FIN 3506 Lecture Notes - Lecture 11: Risk-Free Interest Rate, Delta Gamma, Stochastic Differential Equation

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bronzeturtle979

6 Oct 2016

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One of the most interesting aspects of options is how a financial institution that has written an option will be able to hedge the risks. Black-scholes formula, the price of an option is dependent on certain parameters. The price of an option of a given strike price and expiration depends on the price of the underlying asset, the time to expiration, the volatility and the risk free rate. If a change in these parameters affect the value of the option them is imperative to have an estimate of the change in the value of the option given a change in these parameter. This could lead to a better understanding of the risks. The first thing that we will consider is the change in the price underlying asset and the change in the price of the option and try to understand its effect on the hedging of the option.

Related Questions

The following information is given about options on the stock of a certain company.

Current stock price = $114
Exercise price = $115
Continuously compounded risk-free rate = 0.1 percent Time to expiration = 6 months (T = 0.5) in June
The volatility = 0.28

(1) Find the values of N(d1) and N(d2)

(2) What value does the Black-Scholes-Merton model predict for the call?

(3) What is the delta of the call?

(4) If the stock price goes up by $1 to $115, what would be the call price based on the delta?

(5) Suppose that June 115 call has the gamma of .0176. If the stock price goes up by $1 to $115, what would be the delta of the call based on the gamma?

(6) Suppose that the call rho (Ï) is 25.523. If the continuously compounded risk-free rate goes up by 1 percent to 1.1 percent, what would be the expected call price based on the rho?

(7) Suppose that the call vega (Î½) is 32.106. If the volatility of the continuously return on the stock increase by 1 percent, what would be the expected call price based on the vega?

(8) Suppose that the theta (Î¸) is 9.04. If the time to expiration decreases so that the call expires in 3 months (T = .25), what would be the expected call price based on the theta?

chocolateraven16

Question 1

The following information is given about options on the stock of a certain company.

Current stock price = $23

Exercise price = $20

Continuously compounded risk-free rate = 9.0 percent

Time to expiration = 6 months (T = 0.5)

The variance of the continuously compounded return on the stock = 15.0 percent

The volatility = Sqrt(.15)

d₁ = [ln(S₀/X) + (r_c + Ï²/2)T]/ÏSqrt(T) = [ln(23/20) + (.09 + .15/2)(.5)]/(.3873 x sqrt(.5)) = [.1398 + (.165)(.5)]/.2739 = .2223/.2739 = .8116

d₂ = d₁ - ÏSqrt(T) = .8116 â (sqrt(.15))(sqrt(.5)) = .8116 â .2739 = .5377

No dividends are expected.

Find the values of N(d₁) and N(d₂) using the excel function â=normsdist( ).

What value does the Black-Scholes-Merton model predict for the call?

What is the delta of the call?

If the stock price goes up by $1 to $24, what would be the call price based on the delta?

Suppose that June 20 call has the gamma of .0456. If the stock price goes up by $1 to $24, what would be the delta of the call based on the gamma?

Suppose that the call rho (Ï) is 6.74. If the continuously compounded risk-free rate goes up by 1 percent to 10 percent, what would be the expected call price based on the rho?

Suppose that the call vega (Î½) is 4.67. If the volatility of the continuously return on the stock increase by 1 percent, what would be the expected call price based on the vega?

Suppose that the theta (Î¸) is 3.02. If the time to expiration decreases so that the call expires in 3 months (T = .25), what would be the expected call price based on the theta?

emeraldmole779

As a newly minted MBA. you've taken a management position with Exotic Cuisines, Inc., a restaurant chain that just went public last year. The company's restaurants specialize in exotic main dishes, using ingredients such as alligator, buffalo, and ostrich. A concern you had going in was that the restaurant business is very risky. However, after some due diligence, you discovered a common misperception about the restaurant industry. It is widely thought that 90 percent of new restaurants close within three years; however, recent evidence suggests the failure rate is closer to 60 percent over three years. So, it is a risky business, although not as risky as you originally thought. During your interview process, one of the benefits mentioned was employee stock options. Upon signing your employment contract, you received options with a strike price of $50 for 10,000 shares of company stock. As is fairly common, your stock options have a three-year vesting period and a 10-year expiration, meaning that you cannot exercise the options for a period of three years, and you lose them if you leave before they vest. After the three-year vesting period, you can exercise the options at any time. Thus, the employee stock options are European (and subject to forfeit) for the first three years and American afterward. Of course, you cannot sell the options, nor can you enter into any sort of hedging agreement. If you leave the company after the options vest, you must exercise within 90 days or forfeit. Exotic Cuisines stock is currently trading at $26.12 per share, a slight increase from the initial offering price last year. There are no market traded options on the company's stock. Because the company has only been traded for about a year, you are reluctant to use the historical returns to estimate the standard deviation of the stock's return. However, you have estimated that the average annual standard deviation for restaurant company stocks is about 55 percent. Since Exotic Cuisines is a newer restaurant chain, you decide to use a 65 percent standard deviation in your calculations. The company is relatively young, and you expect that all earnings will be reinvested back into the company for the near future. Therefore, you expect no dividends will be paid for at least the next 10 years. A three-year Treasury note currently has a yield of 1.5 percent, and a 10-year Treasury note has a yield of 2.6 percent.

4. Why do you suppose employee stock options usually have a vesting provision? Why must they be exercised shortly after you depart the company even after they vest?

5. A controversial practice with employee stock options is repricing. What happens is that a company experiences a stock price decrease, which leaves employee stock options far out of the money or "underwater." In such cases, many companies have "repriced" or "restruck" the options, meaning that the company leaves the original terms of the option intact, but lowers the strike price. Proponents of repricing argue that since the option is very unlikely to end in the money because of the stock price decline, the motivational force is lost. Opponents argue that repricing is in essence a reward for failure. How do you evaluate this argument? How does the possibility of repricing affect the value of an employee stock option at the time it is granted?

6. As we have seen, much of the volatility in a company's stock price is due to systematic or marketwide risks. Such risks are beyond the control of a company and its employees. What are the implications for employee stock options? In light of your answer, can you recommend an improvement over traditional employee stock options?

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FIN 3506 Lecture Notes - Lecture 11: Risk-Free Interest Rate, Delta Gamma, Stochastic Differential Equation

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