Problems

К оглавлению1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 
34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 
51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 
68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 
85 86 87 88 89 90 91 92 93 94 95 96 97 

Skill-Building Problems.

1. The current stock price of a company is $37.50, the potential up movement / period of the stock

price is 22.0%, the potential down movement / period of the stock price is -13.00%, the riskfree

rate is 4.0% per period, the exercise price of an one-period, European call option on the stock is

$39.00, the exercise price of an one-period, European put option on the stock is $39.00, the time

to maturity for both options is 0.58 years, and the number of periods for both options is 1.

Determine the replicating portfolio and the current prices of the call and put.

2. The current stock price of a company is $23.75, the potential up movement / period of the stock

price is 27.0%, the potential down movement / period of the stock price is -9.00%, the riskfree

rate is 5.0% per period, the exercise price of an European call option on the stock is $22.00, the

exercise price of an European put option on the stock is $22.00, the time to maturity for both

options is 0.39 years, and the number of periods for both options is 8. Determine the replicating

portfolio on each date and the current prices of the call and put.

3. The current stock price of a company is $43.25, the potential up movement / period of the stock

price is 19.0%, the potential down movement / period of the stock price is -14.00%, the riskfree

rate is 4.0% per period, the exercise price of an European call option on the stock is $45.00, the

exercise price of an European put option on the stock is $45.00, the time to maturity for both

options is 0.83 years, and the number of periods for both options is 8. Determine the risk neutral

proability and the current prices of the call and put.

4. Collect Cisco Systems’ historical stock prices from Yahoo Finance! From the financial media,

collect the current stock price of Cisco Systems, the exercise price of an European call option on

Cisco Systems, the exercise price of an European put option on Cisco Systems, the time to

maturity for both options, and the yield on a riskfree Treasury Bill maturing as close as possible

to the maturity date of the options. Determine:

(a.) What is the annual standard deviation of Cisco Systems stock?

(b.) What is the risk neutral probability and the current prices of the call and put under the

continuous annualization convention?

(c.) What is the risk neutral probability and the current prices of the call and put under the

discrete annualization convention?

Skill-Extending Problems.

5. Extend the Binomial Option Pricing model to incorporate a $2.00 / share dividend that will be

paid out in period 5. In other words, all of the period 5 stock prices will be reduced by $2.00.

Determine the current prices of the call and put.

6. Extend the Binomial Option Pricing model to analyze Digital Options. The only thing which

needs to be changed is the option’s payoff at maturity.

(a.) For a Digital Call, the Payoff At Maturity = $1.00 When Stock Price At Mat > Exercise Price

Or $0.00 Otherwise.

(b.) For a Digital Put, the Payoff At Maturity = $1.00 When Stock Price At Mat < Exercise Price

Or $0.00 Otherwise.

11. Extend the Binomial Option Pricing model to determine how fast the binomial option price

converges to the price in the Black Scholes Option Pricing model. Reduce the Full-Scale model to

a 10 period model and to a 20 period model. Increase the 50 period model to a 100 period model.

Then for the same inputs, compare call and put prices of the 10 period, 20 period, 50 period, 100

period, and Black-Scholes models.

12. Extend the Binomial Option Pricing model to determine how fast the binomial option price with

averaging of adjacent odd and even numbers of periods converges to the price in the Black

Scholes Option Pricing. As you increase the number of periods in the binomial model, it

oscillates between overshooting and undershooting the true price. A simple technique to increase

price efficiency is to average adjacent odd and even numbers of periods. For example, average

the 10 period call price and the 11 period call price. Reduce the Full-Scale model to a 10 period,

11 period, 20 period, and 21 period model. Increase the 50 period model to a 51 period, 100

period, and 101 period model. Then for the same inputs, compare call and put prices of the

average of the 10 and 11 period models, 20 and 21 period models, 50 and 51 period models, 100

and 101 period models, and Black-Scholes model.

Live In-class Problems.

13. Given the partial Single Period spreadsheet BinosinZ.xls, do steps 4 Option Payoffs at

Maturity, 5 Create a Replicating Portfolio, and 6 Calculate the Option Price Now.

14. Given the partial Multi-Period spreadsheet BinomulZ.xls, do step 7 The Option Price Tree.

15. Given the partial Risk Neutral spreadsheet BinoneuZ.xls, do step 2 Risk Neutral Probability

and 3 The Option Price Tree.

