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# Options

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1) A stock price is currently \$100. Over each of the next two three-month periods it is expected to increase by 12% or fall by 12%. Consider a six-month European call option with a strike price of \$105. The risk-free rate is 3%. What is the risk-neutral probability of a 12% rise in both quarters?

2) A stock price is currently \$100. Over each of the next two three-month periods it is expected to increase by 12% or fall by 12%. Consider a six-month European call option with a strike price of \$105. The risk-free rate is 3%. What is the value of the call option?

3) A stock price is \$30, the expected return is 18% per annum and the volatility is 20% per annum. What is the standard deviation of the logarithm of the stock price in two years?

4) A stock price is \$30, the expected return is 18% per annum and the volatility is 20% per annum. What is the lower 95% confidence limit for the logarithm of the stock price in two years?

5) For a call option on a non-dividend paying stock, the strike price is \$40, the stock price is \$37.65, the risk-free rate is 4% per annum, the volatility is 40% per annum and the time to maturity is 6 months. What is the price of the call option?

6) A portfolio of derivatives on a stock has a delta of -1200 and a gamma of -200. What position in the stock in the stock would create a delta-neutral portfolio

7) The delta of a European call option on a non-dividend paying stock is 0.4, its gamma is 0.08 and its vega is 0.2. What is the delta of a European put option with the same strike price and time to maturity as the call option?

#### Solution Preview

1 A stock price is currently \$100. Over each of the next two three-month periods it is expected to increase by 12% or fall by 12%. Consider a six-month European call option with a strike price of \$105. The risk-free rate is 3%. What is the risk-neutral probability of a 12% rise in both quarters?

Probability of upmoves in 2 successive quarterers = Probability of upmove in 1st quarter x Probability of upmove in 2nd quarter

u= 1.12 =1+12%
d= 0.88 =1-12%
r= 3%
t= 3 months= 0.25 years

p= {e ^(rt)-d}/(u-d) 0.53 =EXP(0.03x0.25)-0.88)/(1.12-0.88)

Probability of upmoves in 2 successive quarterers = 0.28 =0.53^2

2 A stock price is currently \$100. Over each of the next two three-month periods it is expected to increase by 12% or fall by 12%. Consider a six-month European call option with a strike price of \$105. The risk-free rate is 3%. What is the value of the call option?

Value of option= \$5.65

Option
Put or call = C =Call
American or European= E =European

Spot price=S= \$100.00
Strike price=X= \$105.00

r=risk free ...

#### Solution Summary

Calculates the value of call option, risk-neutral probability, standard deviation of the logarithm of the stock price, confidence limit for the logarithm of the stock price, position in a delta-neutral portfolio, delta of a European put option etc.

\$2.19
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## Option or combination of options as best course of action

Discuss the option or combination of option you selected as the best course of action for Ford Motor Company and detail your reasons for selecting that option or combination of options

FORD MOTOR CASE STUDY
WHAT WOULD YOU DO?

Ford Motor Company, Dearborn, Michigan. As you look out the window of the "Glass House" headquarters, you comment to yourself that it has not been a fun place to work these past few years. As you contemplate the future opportunities for the company, you are left with the thought that this company will look radically different in the next few years. An icon of U.S. manufacturing, Ford Motor Company has been one of the most prominent automobile producers in the world for almost 100 years. Founded by Henry Ford and eleven business associates in 1903, Ford Motor Company pioneered the moving assembly line, in which workers remained in place performing the same job on each automobile that came down the line. Henry Ford's vision was the production of cars that were affordable to the masses. Today, Ford's product lines include Ford, Lincoln, Mercury, Mazda, Aston Martin, Jaguar, Land Rover, and Volvo.

Despite the proliferation of nameplates, Ford Motor has been losing money for years, particularly in its North American operations. Ford produces more vehicles than it can sell and does so under very difficult personnel conditions that are part of the industry's history. The United Auto Workers (UAW) represents most of the company's production employees, and the contract terms over the years have been designed to provide significant long-term support to those employees. Generous conditions for retirement, benefits, and job protection, developed when the industry was doing well, now weigh down company operations. Another issue that has you concerned is the Premium Automotive Group (PAG), which makes the company's Aston-Martin, Jaguar, Land Rover, and Volvo brands. Originally intended as the platform for the company to enter the luxury car market, this group has lost billions year after year. Although the company has substantial cash reserves (over \$20 billion), operating losses are steadily diminishing the company's financial safety net.

You've identified four options for the company. The first is to close down older plants in an effort to realign production and sales. You don't want to produce more cars than you sell; in fact, it might be good to produce a bit less than you could actually sell. There is an enormous cost to buying out the workers, shuttering the plants, and dealing with the political implications of a closure. Furthermore, you are concerned that a resurgence of buying would leave you with so little capacity that consumers would move to competitors' products. A second option would be to move the company to produce only smaller cars, eliminating or sharply reducing the SUV and truck lines. This has the advantage of getting back to Henry Ford's vision for the company as well as dealing quickly with products that are simply not selling in today's expensive gas environment. Third, you could take the dramatic step of dramatically reducing your North American presence and focus your efforts on international markets where Ford has been very successful. Europe, South America, and China have been very receptive to Ford vehicles and the company is doing well in these markets. The fourth option is to sell the entire PAG group, letting someone else figure out how to make money selling so few vehicles a year. The hand-built PAG automobiles don't fit the Ford way of doing things anyway.

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