Download STAT 473. Practice Problems for Exam 2 Spring 2015 Description

STAT 473. Practice Problems for Exam 2
Spring 2015
Description:
(A) Exam 2 will cover Chapter 10 (Binomial Model), Sections 12.1-12.5 (only Black-Scholes
formulas), and Sections 18.2-18.5 (Log-normal Stock Price Model) from McDonald’s textbook.
(B) Solution key is provided at the end of the exam.
(C) The problems below are intended to serve as preparation or practice for your exam.
(D) You should not expect that your actual exam will cover only the topics or skills reviewed
in this set of problems. Also, you actual exam will be shorter.
(E) You are expected to review and fully understand all the homework problems of Assignments 4 and 5.
(F) In addition, you must review all the examples seen in class.
(G) Any material seen in class could potentially be tested in your actual exam.
1. You are given the following regarding a stock:
(i) The stock is currently selling for $50.
(ii) One-year from now the stock will sell for either $40 or $55.
(iii) The stock pays dividend continuously at a rate proportional to its price. The dividend yield is 10%.
The continuously compounded risk-free interest rate is 5%. If you notice that a one-year
at-the-money European call written on the stock is selling for $1.90.
(a) Use the binomial option pricing model to determine if an arbitrage opportunity
exists.
(b) If one exists, what transactions should you enter into at time 0 to exploit the arbitrage opportunity? Demonstrate the arbitrage opportunity.
2. The following one-period binomial stock price model was used to calculate the price of a
one-year 10-strike call option on the stock. The following informations is given:
(i) The period is one year.
(ii) The true probability p of an up-move is 0.75.
(iii) The stock pays no dividends.
(iv) The price of the one-year 10-strike call is $1.13.
Upon review, the analyst realizes that there was an error in the model construction and
that Sd , the value of the stock on a down-move, should have been 6 rather than 8. The true
probability of an up-move does not change in the new model, and all other assumptions
were correct. Recalculate the price of the call option.
(A) $ 1.13
(B) $ 1.20
(C) $ 1.33
(D) $ 1.40
(E) $ 1.53
3. You use the usual method in McDonald and the following information to construct a one
period binomial tree for modeling the price movements of a non-dividend-paying stock
(the so-called forward tree).
(i) The period is 3 months.
(ii) The initial stock price is $100.
(iii) The stocks volatility is 30%.
(iv) The continuously compounded risk-free interest rate is 4%.
At the beginning of the period, an investor owns an American put option on the stock.
The option expires at the end of the period. Determine the smallest integer-valued strike
price for which an investor will exercise the put option at the beginning of the period.
(A) 114
(B) 115
(C) 116
(D) 117
(E) 118
4. For a 1-year American put option on a stock modeled with a binomial tree:
(i) The tree has two periods.
(ii) The spot stock price is 42.
(iii) The strike price is 42.
(iv) The continuously compounded risk-free rate is 4%.
(v) The stock pays no dividends.
(vi) Volatility of the stock price is 10%.
(a) Determine the option’s premium.
(b) Suppose that a buyer of the option decides not to exercise earlier. Determine the
profit of the writer at expiration.
5. You are to price options on a futures contract. The movements of the futures price are
modeled by a binomial tree. You are given:
(i) Each period is 6 months.
(ii) u/d = 4/3, where u is one plus the rate of gain on the futures price if it goes up,
and d is one plus the rate of loss if it goes down.
(iii) The risk-neutral probability of an up move is 1/3.
(iv) The initial futures price is 80.
(v) The continuously compounded risk-free interest rate is 5%.
(a) Let CEur be the price of a 1-year 85-strike European call option on the futures
contract, and CAmer be the price of an otherwise identical American call option.
Determine CEur − CAmer .
(b) Suppose you notice that the premium of the European call option of part (a) is 3
dollars. Is there an arbitrage-opportunity? If one exists, what transactions should
you enter into to exploit the arbitrage opportunity?.
6. You are given the following information about a non-dividend paying stock:
(i) The current stock price is 100.
(ii) The stock-price process follows the Black-Scholes framework.
(iii) The continuously compounded expected return on the stock is 10%.
(iv) The stocks volatility is 30%.
(v) The risk-free interest rate is 5%.
Consider a nine-month 125-strike American call option on the stock.
Calculate the probability that the call will be exercised.
(A) 24.2%
(B) 25.1%
(C) 28.4%
(D) 30.6%
(E) 33.0%
7. Assume the Black-Scholes framework. Consider a 9-month at-the-money European put
option on a futures contract. You are given:
(i) The continuously compounded risk-free interest rate is 10%.
(ii) The strike price of the option is 20.
(iii) The price of the put option is 1.625.
If three months later the futures price is 17.7, what is the price of the put option at that
time?
(A) 2.09
(B) 2.25
(C) 2.45
(D) 2.66
(E) 2.83
8. Future prices of a stock are modeled with a 2-period Binomial tree. The probabilities of
up movements at all nodes of the tree are equal.
A European option on the stock has the following prices:
62.8
33.8
C0
16.6
6.4
0
Determine the price C0 of the option at the initial node.
Solution Key:
1. (a) Yes, there is an arbitrage opportunity since the market call price of $1.90 is underpriced.
(b) Buy call option, sell 0.3016 shares of IBM, and lend 13.18 dollars (i.e. buy 13.18
dollars in treasuries).
2. (D)
3. (B)
4. (a) 1.05; (b) 0.02171
5. (a) CAm − CEu = 0.08829. (b) Buy the market call for $3, sell 0.4280 forward contracts,
and borrow 3 dollars.
6. (A)
7. (D)
8. C0 = 16.728.