Consider Apple Inc. as the underlying asset, use its daily adjusted closing prices from August 12, 2014 to August 12, 2015 as historical data to estimate standard deviation. Use the rate r = 0.005 as annual risk-free rate and the estimated standard deviation as the volatility of the stock. Assume you want to build a portfolio of options containing one call option with strike K₁= 105, and one put option with strike K₂= 120. Let C₁(t, x) denotes the call option pricing function. Let P₂(t, x) denotes the put option pricing function. Let the maturity T = 12 months. Using the adjusted closing price of August 12, 2015 as the initial stock price.

1. Compute the option prices C₁, P₂on that date.

2. Compute the Delta of this portfolio .

3. Assume we want to build a new portfolio with 3 call options with strike K₁andnput options with strike K₂. Is there a value ofnthat will make this new portfolio delta neutral? If yes findn.

TheApple Inc. (AAPL)Daily Adjusted Closing Prices can be found here:

http://finance.yahoo.com/q/hp?s=AAPL&a=07&b=12&c=2014&d=07&e=12&f=2015&g=d