Some Homework 1 Solutions
Solutions to #1.1, #1.2, #1.3, and book problem 9.4.
#1.1
By put-call parity, we have that
c p = P V (S K).
The LHS of this equation is 0.15 by the second assumption. This gives us
0.15 = S 70e4r
= 60 70e4r .
Solving for r, we ha
Homework 2
Due Wednesday, September 9th at 6:00am
Name
uu
u
90
99
ud
du
d
81
108.9
89.1
dd
72.9
Use the above diagram for problems 2.1-2.5. S is a stock with multiperiod binomial tree
shown above. The following hold:
(i) The stock pays dividends continuou
Some Homework 5 Solutions
#5.1
We want to model this currency exchange. After some translation, we have that the
current price of 1 is $0.007. The strike in this case is $7/700 or $0.01. Using the forward
tree, we have that u = e(0.040.06)0.5+0.4 0.5 = 1.
Homework 1
Due Wednesday, September 2nd at 6am
Name
1.1. Consider a European call option and a European put option on a nondividend-paying
stock. You are given:
The current price of the stock is 60.
The call option currently sells for 0.15 more than the
Some Homework 2 Solutions
#2.1
To solve this problem, we must use the three equations:
Sue0.03 + Be0.0375 = 10
Sde0.03 + Be0.0375 = 0
S + B = C0
(2.1)
(2.2)
(2.3)
The rst two give us that
(Su Sd)e0.03 = 10
This tells us that = 0.53914 Using this value of
Some Homework 6 Solutions
Solutions to #6.1, #6.2, #6.3, #6.5, and #6.6.
#6.1
Under the risk free measure, we know that
dS(t) = (r )S(t)dt + S(t)dZ(t)
This stock pays dividends at 4%, so that means that r = 0.11. The variance gives us that
= 0.5. This st
Homework 2
Due Thursday, January 29th
Name
uu
u
90
99
ud
du
d
81
108.9
89.1
dd
72.9
Use the above diagram for problems 2.1-2.5. S is a stock with multiperiod binomial tree
shown above. The following hold:
(i) The stock pays dividends continuously at a rat
Homework 1
Due Thursday, January 22nd
Name
1.1. Consider a European call option and a European put option on a nondividend-paying
stock. You are given:
The current price of the stock is 60.
The call option currently sells for 0.15 more than the put opti
Homework 7
Due Thursday, March 12th
Name
7.1. Suppose that S is a stock with S(0) = 36. Let V (t, S) be an option on S satisfying the
following:
0.09 2
Vt + 0.03SVs +
S Vss = 0.05V.
2
(a) Determine the price of a dierent option that pays the following in
Some Homework 3 Solutions
Solutions to #3.1, #3.2, #3.3, 10.19, 18.6, and 18.11.
#3.1
We use the standard normal table for part a. The two values we need to use are
and 1.64. We plug these values into the equation
S(1) = 70e0.021+0.3
p
1.64
1N (0,1)
Our p
Homework 3
Due Thursday, February 5th
Name
3.1. You wish to construct a 2-period binomial tree to model the price of a futures contract.
The following hold:
The continuously compounded risk-free interest rate is 9%.
The volatility of the exchange rate i
Homework 4
1.
Name
The following call option prices are observed (all options are on the same stock and with
the same expiration time). Is there any arbitrage opportunity? If so, what would you
do to eect arbitrage?
Strike
Call premium
80
22
100 105
9
5
T
Homework 4
Due Friday, September 25th
Name
4.1. Assume the Black-Scholes framework. Let S be a stock such that S(0) = 10, and the
dividend rate is 4% compounded continuously. Let C be a derivative that pays 100S(2)1
two years from now. In addition, the ri
Homework 3
Due Friday, September 18th
Name
3.1. Let S be a stock whose price is modeled using a lognormal distribution such that
ln S(t) N (ln 70 + .02t, 0.09t).
(a) Determine the 90% lognormal prediction interval for S(1).
