Baran

AIM 3: Explain put-call parity and calculate, using the put-call parity on a nondividend-paying stock, the value of a European and American option, respectively.

 

1、A put option on DCY stock matures six months from today and sells for $0.49. A call option on DCY stock with the same strike price sells for $4.52. Both the put and the call are European options. DCY stock is priced at $55 and the risk-free rate of interest is 4 percent. The strike price of the put and call options is closest to:

A) $51.
 
B) $53.
 
C) $52.
 
D) $50.

Baran

The  correct answer is C

This question can be answered with the put-call parity relation. The relation is p+S0=c+Xe-rT, so rearranging gives X=(p+S0-c)/e-rT=(0.49+55-4.52)/e-0.04(.5)=52.

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2、A European put option on a stock can be replicated with which of the following combined postions?

 

A) Long a European call, long a zero-coupon bond, and short the stock.
 
B) Long a European call, short a zero-coupon bond, and long the stock.
 
C) Short a European call, long a zero-coupon bond, and short the stock.
 
D) Short a European call, short a zero-coupon bond, and long the stock.

Baran

 The  correct answer is A

Using put-call parity, the value of a put is: p=c+Xe-rT-S0. Thus a put is equivalent to being long a call, long a zero-coupon bond, and short the stock.

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3、Ronald Franklin, CFA, has recently been promoted to junior portfolio manager for a large equity portfolio at Davidson-Sherman (DS), a large multinational investment-banking firm. He is specifically responsible for the development of a new investment strategy that DS wants all equity portfolio managers to implement. Upper management at DS has instructed its portfolio managers to begin overlaying option strategies on all equity portfolios. The relatively poor performance of many of their equity portfolios has been the main factor behind this decision. Prior to this new mandate, DS portfolio managers had been allowed to use options at their own discretion, and the results were somewhat inconsistent. Some portfolio managers were not comfortable with the most basic concepts of option valuation and their expected return profiles, and simply did not utilize options at all. Upper management of DS wants Franklin to develop an option strategy that would be applicable to all DS portfolios regardless of their underlying investment composition. Management views this new implementation of option strategies as an opportunity to either add value or reduce the risk of the portfolio.

Franklin gained experience with basic options strategies at his previous job. As an exercise, he decides to review the fundamentals of option valuation using a simple example. Franklin recognizes that the behavior of an option's value is dependent on many variables and decides to spend some time closely analyzing this behavior. His analysis has resulted in the information shown in Exhibits 1 and 2 for European style options.

Exhibit 1: Input for European Options
Stock Price (S)	100
Strike Price (X)	100
Interest Rate (r)	0.07
Dividend Yield (q)	0.00
Time to Maturity (years) (t)	1.00
Volatility (Std. Dev.)(Sigma)	0.20
Black-Scholes Put Option Value	$4.7809

 

Exhibit 2: European Option Sensitivities
Sensitivity	Call	Put
Delta	0.6736	-0.3264
Gamma	0.0180	0.0180
Theta	-3.9797	2.5470
Vega	36.0527	36.0527
Rho	55.8230	-37.4164

Using the information in Exhibit 1, Franklin wants to compute the value of the corresponding European call option. Which of the following is the closest to Franklin's answer?

A) $4.78.

B) $5.55.

C) $11.54.

D) $12.07.

Baran

 

The  correct answer is C

This result can be obtained using put-call parity in the following way:

Call Value = Put Value – Xe^-rt + S = $4.78 - $100.00e^{(-0.07 * 1.0)} + 100 = $11.54

The incorrect value of $4.78 does not discount the strike price in the put-call parity formula. The value $12.07 results from using the binomial model.

Baran

 

Franklin is interested in the sensitivity of the European call option to changes in the volatility of the underlying equity's returns. What happens to the value of the call option if the volatility of the underlying equity's returns decreases?

The call option value:

A) decreases.

B) increases.

C) stays the same.

D) increases or decreases.

Baran

 

The  correct answer is A

 

Due to the limited potential downside loss, changes in volatility directly affect option value. Vega measures the option’s sensitivity relative to volatility changes.

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4、Which of the following best explains put-call parity?

A) A stock can be replicated using any call option, put option and bond.
 
B) No arbitrage requires that using any three of the four instruments (stock, call, put, bond) the fourth can be synthetically replicated.
 
C) A stock can be replicated using any at the money call and put options and a bond.
 
D) No arbitrage requires that only the underlying stock can be synthetically replicated using at the money call and put options and a zero coupon bond with a face value equal to the strike price of the options.

Baran

The  correct answer is B

A portfolio of the three instruments will have the identical profit and loss pattern as the fourth instrument and therefore the same value by no arbitrage. So the fourth security can be synthetically replicated using the remaining three.