Question

INSTRUCTION: Use “OptAll” program for the following questions. When you open “OptAll” program, be sure you enable macro. The tabs of the worksheets indicate the corresponding option pricing models or exotic options. As demonstrated in the past two classes, the “OptAll” program is fairly straight forward to use. Enter the required inputs and the results will be automatically calculated. We have studied these options in Module 3. You may (and should) apply the “OptAll” program to the previous Module 3 PowerPoint Slide/Excel example questions and/or Assignment questions first to ensure you are able to use this program correctly.

(Complete the following 7 Questions. Each question is worth 15 Points (that is, 5% extra credit).)

Let S = $100, K = $95, r = 8% (continuously compounded), ? = 30%, ? = 0, T = 1 year, and n = 3 (# of steps).
a. Use “OptAll” program to calculate the binomial option price for a European call option as well as the binomial option price for an American call option. What are their prices? Why their prices are the same (or not the same)?
b. Use “OptAll” program to calculate the binomial option price for a European put option. What is the price?
c. Use “OptAll” program to calculate the binomial option price for an American put option. What is the price?

Repeat the previous problem assuming that the stock pays a continuous dividend of 8% per year (continuously compounded). Calculate the prices of the American and European puts and calls.

Suppose S = $100, K = $95, ? = 30%, r = 0.08, ? = 3%, and T = 0.75. What is the Black-Scholes price of a European call? What is the Black-Scholes price of a European put?

Consider a bull spread where you buy a 40-strike call and sell a 45-strike call.
Suppose ? = 0.30, r = 0.08, ? = 0, and T = 0.5.
a. Suppose S = $40. What are delta, gamma, vega, theta, and rho for each call and the bull spread? (Note: The greeks of the bull spread are simply the sum of the greeks of the individual options. The greeks of the call with a strike of 45 enter with a negative sign because this option was sold.)
b. Suppose S = $45. What are delta, gamma, vega, theta, and rho for each call and the bull spread?

Refer to the Asian Options example on PowerPoint slide 14-9. If N=100, what are the premiums of at-the-money geometric average price and geometric average strike calls and puts?

Let S = $40, K = $45, ? = 0.30, r = 0.08, ? = 0, and T = 0.25. What are the prices of knock-out options (up-and-out call and put and down-and-out call and put) with a barrier of $38? What are the prices of knock-in options (up-and-in call and put and down-and-in call and put) with a barrier of $38? What are the Black-Scholes Call and Put Prices?

Let S = $40, K = $40, ? = 0.30, r = 0.08, ? = 3%. (a) What is the price of a standard European call with 2 years to expiration? (b) Suppose you have a compound call giving you the right to pay two dollars 1-year from today to buy the option in part (a). For what stock prices in 1-year will you exercise this option? (c) What is the price of this compound call (call on call)? (d) Suppose you have a compound put giving you the right to pay two dollars 1-year from today to sell the option in part (a). For what stock prices in 1-year will you exercise this option? (e) What is the price of this compound put (put on call)?

Answer 1

a) American Call Option: Price calculated is 19.111. European Call Option: Price calculated is 19.084. The prices are different due to the style of option. Assuming everything else is constant, the only difference that is causing change in price of both the options is the time of exercise - American style option can be exercised at any time during the holding period, hence the value at each node shall be calculated as Max (Binomial node value,Option Exercise value). Whereas, the value at each node for European style call option is always Binomial node value. Hence the difference in price.

b) For European Put option, using the same information, price derived is 6.807.

c) For American Put option, using the same information, price derived is 7.321.

Considering the dividend payment of 8.0%, American Call is priced at 14.183 & American Put at 9.505; European Call at 13.942 and Put at 9.326.

Considering the time to maturity is 9 months (or 0.75 of the year), European Call and Put prices as per Black and Scholes model are 14.414 and 6.093 respectively.

As I did not have the "OptAll" Add-in information, I have used a model to derive the prices.