Probability for Math 441
November 2, 2010
Outline
Deﬁnition of Probability
Random variables
Expectation for a ﬁnite sample space
Variance and standard deviation
c.d.f and p.d.f
Expectation (revisited)
The normal variable
Outline
Deﬁnition of Probability
Chapter 9
Binomial trees and risk-neutral
pricing
9.1 One step model
The one-step binomial tree simplifies things further with the
following assumptions:
There are only two times: the present time t = 0 and the
expiration date of the option at t = T.
If
4.2 Derivatives
Companies use derivatives to manage their exposure to risks, namely
interest- rate risks, currency risks, and inflationary risks. Derivatives refer to
a financial instrument whose value is derived from another instrument or
asset. If we th
Chapter 6 Options
6.1 Definitions
CHAPTER 6. OPTIONS
buying the asset at K is a losing proposition, i.e., the asset is
trading for less than K on the market.
In a European call option the buyer can only exercise the option at
the expiration of the option,
Options
There are two types of options, call options and put options, and each of
these has two flavors, American or European. In all these variations there are
always two parties. There is the buyer of the option and the seller of the
option. The buyer p
Interest Rates
1.1 Rate of return
Examples:
An initial in- vestment of $1000 grows to $1250 after 1 month. We
have earned $250. Since we expect many investments to earn
money in proportion to the amount of initial money invested, we
might express our earn
Bonds
2.1 Factors affecting interest rates
Example 2.1. The six-month risk-free rate today is 1.7%. The 1year risk- free rate is 2.1%. How much will an investment of $100
be worth in 6 months if it invested today at the risk-free rate? What
if it is borro
Problem Set 8
!
1.
!
Show that if A,B = 1, then =
in Equation 8.2.
!
Bu the efficient frontier lies on a line
Conclude that
connecting A and B in mean-standard deviation space. Show
that the minimal variance portfolio occurs when the portfolio
is short
Problem Set 2
1.
A 2-year T-note has a par value of $1000. It pays a coupon of
4.5% per annum, paid semiannually. If the bond is issued
today, calculate the price of the bond if we assume all riskfree zero rates are 4%.
2.
In the previous problem, suppose
Problem Set 1
1.
An investment of $3000 grows to $4000. What is the rate of
return on the investment?
2.
A certain investment lasting 2 years guarantees a 9% rate of
return on the investment. If the principal for this investment
is $20,000, how much is th
Chapter 3
Swaps and more on bonds
3.1 Bootstrapping
Example 3.1.
Bond A is a zero coupon bond that matures in 4 months. Its face value is
$1000 and its market price is $992. Bond B is a bond that matures in 10
months, with semiannual payments and a coupon