Math 489/889
Stochastic Processes and
Advanced Mathematical Finance
Practice Problems Homework 10
Steve Dunbar
Monday, December 7, 2009
Initialization of Necessary Packages
(1.1)
_R
(1.2)
Problem 1
Find the mode (the value of the independent variable with the highest probability)
Note very carefully that Maple uses different parameters than I do! I define the
So be careful to get the distribution you really want!
(2.1)
(2.2)
e
(2.3)
Problem 2
A call option on a stock is said to be in the money if the value of the stock is higher
than the strike price, so that selling or exercising the call option would result in a
profit (ignoring transaction costs.) Consider a stock (or more properly an index
fund) whose value S(t) is described by the stochastic differential equation
corresponding to a non-zero compounded growth and pure market fluctuations
proportional to the stock value and has a current market price of ? What is the
probability that a call option is in the money based on a strike price K = 1.25
at
expiration
and also for
time units later? Evaluate for
with the same parameters.
1.25
0.5
0.04
0.10
0.01750000000
,
,
, and
0.07071067812
_R0
(3.3)
0.0018173522
(3.4)
See the full solution derivation to see how to express the probability that the stock
is in the money expressed in terms of the Standard Normal CDF. The calculation
is:
_R1
(3.5)
0.0018173522
(3.6)
1
(3.7)
0.5977344689
(3.8)
0.5977344689
(3.9)
Problem 3
What is the price of a European call option on a non-dividend-paying stock when
the stock price is $52, the strike price is $50, the risk-free interest rate is 12% per
annum (compounded continuously), the volatility is 30% per annum, and the
time to maturity is 3 months?
(4.1)
52.00
50.00
0.12
0.25
0.30
(4.2)
5.057386760
(4.3)
0.5364714210
(4.4)
0.3864714210
(4.5)
0.3864714210
(4.6)
0.704183608830784014
(4.7)
0.650426217823152353
(4.8)
5.05738676
(4.9)
Problem 4
What is the price of a European call option on a non-dividend paying stock when
the stock price is $30, the exercise price is $29, the risk-free interest rate is 5%, the
volatility is 25% per annum, and the time to maturity is 4 months?
30.00
29.00
0.05
1
3
0.25
(5.1)
2.525146967
(5.2)
0.4225156800
(5.3)
0.2781781127
(5.4)
0.2781781127
(5.5)
0.663675670914689264
(5.6)
0.609562182219208548
(5.7)
2.52514697
(5.8)
Problem 5
(6.1)
0
(6.2)
(6.3)
0
(6.4)
Problem 6
Black Scholes theory does not apply. The Black-Scholes derivation explicitly
assumes that the changes in price are continuous. Here the price will immediately
change discontinuously by the amount $10.00.
Problem 7
(8.1)
(8.2)
(8.3)