kmf229

A bond analyst decides to use the BSM model to price options on bond prices. This model will most likely be inadequate because:

A) the risk free rate must be constant and known.

B) BSM cannot be modified to deal with cash flows like coupon payments.

C) the price of the underlying asset follows a lognormal distribution.

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The BSM model is not useful for pricing options on bond prices and interest rates. In those cases, interest rate volatility is a key factor in determining the value of the option. BSM can be modified to deal with cash flows like coupon payments. The assumption that “the price of the underlying asset follows a lognormal distribution” is not applicable.

kmf229

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 $0.03 per share by selling the call and borrowing the remaining funds needed for the position at the risk-free rate.

B) earn an arbitrage profit of $0.30 per share by selling the call and lending $57.20 at the risk-free rate.

C) not earn an arbitrage profit because he should short the protective put position.

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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).

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The one-year call option on Dale Corporation:

A) is underpriced.

B) may be over or underpriced. The given information is not sufficient to give an answer.

C) is overpriced.

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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.

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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 concerning the distribution of stocks but incorrect concerning the risk-free rate.

B) incorrect for both reasons.

C) correct for both reasons.

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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.

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Which of the following is least accurate regarding the limitations of the BSM model?

A) The BSM is not useful in pricing options on bonds and interest rates.

B) The BSM is designed to price American options but not European options.

C) The BSM is not useful in situations where the volatility of the underlying asset changes over time.

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The following are limitations of the BSM:

• The assumption of a known and constant risk free rate means the BSM is not useful for pricing options on bond prices and interest rates.
• The assumption of a known and constant asset return volatility makes the BSM not useful in situations where the volatility is not constant which occurs much of the time.
• The assumption of no taxes and transaction costs makes the BSM less useful.
• The BSM is designed to price European options and not American options.

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If Bingly forecasts the volatility for a stock and find that it is significantly greater than that implied by the prices of the puts and calls of the stock, he would conclude that:

A) puts and calls are underpriced.

B) puts and calls are overpriced.

C) the puts are overpriced and the calls are underpriced.

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There is a positive relationship between the volatility of the stock and the price of both puts and calls. A higher estimate of volatility implies that the prices of both puts and calls should be higher.

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All else being equal, the greater the dividend paid by a stock the:

A) higher the call price and the lower the put price.

B) lower the call price and the lower the put price.

C) lower the call price and the higher the put price.

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When dividend payments occur during the life of the option, the price of the underlying stock is reduced (on the ex-dividend date). All else being equal, the lower price reduces the value of call options and increases the value of put options.

kmf229

stock is priced at 40 and the periodic risk-free rate of interest is 8%. 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 is closest to:

A) $9.25.

B) $3.57.

C) $9.13.

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First, calculate the probability of an up move or a down move:
U = 1.20 so D = 0.833
Pu = (1 + 0.08 − 0.833) / (1.20 − 0.833) = 0.673
Pd = 1 − 0.673 = 0.327
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(0.673) + 3(0.327)] / 1.08 = 13.745
C- = [3(0.673) + 0 (0.327)] / 1.08 = 1.869
Call value today = [13.745(0.673) + 1.869(0.327)] / 1.08 = 9.13

kmf229

A two-period interest rate tree has the following expected one-period rates:

t = 0		t = 1		t = 2
				7.12%
		6.83%
6.00%				6.84%
		6.17%
				6.22%

The price of a two-period European interest-rate call option on the one-period rate with a strike rate of 6.25% and a principal amount of $100,000 is closest to:

A) $725.86.

B) $423.89.

C) $449.33.

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• Calculate the payoffs on the call in percent for I++ and I+− (= I−+):
  I++ value = (0.0712 − 0.0625) / 1.0712 = 0.00812173.
  I+− value = (0.0684 − 0.0625) / 1.0684 = 0.00552228.
  Remember that the payoff on the call value is the present value of the interest rate difference based on the rate realized at t = 2 because the payment is received at t = 3.
• Calculate the t = 1 values (the probabilities in an interest rate tree are 50%):
  At t = 1 the values are I+ = [0.5(0.00812173) + 0.5 (0.00552228)] / 1.0683 = 0.00638585.
  At t = 1 the values are I− = [0.5(0) + 0.5 (0.00552228)] / 1.0617 = 0.00260068.
• Calculate the t = 0 value:
  At t = 0 the option value is [0.5(0.00638585) + 0.5(0.00260068)] / 1.06 = 0.00423893 0.00423893 × 100,000 = $423.89.

kmf229

A stock is priced at 38 and the periodic risk-free rate of interest is 6%. 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%?

A) $0.57.

B) $0.64.

C) $2.58.

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Given an up factor of 1.15, the down factor 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. You are directly given the probability of up = 0.68. The down probability = 0.32. The value of the put option is [0.322 × 6.27] / 1.062 = $0.57.

kmf229

Referring to put-call parity, which one of the following alternatives would allow you to create a synthetic stock position?

A) Sell a European call option; buy a European put option; short the present value of the exercise price worth of a riskless pure-discount bond.

B) Buy a European call option; buy a European put option; invest the present value of the exercise price in a riskless pure-discount bond.

C) Buy a European call option; short a European put option; invest the present value of the exercise price in a riskless pure-discount bond.

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According to put-call parity we can write a stock position as: S0 = C0 – P0 + X/(1+Rf)TWe can then read off the right-hand side of the equation to create a synthetic position in the stock. We would need to buy the European call, sell the European put, and invest the present value of the exercise price in a riskless pure-discount bond.

kmf229

Referring to put-call parity, which one of the following alternatives would allow you to create a synthetic European call option?

A) Buy the stock; sell a European put option on the same stock with the same exercise price and the same maturity; short an amount equal to the present value of the exercise price worth of a pure-discount riskless bond.

B) Sell the stock; buy a European put option on the same stock with the same exercise price and the same maturity; invest an amount equal to the present value of the exercise price in a pure-discount riskless bond.

C) Buy the stock; buy a European put option on the same stock with the same exercise price and the same maturity; short an amount equal to the present value of the exercise price worth of a pure-discount riskless bond.

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According to put-call parity we can write a European call as: C0 = P0 + S0 – X/(1+Rf)TWe can then read off the right-hand side of the equation to create a synthetic position in the call. We would need to buy the European put, buy the stock, and short or issue a riskless pure-discount bond equal in value to the present value of the exercise price.