(b) Determine the 95% lognormal
Homework 6
Due Thursday, March 5th
Name
6.1. Assume the Black-Scholes framework. Let S be a stock such that S(0) = 10, and the
dividend rate is 4% compounded continuously. Let C be a derivative that pays 100S(2)1
two years from now. Suppose that S satises
Math 5632: Financial Economics for Actuaries
Spring Semester 2015 The Ohio State University
Tuesday/Thursday 3:55-5:15pm
Lecturer:
Dr. Bradley Waller
Ofce:
Mathematics Tower 529
Email:
[email protected]
Website:
https:/people.math.osu.edu/waller.
Homework 6
Due Tuesday, November 4th
Name
33. This is a continuation of #32 from homework 5. Suppose that the price of the stock has
changed to 35 in 4 months.
(a) What is the change in value of the portfolio?
(b) What is the current of the portfolio?
34.
Approximation
Let S denote a stock with price 60 today. The price of a one year, strike 50 European
call on this stock is 13.14. The dividend rate is = 0.02, the risk free rate is r = 0.04, the
volatility is = 0.3, and d1 = 0.82.
a) What is
c?
d2
1
b) Wha
Simulation
You are given the four uniform random numbers: 0.03515, 0.26109, 0.75804, and 0.08691.
Also, todays stock price for Volkswagen is $27.13. You assume that the following hold:
r = 0.04, = 0.1, = 0.02, and = 0.4. Determine the following:
(a) The s
Homework 9
Due Thursday, April 21th
Name
12%
9%
8%
7%
6%
5%
4%
9.1. Use the interest rate binomial tree shown above for this problem. Assume that the
probability of an up move is 0.52, each step is one year, and the interest rates are annual
effective rat
Homework 5
Due Tuesday, March 1st
Name
5.1. Suppose that the price of a nondividend paying stock is 25, and the price of a put option
on this stock is 3.29. You also know that = 0.32 and = 0.13. Estimate the new value
of the put if the stocks value change
Homework 7
Due Tuesday, April 5th
Name
7.1. Use a two period forward (binomial) tree with six month steps to determine the price
of a one year Asian arithmetic average strike put option given that
S(0) = 48
r = 0.07
= 0.03
= 0.4.
Do not forget to use ri
Homework 8
Due Tuesday, April 12th
8.1. Let dX(t) = 3e
3t
Name
X(t)dt
X(t)dZ(t) and Y (t) = t
p
X(t). Suppose that
dY (t) = a(t, Y (t)dt + b(t, Y (t)dZ(t)
Calculate a(1, 1).
8.2. The prices of two stocks are governed by the following:
dX(t)
= 0.05dt + 0.3
Homework 6
Due Tuesday, March 8th
Name
Use the Theta Interpretation spreadsheet for the following two problems.
6.1. Suppose that a stock satisfies the following
S(0) = 32
= 0.03
r = 0.09
= 0.2
You purchase 50 eight month calls with strike 33 on this st
Some Homework 5 Solutions
Solutions to #5.1, #5.2, #5.3, #5.4, #5.5, #12.20, and #13.1.
#5.1
This is an application of the Approximation. In this case, = 2.
V (t, 23) V (t, 25) + (0.32)(2) + (
0.13
)(2)2
2
= 4.19
#5.2
This is similar to #5.1. We only need
Some Homework 4 Solutions
Solutions to #4.1, #4.2, and #4.3.
#4.1
The assumptions give us
S(t) = 10e(0.07
0.125)t+0.5Z(t)
p
S(2) = 10e 0.11+0.5 2N (0,1)
p
1
S(2) 1 = e0.11 0.5 2N (0,1)
10
b2
Using our favorite formula E(Sea+bN (0,1) ) = Sea+ 2 , we have t
Homework 5
Due Thursday, October 23rd
There will be a quick quiz over this!
Name
27. A stock, S, has the following characteristics:
The rate of return is 0.14.
The volatility is 0.2.
The Sharpe ratio is 0.4.
The current price of S is 14.
In addition,
Some Homework 2 Solutions
Solutions to #2.1, #2.2, #2.3, #2.4, #2.5, #2.6, and book problem 10.18a.
#2.1
To solve this problem, we must use the three equations:
Sue0.03 + Be0.0375 = 10
Sde0.03 + Be0.0375 = 0
S + B = C0
(2.1)
(2.2)
(2.3)
The rst two give u