For Apple, Inc (AAPL) What are the S, E, R, 2 and T in your Black-Scholes formul

Question

For Apple, Inc (AAPL)

What are the S, E, R, 2 and T in your Black-Scholes formula for each of your two chosen options? Show calculations, where necessary, and explain in words how you got each number. Attach print-outs of website pages you used that indicate where the information is coming from; highlight the important numbers on your print-outs. (Keep in mind that the information on the websites gets constantly updated, and so if the information is not printed out at approximately the same time the option price results will not be correct.)

What are the intrinsic values of your two chosen call options? Calculate. What do they mean?

Are your two call options priced correctly, or are they overpriced or underpriced, according to the Black-Scholes option pricing model? By how much? Calculate and explain. Can you explain why this may be the case?

For each of your two options create a graph (like the one below) and label and put the dollar amounts of

current stock price for the stock you picked

exercise prices that you picked

current call option values that you have calculated

intrinsic values that you have calculated

time value premiums based on your calculations

In order to replicate the payoff of your two call options at the expiration date that you selected, how many shares of stock should you buy today, and how much should you borrow at the risk-free rate? Calculate and explain.

Are the put options for your stock overpriced or underpriced, according to the Black-Scholes option pricing model? Calculate and explain.

Explanation / Answer

The Black-Scholes model is used to calculate the theoretical price of European put and call options, ignoring any dividends paid during the option's lifetime.

Formula:

C = SN(d1)-Ke(-rt)N(d2) where,

C = Theoretical call premium S = Current stock price t = time K = option striking price r = risk free interest rate N = Cumulative standard normal distribution e = exponential term (2.7183) d1 = ( ln(S/K) + (r + (s2/2))t ) / st d2 = d1 - st s = standard deviation of stock returns

Example :

A company currently sells for $210.59 per share. The annual stock price volatility is 14.04%, and the annual continuously compounded risk-free interest rate is 0.2175%. Find the value of d1 in the Black-Scholes formula for the price of a call on a company's stock with strike price $205 and time for expiration of 4 days.

Given,

S= $210.59, K= $205 t = 4 days r = 0.2175% s = 14.04%

To Find,

Call option priced1

Solution :

Step 1:

Substitute the given value in the formula, d1 = ( ln(210.59/205) + (0.002175+(0.14042) / 2)(0.01096) ) / 0.1404*(0.01096) d1 = 1.8394

Step 2:

d2 = 1.8394 - 0.1404*(0.01096) d2 = 1.8247

Step 3:

Substitute the value of d1 and d2 in the Call option (C) formula C = 210.59 * - 205 * SN(d1)-Ke(-rt)N(d2)

C = -8.1313

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