Problems
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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.