Portfolio Management【 Reading 56】Sample 2012, create, following, one

kmf229

Referring to put-call parity, which one of the following alternatives would allow you to create a synthetic riskless pure-discount bond?

A) Buy a European put option; sell the same stock; sell a European call option.

B) Buy a European put option; buy the same stock; sell a European call option.

C) Sell a European put option; sell the same stock; buy a European call option.

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According to put-call parity we can write a riskless pure-discount bond position as:
X/(1+Rf)T = P0 + S0 – C0.

We can then read off the right-hand side of the equation to create a synthetic position in the riskless pure-discount bond. We would need to buy the European put, buy the same underlying stock, and sell the European call.

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.

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

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

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

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

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

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

Rachel Barlow is a recent graduate of Columbia University with a Bachelor’s degree in finance. She has accepted a position at a large investment bank, but first must complete an intensive training program to gain experience in several of the investment bank’s areas of operations. Currently, she is spending three months at her firm's Derivatives Trading desk. One of the traders, Jason Coleman, CFA, is acting as her mentor, and will be giving her various assignments over the three month period.
One of the first projects he asks her to do is to compare different option trading strategies. Coleman would like Barlow to pay particular attention to strategy costs and their potential payoffs. Barlow is not very comfortable with option models, and knows she needs to be able to fully understand the most basic concepts in order to move on. She decides that she must first investigate how to properly price European and American style equity options. Coleman has given her software that provides a variety of analytical information using three valuation approaches: the Black-Scholes model, the Binomial model, and Monte Carlo simulation. Barlow has decided to begin her analysis using a variety of different scenarios to evaluate option behavior. The data she will be using in her scenarios is provided in Exhibits 1 and 2. Note that all of the rates and yields are on a continuous compounding basis.

Exhibit 1

Stock Price (S)	$100
Strike Price (X)	$100
Interest Rate (r)	7%
Dividend Yield (q)	0%
Time to Maturity (years)	0.5
Volatility (Std. Dev.)	20%

Exhibit 2

Stock Price (S)	$110
Strike Price (X)	$100
Interest Rate (r)	7%
Dividend Yield (q)	0%
Time to Maturity (years)	0.5
Volatility (Std. Dev.)	20%
Value of European Call	$14.8445

Barlow notices that the stock in Exhibit 1 does not pay dividends. If the stock begins to pay a dividend, how will the price of a call option on that stock be affected?

A) Increase.

B) Increase or decrease.

C) Decrease.

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The call option value will decrease since the payment of dividends reduces the value of the underlying, and the value of a call is positively related to the value of the underlying. (Study Session 17, LOS 60.g)

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Barlow determines the price of an American call option on the stock shown in Exhibit 2. Which of the following is the most accurate?

A) $14.72.

B) $15.41.

C) $14.84.

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The value of the American-style call option is the same as the value of the equivalent European-style call option. Since the underlying stock does not pay a dividend, it is never optimal to exercise the American option early. Hence the early-exercise option imbedded in the American-style call has no value in this case. This makes the American option worth exactly the same as the European option. (Study Session 17, LOS 60.j)

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Using the information in Exhibit 2, Barlow computes the value of a European put option. Which of the following is closest to Barlow's answer?

A) $1.41.

B) $0.98.

C) $4.84.

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Using the information in Exhibit 2, this value can be determined from put-call parity as follows:
Put = Call + Xe−rt − S
So we have Put = $14.8445 + $100.00e(−7.00% × 0.5) − $110.00 = $1.4050
(Study Session 17, LOS 60.c)

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Barlow notices that the stock in Exhibit 2 does not pay dividends. If the stock starts to pay a dividend, how will the price of a put option on that stock be affected?

A) Increase or decrease.

B) Increase.

C) Decrease.

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The put option value will increase since the payment of dividends reduces the value of the underlying, and the value of a put is negatively related to the value of the underlying. (Study Session 17, LOS 60.g)

HuskyGrad2010

Which of the following is NOT one of the assumptions of the Black-Scholes-Merton option-pricing model?

A) There are no taxes and transactions costs are zero for options and arbitrage portfolios.

B) There are no cash flows over the term of the options.

C) The yield curve for risk-free assets is fixed over the term of the option.

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The yield curve is assumed to be flat so that the risk-free rate of interest is known and constant over the term of the option. Having a fixed yield curve does not necessarily imply that the yield curve is flat.