1. Consider a stock with an initial price of $69, an expected return or risk-free rate of 5% per year, and a volatility of 35% per year is held for 6 months. European call and put options, both with strike prices of $70, have been purchased on this stock. a. Use Black- Scholes to solve for and graph the call delta, then interpret its meaning. b. Use Black-Scholes to solve for and graph the put delta, then interpret its meaning. c. Use the put-call parity relationship to test the Black-Scholes pricing model predictions. 2. COVID-19 and Stock Market Volatility a. What may cause a stock price today to be different from what it was the previous day? Pick a theoretical model and provide the basic explanations consistent with that model. b. Does COVID-19 represent a change in fundamentals or speculation? Carefully explain your answer. c. Explain the KEY difference between historical and implied volatility. d. Explain the logic behind creating "spreads" with options. e. WHY might Investor Sam be more inclined create a bear spread (as opposed to a bull spread)? Next, assume Sam's creates a bear spread using two put options, X1 and X2. Sketch a graph of this position. (You may use a single or 3 separate graphs. Please be sure to use dotted lines or a color pencil to illustrate the synthetic position.) f. Create a bear spread using two call options. (You may use 3 separate graphs. If, however, you use a single graph, be sure to use dotted lines or a color pencil to illustrate the synthetic position.) g. Assume Investor Casey is optimistic that a vaccine for COVID-19 will be found soon, thus has estimated the following probabilities pertaining to the S&P performance: 16% that S&P falls below the low strike price, X1, so Casey pays Sam; 71% that the S&P remains between the strike prices, so Casey makes no payment; and 13% that the S&P will move at and above the higher strike price, X2, so Sam pays Casey. Graph the lognormal probability distribution, with the S&P index movements represented on the x-axis and the probability of investment returns on the y-axis. (NOTE: Probability distributions and profit-loss diagrams are different, but they arguably convey similar information.) h. What is the total probability that the S&P will fall below the higher strike price? i. Is your answer in part h more likely to represent the delta for the put option at X2 (part e) or the call option at X2 (part f)? Explain your logic, then compute the delta for the corresponding option at that point. SHOW YOUR WORK. j. Based upon the probability that no payment is made, what can be inferred about the gammas for options X, and X2? k. Place check marks in the appropriate boxes in the following table: Spread Type Sam Makes Initial Sam Receives Net Premium Bullish Bearish Net Payment Put Debit Spread Call Debit Spread Put Credit Spread Call Credit Spread I. Because Sam is bearish, which type of spreads (from part k) would he use to enter into an arrangement with Casey? (What are the types of spreads you drew in part e and part f?) m. Which spread type would Casey be likely to require? Explain your logic.