What does the Black-Sholes-Merton stock option pricing model assume about the probability distribution of the stock price in one year? 

• What does it assume about the continuously compounded rate of return on the stock during the year?

1. The volatility of a stock price is 30% per annum. What is the standard deviation of the percentage price change in one trading day?

3. 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.

3. 
   
   1. 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

• Draw profit diagrams for the following positions:
  • 2350 Strike SPX Straddle
  • A Butterfly Spread using the 2350, 2355 and 2360 Strike call options.
  • A Bull Spread involving the 2350 Strike call Spread.
    • For all graphs indicate your Minimum Profit
    • Maximum Profit
    • 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?

• 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.

• What is the term commonly used for this type of strategy/position that you have taken?
• Determine the value at expiration and the profit for your strategy under the following outcomes;
  • The price of the underlying at expiration is $70
• Determine the following:
  • The maximum profit
  • The maximum loss

• Determine the breakeven price at expiration

Question 5:

For a 1-month European call option on Jet Blue’s stock (JBLU), you are given:

• The current stock price is $27
• The strike price is $30
• The continuously compounded risk-free rate is 8%
• The stock pays a continuous dividends of 2%
• The volatility of the stock is 0.2
  • Calculate d₁ used in the Black Scholes formula for the price of this option
  • Determine the Black-Scholes premium for the
    • call option
    • put option
• What is the Call option delta?
• 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:

• The strike price is $50
• The continuously compounded risk-free rate is 10%
• The annual volatility on the stock is 0.3
• The stock follows the Black-Scholes framework
• d₂ = -0.1086
  • Determine Jet Blue’s current stock price
  • Determine the Black-Scholes price of the Call option on the stock
  • Determine the Black-Scholes delta for the call option
  • What is the option’s vega?

Question 7:

For a 1 year European call option of Microsoft’s stock, you are given:

• The stock’s price is $45.00
• The stock pays a continuous dividend of 2%
• The stock’s annual volatility is 0.1
• The continuously compounded risk-free rate is 0.04
• The Black-Sholes Delta = 0.5
  • 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:

• The stock price is $50.00
• Time to Expiry is t
• The strike price is 50e^0.04t
• The continuously compounded interest rate is 0.04
• 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

• Calculate the implied volatilities of options for these 3 periods using the Black-Scholes model.
• Did the implied volatilities increase or decrease as the period to expiry increase?

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