needs a series of assumptions
to hold in order to be able to operate in it. For instance, if the assumption t

Financial Engineering
MRM 8610, Spring 2014
Practice Exercises for the Final Exam
1. Show that Xt =
1
1+t Wt
solves
dXt =
1
1
Xt dt +
dWt ,
1+t
1+t
X0 = 0.
2. Solve the stochastic dierential equation
dXt = Xt dt + et dWt .
Hint: Use Its formula for et Xt

Financial Engineering
MRM 8610, Spring 2014
Solution of Midterm exam
1. We have that, due to the dierent risk free rates, the p for each period are dierent.
Indeed,
p (0) =
p (1) =
p (2) =
1.05 0.5
= 0.55
1.5 0.5
1.08 0.5
= 0.58
1.5 0.5
1.06 0.5
= 0.56.
1

Financial Engineering
MRM 8610, Spring 214
Practice exercises for the Midterm
1. Exercise 1.3 [S1]. Consider a one-period Binomial model with S0 = 4, u = 2, d = 1/2
and r = 1/4. Suppose that we want to determine the price at time 0 of the derivative
secur

Module 5
Revenue Recognition
and Operating Income
DISCUSSION QUESTIONS
Superscript A, B denotes assignments based on Appendix 5A, 5B.
Q5-1.
Revenue must be realized or realizable and earned before it can be reported in the
income statement. Realized or re

3. Analyze and interpret accounting
adjustments and their financial
statement effects. (p. 2-23)
2. Analyze and interpret transactions
using the financial statement
effects template. (p. 2-20)
4. Construct financial statements from
account balances. (p

Financial Engineering
MRM 8610, Spring 2014
Risk Neutral Pricing
We rely on the intuition we obtained from the Binomial Asset Pricing Model in order to
obtain a nice pricing representation for more general pricing model, in particular for the BlackScholes

Financial Engineering
MRM 8610, Spring 2014
The Black-Scholes Option Pricing Model
We mentioned before that the Bachelier model, the simplest continuous-time mathematical
model for the stock market, assumes that the stock price follows a Brownian motion w

Financial Engineering
MRM 8610, Spring 2014
Risk neutral pricing (contd)
So far we have seen that the price of a European derivative at time t which we obtained
based on two fundamental ideas: that the market is arbitrage-free, and that replication argume

Financial Engineering
MRM 8610, Spring 2016
Homework 3
Daniel Bauer
Due Thursday, 02/04/2016 before class.
1. Brownian Motion and Martingales: (1+1+1 points)
Let (Wt )t0 and (Zt )t0 be two independent Brownian motions. Use the definition of Brownian motio

Financial Engineering
MRM 8610, Spring 2016
Homework 2
Daniel Bauer
Due Thursday, 01/28/2016 before class
1. Review Taylors Theorem from Calculus.
2. Brownian Motion: (3 points)
Let (Wt )t0 and (Zt )t0 be two independent standard Brownian motions. Use the

Financial Engineering
MRM 8610, Spring 2014
Preliminaries in Probabilities: Brownian Motion
1
Information and -algebras
Consider a two-period Binomial model as shown in Figure 1. The set of possible paths that
the stock price can follow is cfw_1 , 2 , 3 ,

Financial Engineering
MRM 8610, Spring 2014
Practice Exercises for the Final Exam
1. Show that Xt =
1
1+t Wt
solves
dXt =
1
1
Xt dt +
dWt ,
1+t
1+t
X0 = 0.
2. Solve the stochastic dierential equation
dXt = Xt dt + et dWt .
Hint: Use Its formula for et Xt

Information and -algebras
Conditional expectation
Brownian motion
MRM 8610 - Financial Engineering
Preliminaries in Probability
Department of Risk Management and Insurance
Georgia State University
Atlanta, February 6, 2014
Preliminaries in Probability
Inf

Lecture 10: Quadrature Methods
Ajay Subramanian
March 22, 2012
Ajay Subramanian ()
Lecture 10: Quadrature Methods
March 22, 2012
1 / 29
Gaussian Quadrature Formulae
Formula can be extended to integrate the product of polynomials with
any function .
Need t

Lecture 3: Dynamic Monto Carlo
Ajay Subramanian
January 26, 2012
Ajay Subramanian ()
Lecture 3: Dynamic Monto Carlo
January 26, 2012
1 / 31
Main Issues
Algorithms for simulating paths of a random process
Simulating samples of continuous-time random proces

Lecture 8: Numerical Solution of Linear Systems
Ajay Subramanian
March 8, 2012
Ajay Subramanian ()
Lecture 8: Numerical Solution of Linear Systems
March 8, 2012
1 / 24
Introduction
Methods to solve linear systems of the form Ax = b where A is an
(m m) mat

Lecture 11: Copulas
Ajay Subramanian
April 14, 2011
Ajay Subramanian ()
Lecture 11: Copulas
April 14, 2011
1 / 16
Introduction
The word copula denotes linking or connecting between parts
Adopted in statistics to denote a class of functions allowing us to

Case 1: An American Monte Carlo
Ajay Subramanian
February 17, 2011
Ajay Subramanian ()
Case 1: An American Monte Carlo
February 17, 2011
1/9
Introduction
Pricing American-style options using Monte-Carlo simulation using a
method proposed by Rogers (2002).