ACTSC/STAT 446/846 Problems for Week 6 Solutions
Solution 1:
2
if K 1/2
1
Payoff:
if 1/2 < K 2
1
1/2
1
0
if K > 2
0
0
%
1
&
- if K 12 , then the price is e0.03 = 0.9704 since the payoff is certain and always 1. Can also
construct the replicating portfolio
ACTSC 446/846 Practice Problems for Final Exam
1. An investor sells a put option on a stock and receives a premium (whose value is the price of the
option). The stock is assumed to follow the Black-Scholes model with volatility 0.3 and initial
price S0 =
ACTSC 446/846 Problem Set 7 Solutions
1. Problem 1 Solution:
(a) A nonzero payoff is only possible if the volatility is high and the stock jumps up. This is the
case if and only if the state of the world at time t = 1 is given by = 1 . The contingent clai
ACTSC 446/846 Problems for Week 9
1. McDonald, Problems 20.2, 20.4, 20.6 and 20.9 1 .
2. Explain why the following statement is wrong: If cfw_Wt , t 0 is a std Brownian motion, then
Wt Ws N (0, t + s) since Ws N (0, s) and Wt N (0, t)?
3. Let W = cfw_Wt ,
ACTSC 446/846 Problems for Week 11 Solutions
1. Assuming the Black-Scholes model, determine a formula for the price at time 0 of a geometric
Asian call option, whose payoff is max(0, (S1 S2 S3 )1/3 K) at its maturity date T = 3.
Solution: Under the risk-n
ACTSC/STAT 446/846 Problems for Week 11
1. Assuming the Black-Scholes model for cfw_St , t 0, determine a formula for the price at time 0 of
a geometric Asian call option, whose payoff is max(0, (S1 S2 S3 )1/3 K) at its maturity date T = 3.
2. Assume that
ACTSC 446/846 Problem Set 8 Solutions
1. The current price of an asset is $20 and it has a volatility of 15% per year. We model the evolution
of its price with a binomial tree with monthly time-steps. The continuously compounded interest
rate is 7% per ye
ACTSC 446/846 Problems for Week 9 Solutions
1. Problem 20.2
If y = S 2 then S = y and dy = 2S (S, t) + (S, t)2 dt + 2S (S, t) dZt where (S, t) is the
drift of S and (S, t) is the volatility of S. For the three specifications:
(a) dy = 2 y + 2 dt + 2 ydZt
ACTSC 446/846 Problems for Week 10
1. McDonald 13.1, 13.2, 13.15, 13.16, 21.5, 21.6, 21.7
2. Let cfw_St , t 0 be a stock price process satisfying
dSt = ( )St dt + St dWt ,
where (< ) is the continuous dividend rate paid by the stock. Let Bt = ert be the v
ACTSC 446/846 Problems for Week 12
1. (Problem started in class on Dec. 1) Consider a stock price with dynamics
dSt = t St dt + dWt (under P )
where
(
0.06
t =
0.08
if 0 t < 5
if t 5,
and a riskless bond with value Bt = ert at time t.
(a) Show there exist
ACTSC 446/846 Problems for Week 7
1. We consider the following market model. It consists of two tradeable assets, one money market
account B(t) and one stock S(t) (t = 0, 1), as well as third object which we call the volatility
v. The volatility determine
ACTSC/STAT 446/846 Problems for Week 8
1. The current price of an asset is $20 and it has a volatility of 15% per year. We model the evolution
of its price with a binomial tree with monthly time-steps and the Cox-Ross-Rubinstein model.
The continuously co
ACTSC 446/846 Problems for Week 6
1. Consider the binomial market model with parameters: r = 0.03 (continuously compounded
interest rate), S0 = 1, u = 2, d = 12 . Compute the price of a digital call option with strike price
K which is characterized by the
ACTSC 446/846 Problems for Week 12 Solutions
1. (Problem started in class on Dec. 1) Consider a stock price with dynamics
dSt = t St dt + dWt (under P )
where
(
0.06
t =
0.08
if 0 t < 5
if t 5,
and a riskless bond with value Bt = ert at time t.
