After this position has been taken what position in the stock is then necessary for delta neutrality?

Question 1: What, to the nearest cent, is the lower bound for the price of a two-year European call option on a stock when the stock price is $20, the strike price is $15, and the risk-free interest rate with continuous compounding is 5% and there are no dividends?Question 2: A three-month call with a strike price of $25 costs $2. A three-month put with a strike price of $20 and costs $3. A trader uses the options to create a strangle. For what two values of the stock price in three months does the trader breakeven with a profit of zero?

Question 3: A portfolio of derivatives on a stock has a delta of 2400 and a gamma of –100. An option on the stock with a delta of 0.6 and a gamma of 0.04 can be traded.
1. What position in the option creates a portfolio that is gamma neutral? Give size of position and state whether it is long or short
2. After this position has been taken what position in the stock is then necessary for delta neutrality? Give size of position and state whether it is long or short

Question 4: The current price of a non-dividend-paying stock is $30. Over the next six months it is expected to rise to $36 or fall to $26. Assume the risk-free rate is zero
1. What long position in the stock is necessary to hedge a short call option when the strike price is $32? Give the number of shares purchased as a percentage of the number of options that have been sold
2. What is the value the call option?

Question 5: A portfolio manager in charge of a portfolio worth $10 million is concerned that the market might decline rapidly during the next six months and would like to use options on the S&P 100 to provide protection against the portfolio falling below$9.5 million. The S&P 100 index is currently standing at 500 and each contract is on 100 times the index.
1. If the portfolio has a beta of 1, how many put option contracts should be purchased?
2. If the portfolio has a beta of 0.5, how many put options should be purchased?