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 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.
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
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
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 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
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 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
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 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
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:
waller@math.ohio-state.edu
Website:
https:/people.math.osu.edu/waller.
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
Homework 1
Due: Thursday, January 21st at 2pm
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
Math 5632
Autumn 2013
1.
Homework 3
Name:
You observed the following call option prices for dierent strike prices.
Strike Price
Call Premium
100 105
10 16
Is there an arbitrage opportunity? If so, build a portfolio to show it.
The call price C as a functi
Math 632 Notes
Chapter 9
Review:
Derivative
forward
payos
ST F0,T (S)
Long and short positions
call option
(ST K)+
put option
(K ST )+
P
F0,T (S) = e(r)T S0 , F0,T (S) = eT S0
Payo diagrams
Synthetic forward and its payo
No-arbitrage, one price prin
Math 632 Notes
Chapter 13
Delta Hedging
Let Ct = C(St , K, , r, T t, ) and suppose Si+h Si = . By Taylors formula
1 2 Ci
Ci
Ci
(Si+h Si ) +
h
(Si+h Si )2 +
2
Si
2 Si
t
1
= Ci + i + 2 i + i h
2
Ci+h Ci +
(1)
Equation (1) is called the - approximation of
Adil Mutlak
Energy of a System
Chapter7
1
The Chapters Goal
To introduce the new concepts of work and energy to solve problems in physics.
we will introduce the following
Energy transfer from one object to another
The concepts of kinetic and potential e
Math 632 Notes
Chapter 12
Review of normal distribution N (, 2 ).
Black-Scholes Formula
Assumptions:
The return on the stock is normally distributes and independent overtime.
The volatility of the return is a constant .
The dividend yield is a consta
Math 632 Notes
Chapters 10, 11
One Period Binomial Tree:
Suppose the (time) duration of a period is h. Suppose the current price of a stock is
S0 = S, and one period later at time h, the price of the stock Sh can be either uS or
dS. Suppose an option C o
Homework 6
1.
Name
In a 1-period binomial model, the stock price S(h) has two values Su = uS0 and Sd =
dS0 with (risk-neutral) probabilities P[S(h) = Su ] = p and P[S(h) = Sd ] = 1 p
respectively. (S0 = S(0) is the stock price at time 0.)
(a) In a 2-perio
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
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
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.
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,
Homework 5
1.
Name
In a one-period binomial model with h = 1, the current price of a non-dividend paying
stock is 50, u = 1.2, d = 0.8, and the continuous interest rate is 2%. Consider a call
option on the stock with strike K = 50. What position in the st
Homework 2
1.
Name
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 option.
Both the call option and
Math 5632
Autumn 2013
1.
Homework 1
Name:
The current price of a non-dividend paying stock is S0 = 100. The interest rate is 2% per
year compounded continuously. Consider a forward contract on the stock that expires
at time T = 1/2.
(a) Find the forward p
Homework 3
Due Thursday, October 2nd
Name
15. Suppose that S is a stock that is lognormally distributed and that S satisfies the following:
S(0) = 22.
4
The expected price of the stock in 16 months is E S
= 24.
3
The volatility of S is = 0.3
4
(a)
Homework 1
Due Thursday, September 4th
Name
1. Consider a portfolio that contains:
A long position in a 50-strike put option on a stock
A long position in a forward contract on the same stock with a forward price of F = 50.
Both the option and the forwa
Homework 4
Due Thursday, October 16th
Name
22. Suppose that the stock S satisfies the Black-Scholes framework and the dynamics of S
are
dS(t) = 0.12S(t)dt + 0.3S(t)dZ(t).
Additionally, the initial price of the stock is 15, S pays no dividends, and the sha
Homework 2
Due Tuesday, September 16th Name
133.1
uu
\
ud
du
121
/
110 108.9
K
99
dd
/
891
Use the above diagram for problems 711. S is a stock with multiperiod binomial tree shown
above. The following hold:
(i) The stock pays diVidends continuously at
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 70e 4r
= 60 70e 4r .
1
Solving for r, w