Question

Assume the Black-Scholes framework for options pricing. You are a portfolio manager and already have a long position in Apple (ticker: AAPL). You want to protect your long position against losses and decide to buy a European put option on AAPL with a strike price of $180.15 and an expiration date of 1-year from today. The continuously compounded risk free interest rate is 8% and the stock pays no dividends. The current stock price for AAPL is $200 and its volatility is 30%. Assume all options premiums are exactly equal to the Black-Scholes options prices. Once you discover how expensive the put option premium is, you decide to sell a 1-year call European call option on AAPL to help pay for the put option. You are free to choose the strike price of the call option. At what strike price will the call option you sell pay for the put option you buy? $280.15 O $270.15 $180.15 $200

Answer 1

As per Black Scholes Model, value of put option

p = K*e^(-r*t) * N(-d2) - S *N(-d1)

where

Spot price S =$200

Strike price K = $180.15

risk free rate r =0.08

standard deviation s=0.3

time in years t= 1

d1= ( ln(S/K) + (r + s^2/2) *t ) / (s*t^0.5) = 0.765

d2 =d1-s*t^0.5 = 0.465

So, N(-d1) = 0.2221 (Can be calculated by NORMSDIST function in EXCEL)

and N(-d2) =0.3209

Putting the above values , we get the value of put option as

p = $8.949

As , the value of put option with strike $180.15 is $8.949

the call option with strike price $180.15 cant have a premium of $8.949 as it has an intrinsic value of around $20 and also time value for one year

So, possible strike prices are $200, $270.15 and $280.15

The price of a call option as per BSM model

C=S*N(d1)-K*e^(-r*t) * N(d2), putting the values in this equation for K =$200 we get

C = $31.42

The price of a call option as per BSM model

C=S*N(d1)-K*e^(-r*t) * N(d2), putting the values in this equation

For K =$200 we get

C = $31.42

For K =$270.15 we get

C = $8.951

So, the correct option is $270.15