STAT 473. Practice Problems for Exam 2 Spring 2015

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 Assign-

ments 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 divi-

dend 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.

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