Question

Problem 1:

The following information is given about options on the stock of a certain company:

            S₀ = $80, X =$70, r =10% per year (continuously compounded), T = 9 months, s= 0.30

No dividends are expected. One option contract is for 100 shares of the stock. All notations are used in the same way as in the Black-Scholes-Merton Model.

1. What is the European call option price and European put option price, according to the Black-Scholes model?
2. What is the cost of buying a protective put?
3. What is the cost of writing a covered call?
4. What will be the payoff and profit of the protective put if the stock price on maturity is $60, $70, $76, $80, $86?
5. What will be the payoff and profit of the covered call if the stock price on maturity is $60, $70, $76, $80, $86?

Answer 1

Price of a European call option as per black sholes model os given by

C= S0× N(d₁) - K×e^-rt N(d₂)

d₁ = ( ln(S0/K) + (r + (σ²/2)) t ) / σ √t

d₂ = ( ln(S0/K) + (r - (σ²/2)) t ) / σ √t = d₁ - σ √t

where

C = price of call

S0= current stock price = 80

K= X= Strike price = 70

r= risk free rate= 10% continueous

t= T/12 = 9/12 = 3/4 = .75 year

= s= standard deviation = .30

In(S0/k)= ln(80/70)/(.30×(.75)^.5 )=.51396

( (r+s^2/2)t)/s×(t)^.5 = ((.10 +.30^2/2)×.75)/(.30×(.75)^.5

= .41858

d1= .51396 + .41858 = .93254

d2= d1 - s×(t)^.50 =.93254 - .30× (.75)^.5 = .67273

N(d1) = N(.93254) = is the cumulative normal distribution probability for z = .93254 as per normal distribution table this is .82447

And similarly N(.67273) = .74944

Therefore

C= 80× .82447 - 70 ×e^-(.10×.75)×.74944 = $17.29

Put price can be calculated by using put call perity PCP.

PCP is given by

C - P = S0 - K × e^(-rt)

17.29 -P =80 - 70 ×e^(-.10×.75)

P= $2.23 this is the put price

Part b

Protective put means buying a put and stock so that when the price price goes down put can protect the holder.

Payment for buying put = 2.23×100= 223

Payment for Buying share = 80 ×100 = 8000

Total cost = $8223

Covered call means selling a call long with buying stock

Amount received in selling call= 17.29×100 =$1729

Amounts paid to buy stock= 80×100=$8000

Total cost = 9729$

Part 3

Receivable Pay off of put = max((K-S),0)×m

Where S = different stock prices

m = lot size= 100

Pay off from stock= S×100

Profit = 100S - max((70 - S),0)×100 - 8223

Table is as follows

Call pay off payble= max((S-K),0)

stock payoff receivable= 100S

Profit = 100S- max((S-70),0)×100 -9729

Table is