(a) Show t
ACTSC 446/846 Problems for Week 10 Solutions
1. McDonald 13.1
The delta of the option is .2815. To delta hedge writing 100 options we must purchase 28.15
shares for a delta hedge. The total value of this position is 1028.9 which is the amount we will
init
Perpetuities
A perpetuity is an annuity whose payments continue forever. Examples are scholarship endowment funds,
dividends on preferred shares, etc.
In discussing perpetuities, the accumulated value is meaningless since there is no end to the payments.
Math 137: Calculus 1 for Honours Mathematics
Fall 2015
Week 1
Alain Gamache
Office: MC 6238
agamache@uwaterloo.ca
Phone ext: 38041
September 10, 2015
Maths 137: Calculus 1 for Honours Mathematics
Week 1
Housekeeping items
Assignments due:
Friday by noon i
Continuous Compounding
We see the effect as interest moves from annual compounding to quarterly compounding to daily compounding.
This begs the question: is it possible to have continuous compounding and what effect would this have on the
annual return?
C
University of Waterloo
Midterm #1
ACTSC 371 Introduction to Investments
Instructors: Banerjee (Section 2); Blake (Section 1)
Date: Friday, October 7, 2016
Time: 8:30-9:50.
Duration: 80 minutes
Number of Pages: (including the cover page) 10
Aids: Faculty A
ACTSC 371
Fall, 2016 Quiz 3
Draft Questions
1. Calculate the Macaulay duration and modified duration for the following two $1,000 bonds:
Bond A three year zero coupon bond
Bond B four year annual coupon bond, paying annual coupons at the annual rate of 7%
ACTSC 371
Review for Midterm 2
Be comfortable with the calculations in the time value of money handout.
Know what is meant by the term utility value and that an equation to derive it must incorporate expected return
and a value for risk the standard devia
ACTSC 372 Winter 2017
Written Assignment 2
Due on Feb 3rd
Question 1: Expected Return v.s. Holding Period Return (25pt)
When convincing investors to invest in a particular stock, fund manager often shows the stocks expected
return estimated from its histo
ACTSC 372
Corporate Finance II
Chapter 6: Capital Structure
R eview
To finance a new project a firm can choose to use
Internal cash flow
External cash flow
New Equity
New Debt
Different external financing options have different features
Voting righ
ACTSC 372 Winter 2017
Written Assignment 3
Due on Feb 17th
Question 1 (25pt): Risk, Return, and Capital Budgeting (Multiple Choices)
In these multiple choice questions, circle the correct answer. No justification is required.
(1) (5pt) The NPV formula for
Lectures week 12
Chapter 9 Asset-Liability Management
In an ideal world we would pick our assets, such that
the inflow of income from these assets exactly
matches our outflow of expenses and other liabilities.
This is called exact matching and is not alwa
Lectures week 11
6.6-6.7 Price between coupon dates
Book values. These are all calculated at the investors
yield rate j and we assume that the investor
purchased the bond at issue.
B 0 P Fra n|j Cv n purchase price at issue.
B 1 1 jB 0 Fr Fra n1|j Cv n1 b
Lectures week 8
4.6 Continuous Annuities
Notation a n| lim m m1 a mn|i im
m
1
v e and ln1 i
1v n
, where
This is the PV of an n-year annuity which pays
continuously at a rate of $1 per year.
Example: Find the PV of an annuity which pays
continuously at
Lectures week 7
3.9 Payments in Arithmetic Progression
Payments starting at P and each subsequent
payment is Q more.
PV Pa n| Qv 2 2Qv 3 n 1Qv n
(1)
1 iPV 1 iPa n| Qv 2Qv 2 n 1Qv n1 (
1 2
iPV iPa n| Qv v 2 v n nQv n
iPa n| Qa n| nQv n
a n| nv n
PV Pa n|
ACTSC 371
Chapter 3 Trading on Securities Markets
How firms issue securities to the public(page13)
When a corporation needs to raise money, its basic choice is to _raise money/borrow equity_. It can do both
privately (bank loans, private debt or equity pl
ACTSC 371
Chapter 5 Capital Allocation to Risky Assets
When it comes to the construction of an investors portfolio, the goal is to construct a portfolio that will
_maximum return and minimize risk. We are also trying to determine if we can use mathematica