PROBLEM: A delta-neutral hedge on an asset will adjust the number of options over time to match the risk of the underlying position. Suppose you have just been hired to manage a fund which has a portfolio of large-cap stocks identical to the S&P 500. The investment objective of the fund is to generate income from the dividends on the stocks in the portfolio while protecting the value of the fund against a stock market decline. Show your work for each step of this Problem.

The fund holds 100,000 shares of the S&P 500 and the price of each share is the same as the S&P 500 index value. You hedge these stocks using S&P 500 put options, where each option represents 100 shares of the S&P 500. Remember that + indicates long and indicates short.

1. On Tuesday, April 13 your first day on the job the S&P 500 is trading at 4132.80 index points per share. You decide to use at-the-money (4130 strike) puts on the S&P 500 with a premium of 101.30 index points per share of the S&P 500. These puts have a delta of 0.482.

1. How many of these puts do you need to become delta neutral (i.e., delta for the combined stocks + puts = zero)? Round your answer to the nearest whole number, since there are no fractional options. (1 point)

2. How much is the total premium (in index points) for these puts? Do you pay or receive this premium? (2 points)

2. On Wednesday, April 14, the S&P 500 is trading at 4117.90 index points. The puts now have a premium of 108.20 index points and a delta of .505

1. How much did the total value (in index points) of the stocks change between Tuesday, April 13 and Wednesday, April 14? Did the value of the stocks increase or decrease? (2 points)

2. How much did the total value (in index points) of the puts change between Tuesday, April 13 and Wednesday, April 14? Did the value of the puts increase or decrease? Did the puts move toward in the money or out of the money? (3 points)

3. How much is the total gain or loss (in index points) on the combined stocks + puts position between Tuesday, April 13 and Wednesday, April 14? Describe how theta might have contributed to this difference. (2 points)

4. To adjust/re-balance your hedge, answer the following questions: (4 points)

• How many puts do you need now (as of Wednesday, April 14) to hedge these stocks? Round your answer to the nearest whole number, since there are no fractional options.
• How many puts do you already have?
• Are you now over-hedged (i.e., too many puts) or under-hedged (i.e., too few puts)?
• How many puts should you buy or sell to become delta neutral again?

3. On Thursday, April 15, the S&P 500 is trading at 4162.40 index points. The puts now have a premium of 88.10 index points and a delta of .436

1. How much did the total value (in index points) of the stocks change between Wednesday, April 14 and Thursday, April 15? Did the value of the stocks increase or decrease? (2 points)

2. How much did the total value (in index points) of the puts change between Wednesday, April 14 and Thursday, April 15? Did the value of the puts increase or decrease? Did the puts move toward in the money or out of the money? (3 points)

3. How much is the total gain or loss (in index points) on the combined stocks + puts position between Wednesday, April 14 and Thursday, April 15? Describe how vega might have contributed to this difference? (2 points)

4. To adjust/re-balance your position, answer the following questions: (4 points)

• How many puts do you need now (as of Thursday, April 15) to hedge these stocks? Round your answers to the nearest whole number, since there are no fractional options.
• How many puts do you already have?
• Are you now over-hedged (i.e., too many puts) or under-hedged (i.e., too few puts)?
• How many puts should you buy or sell to become delta neutral again?