16. Given the partial Full-Scale Real Data spreadsheet BinofulZ.xls, do step 7 Calculate the New

Outputs.

Skill-Building Problems.

1. The current stock price of a company is $37.50, the potential up movement / period of the stock

price is 22.0%, the potential down movement / period of the stock price is -13.00%, the riskfree

rate is 4.0% per period, the exercise price of an one-period, European call option on the stock is

$39.00, the exercise price of an one-period, European put option on the stock is $39.00, the time

to maturity for both options is 0.58 years, and the number of periods for both options is 1.

Determine the replicating portfolio and the current prices of the call and put.

2. The current stock price of a company is $23.75, the potential up movement / period of the stock

price is 27.0%, the potential down movement / period of the stock price is -9.00%, the riskfree

rate is 5.0% per period, the exercise price of an European call option on the stock is $22.00, the

exercise price of an European put option on the stock is $22.00, the time to maturity for both

options is 0.39 years, and the number of periods for both options is 8. Determine the replicating

portfolio on each date and the current prices of the call and put.

3. The current stock price of a company is $43.25, the potential up movement / period of the stock

price is 19.0%, the potential down movement / period of the stock price is -14.00%, the riskfree

rate is 4.0% per period, the exercise price of an European call option on the stock is $45.00, the

exercise price of an European put option on the stock is $45.00, the time to maturity for both

options is 0.83 years, and the number of periods for both options is 8. Determine the risk neutral

proability and the current prices of the call and put.

4. Collect Cisco Systems’ historical stock prices from Yahoo Finance! From the financial media,

collect the current stock price of Cisco Systems, the exercise price of an European call option on

Cisco Systems, the exercise price of an European put option on Cisco Systems, the time to

maturity for both options, and the yield on a riskfree Treasury Bill maturing as close as possible

to the maturity date of the options. Determine:

(a.) What is the annual standard deviation of Cisco Systems stock?

(b.) What is the risk neutral probability and the current prices of the call and put under the

continuous annualization convention?

(c.) What is the risk neutral probability and the current prices of the call and put under the

discrete annualization convention?

Skill-Extending Problems.

5. Extend the Binomial Option Pricing model to incorporate a $2.00 / share dividend that will be

paid out in period 5. In other words, all of the period 5 stock prices will be reduced by $2.00.

Determine the current prices of the call and put.

6. Extend the Binomial Option Pricing model to analyze Digital Options. The only thing which

needs to be changed is the option’s payoff at maturity.

(a.) For a Digital Call, the Payoff At Maturity = $1.00 When Stock Price At Mat > Exercise Price

Or $0.00 Otherwise.

(b.) For a Digital Put, the Payoff At Maturity = $1.00 When Stock Price At Mat < Exercise Price

Or $0.00 Otherwise.

11. Extend the Binomial Option Pricing model to determine how fast the binomial option price

converges to the price in the Black Scholes Option Pricing model. Reduce the Full-Scale model to

a 10 period model and to a 20 period model. Increase the 50 period model to a 100 period model.

Then for the same inputs, compare call and put prices of the 10 period, 20 period, 50 period, 100

period, and Black-Scholes models.

12. Extend the Binomial Option Pricing model to determine how fast the binomial option price with

averaging of adjacent odd and even numbers of periods converges to the price in the Black

Scholes Option Pricing. As you increase the number of periods in the binomial model, it

oscillates between overshooting and undershooting the true price. A simple technique to increase

price efficiency is to average adjacent odd and even numbers of periods. For example, average

the 10 period call price and the 11 period call price. Reduce the Full-Scale model to a 10 period,

11 period, 20 period, and 21 period model. Increase the 50 period model to a 51 period, 100

period, and 101 period model. Then for the same inputs, compare call and put prices of the

average of the 10 and 11 period models, 20 and 21 period models, 50 and 51 period models, 100

and 101 period models, and Black-Scholes model.

Live In-class Problems.

13. Given the partial Single Period spreadsheet BinosinZ.xls, do steps 4 Option Payoffs at

Maturity, 5 Create a Replicating Portfolio, and 6 Calculate the Option Price Now.

14. Given the partial Multi-Period spreadsheet BinomulZ.xls, do step 7 The Option Price Tree.

15. Given the partial Risk Neutral spreadsheet BinoneuZ.xls, do step 2 Risk Neutral Probability

and 3 The Option Price Tree.

16. Given the partial Full-Scale Real Data spreadsheet BinofulZ.xls, do step 7 Calculate the New

Outputs.