Baran

6、A stock is priced at 40 and the periodic risk-free rate of interest is 8 percent. What is the value of a two-period European call option with a strike price of 37 on a share of stock using a binomial model with an up factor of 1.20 and a (risk-neutral) up probability of 67 percent?

A) $20.60.
 
B) $3.57.
 
C) $9.25.
 
D) $9.07. 

Baran

The  correct  answer  is D

Two up moves produce a stock price of 40 × 1.44 = 57.60 and a call value at the end of two periods of 20.60. An up and a down move leave the stock price unchanged at 40 and produce a call value of 3. Two down moves result in the option being out of the money. The value of the call option is discounted back one year and then discounted back again to today. The calculations are as follows:

C+ = [20.6(0.67) + 3(0.33)] / 1.08 = 13.6962

C- = [3(0.67) + 0 (0.33)] / 1.08 = 1.8611

Call value today = [13.696(0.67) + 1.8611(0.33)] / 1.08 = 9.07

Baran

7、A stock is priced at 38 and the periodic risk-free rate of interest is 6 percent. What is the value of a two-period European put option with a strike price of 35 on a share of stock using a binomial model with an up factor of 1.15 and a risk-neutral probability of 68 percent?

A) $2.58.
 
B) $0.64. 
 
C) $0.57. 
 
D) $2.90. 

Baran

The  correct  answer  is C

Given an up probability of 1.15, the down probability is simply the reciprocal of this number 1/1.15=0.87. Two down moves produce a stock price of 38 × 0.872 = 28.73 and a put value at the end of two periods of 6.27. An up and a down move, as well as two up moves leave the put option out of the money. The value of the put option is [0.322 × 6.27] / 1.062 = $0.57.

Baran

8、Al Bingly, CFA, is a derivatives specialist who attempts to identify and make short-term gains from trading mispriced options. One of the strategies that Bingly uses is to look for arbitrage opportunities in the market for European options. This strategy involves creating a synthetic call from other instruments at a cost less than the market value of the call itself, and then selling the call. During the course of his research, he observes that Hilland Corporation’s stock is currently priced at $56, while a European-style put option with a strike price of $55 is trading at $0.40 and a European-style call option with the same strike price is trading at $2.50. Both options have 6 months remaining until expiration. The risk-free rate is currently 4 percent.

Bingly often uses the binomial model to estimate the fair price of an option. He then compares his estimated price to the market price. He observes that Dale Corporation’s stock has a current market price of $200, and he predicts that its price will either be $166.67 or $240 in one year. The risk-free rate is currently 4 percent. He also observes that the price of a one-year call with a $220 strike price is $11.11.

Bingly also uses the Black-Scholes-Merton model to price options. His stated rationale for using this model is that he believes the prices of the stocks he analyzes follow a lognormal distribution, and because the model allows for a varying risk-free rate over the life of the option. His plan is to use a statistical technique to estimate the volatility of a stock, enter it into the Black-Scholes-Merton model, and see if the associated price is higher or lower than the observed market price of the options on the stock.

Bingly wishes to apply the Black-Scholes-Merton model to both non-dividend paying and dividend paying stocks. He investigates how the presence of dividends will affect the estimated call and put price.

In the case of the options on Hilland Corporation’s stock, if Bingly were to establish a long protective put position, he could:

A) earn an arbitrage profit of $0.30 per share by selling the call and lending $57.20 at the risk-free rate. 
 
B) earn an arbitrage profit of $0.03 per share by selling the call and borrowing the remaining funds needed for the position at the risk-free rate.
 
C) not earn an arbitrage profit because he should short the protective put position.
 
D) not earn an arbitrage profit because the position is in equilibrium.

Baran

The  correct answer is B

Under put-call parity, the value of the call = put + stock – PV(exercise price). Therefore, the equilibrium value of the call = $0.40 + $56 - $55/(1.040.5) = $2.47. Thus, the call is overpriced, and arbitrage is available. If Bingly sells the call for $2.50 and borrows $53.93= $55/(1.040.5), he will have $56.43 > $56.40 (= $56 + $0.40), which is the price he would pay for the protective put position. The arbitrage profit is the difference ($0.03 = $56.43 - $56.40).

Baran

The one-year call option on Dale Corporation:
A) is underpriced.
 
B) is overpriced.
 
C) is fairly priced.
 
D) may be over or underpriced. The given information is not sufficient to give an answer.

Baran

The  correct answer is B

The up movement parameter U=1.20, and the down movement parameter D=0.833. We calculate the probability of an up move πU = (1 + 0.04 – 0.833)/(1.2 – 0.833) = 0.564. The call is out of the money in the event of a down movement, and has an intrinsic value of $20 in the event of an up movement. Therefore, the estimated value of the call is C = (0.564) × $20 / (1.04) = $10.85. Thus, the price of $11.11 is too high and the call is overpriced.

Baran

Bingly’s sentiments towards the Black-Scholes-Merton (BSM) model regarding a lognormal distribution of prices and a variable risk-free rate are:
A) correct for both reasons.
 
B) incorrect for both reasons. 
 
C) correct concerning the distribution of stocks but incorrect concerning the risk-free rate.
 
D) incorrect concerning the distribution of stocks but correct concerning the risk-free rate.

Baran

The  correct answer is C

The model requires many assumptions, e.g., the distribution of stock prices is lognormal and the risk-free rate is known and constant. Other assumptions are frictionless markets, the options are European, and the volatility is known and constant.