Consider Apple Inc. as the underlying asset, use its daily adjusted closing pric

Question

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, P2 on that date.

2. Compute the Delta of this portfolio .

3. Assume we want to build a new portfolio with 3 call options with strike K1 and n put options with strike K2. Is there a value of n that will make this new portfolio delta neutral? If yes find n.

The Apple 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

Explanation / Answer

Use the STDEV.S() function in excel to get the Standard Deviation.

The data (daily adjusted closing price) downloaded from the given link is used along with the above mentioned fuction. It gives the standard deviation = 11.4343, Mean =115.78

Now we can use Black–Scholes model to calculate option price.

Given data :

Underlying price = 113.49 (adjusred closing price of August 12, 2015)

Risk free rate = 0.005

Volatality = Standard Deviation/mean - 11.4343/115.78 = 0.098

Strike Rate = For Call, K1=105, for Put K2=120

Time to Maturity = Days until expiration = 12 Months = 1 Year = 365 Days

Using these values in any online or excel based Black–Scholes model calculator will give you price for

call C1 = 10.1982 ( http://www.danielsoper.com/fincalc/calc.aspx?id=37)

Put P2= 8.103 (http://www.danielsoper.com/fincalc/calc.aspx?id=38)

Note: Submit other questions as new questions. Do not combine multiple questions in one question.

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