Question 1: What does the Black-Sholes-Merton stock option pricing model assume about the probability distribution of the stock price in one year?
a. What does it assume about the continuously compounded rate of return on the stock during the year?
b. The volatility of a stock price is 30% per annum. What is the standard deviation of the percentage price change in one trading day?
c. Calculate the price of a three-month European put option on a non-dividend-paying stock with a strike price of $60 when the current stock price is $65, the risk-free interest rate is 8% per annum, and the volatility is 32% per annum.
a. What difference does it make to your calculations in part c if a dividend of $2.95 is expected in three months?
Question 2: For the following problems assume the effective interest rate is 1.156%. The current value of the S&P SPX index is $2,348.45. Using the data on the SPX premiums below for the SPX options with 30 days to expiration.
|
Strike
|
Call
|
Put
|
|
$2,340
|
$35.95
|
$27.60
|
|
$2,345
|
$33.40
|
$30.60
|
|
$2,350
|
$29.75
|
$33.05
|
|
$2,355
|
$26.60
|
$33.10
|
|
$2,360
|
$25.65
|
$35.35
|
(a) Draw profit diagrams for the following positions:
a. 2350 Strike SPX Straddle
b. A Butterfly Spread using the 2350, 2355 and 2360 Strike call options.
c. A Bull Spread involving the 2350 Strike call Spread.
i. For all graphs indicate your Minimum Profit
ii. Maximum Profit
iii. Break-even point
Question 3: Consider a position consisting of a $100,000 investment in asset A and a $100,000 investment in asset B. Assume that asset A has a daily volatility of 1.2% and asset B has a daily volatility of 1.8%, and that the coefficient of correlation between their returns is 0.3. What is the 5-day 97% VaR for the portfolio?
(i) What is the standard deviation of the return from stock A over 4 days?
(ii) What is the standard deviation of the return from stock B over 4 days?
Question 4: You simultaneously purchase and write both a call option and a put option with exercise prices of $65 and $71, respectively.
(a) What is the term commonly used for this type of strategy/position that you have taken?
(b) Determine the value at expiration and the profit for your strategy under the following outcomes;
a. The price of the underlying at expiration is $70
(c) Determine the following:
a. The maximum profit
b. The maximum loss
(d) Determine the breakeven price at expiration
Question 5: For a 1-month European call option on Jet Blue's stock (JBLU), you are given:
(a) The current stock price is $27
(b) The strike price is $30
(c) The continuously compounded risk-free rate is 8%
(d) The stock pays a continuous dividends of 2%
(e) The volatility of the stock is 0.2
a. Calculate d1 used in the Black Scholes formula for the price of this option
b. Determine the Black-Scholes premium for the
i. call option
ii. put option
c. What is the Call option delta?
d. Explain what the delta value obtained in part c tells us about the option.
Question 6: You are considering the purchase of a 3 month European put option on Jet Blue's stock which has an announced dividend payment of 1.50 in two months. You are given the following information:
(a) The strike price is $50
(b) The continuously compounded risk-free rate is 10%
(c) The annual volatility on the stock is 0.3
(d) The stock follows the Black-Scholes framework
(e) d2 = -0.1086
a. Determine Jet Blue's current stock price
b. Determine the Black-Scholes price of the Call option on the stock
c. Determine the Black-Scholes delta for the call option
d. What is the option's vega?
Question 7: For a 1 year European call option of Microsoft's stock, you are given:
(a) The stock's price is $45.00
(b) The stock pays a continuous dividend of 2%
(c) The stock's annual volatility is 0.1
(d) The continuously compounded risk-free rate is 0.04
(e) The Black-Sholes Delta = 0.5
a. Determine the stock's strike price.
Question 8: For a European call option on the non-dividend paying OEX index we have the following information:
(a) The stock price is $50.00
(b) Time to Expiry is t
(c) The strike price is 50e0.04t
(d) The continuously compounded interest rate is 0.04
(e) The following prices were observed for various times to expiry
Time to Expiry - Price
3 Months - 3.98
6 Months - 5.96
9 Months - 7.14
(i) Calculate the implied volatilities of options for these 3 periods using the Black-Scholes model.
(ii) Did the implied volatilities increase or decrease as the period to expiry increase?