Question

Use the Black-Scholes formula to value the following options:

a. A Call option written on a stock selling for $100 per share with a $110 exercise price. The stock's standard deviation is 15% per quarter. The option matures in three months. The risk free interest is 3% per quarter.

b. A put option written on the same stock at the same time, with the same exercise price and expiration date.

Now for each of these options find the combination of stock and risk-free asset that would replicate the option.

Answer 1

So current stock price = 100

K strike price = 110

r risk free rate = 3% = 0.03

s: standard deviation = 15% per quarter

s: standard deviation per annum = 15%*(40.5) = 15%*2 = 30%

t: time to maturity = 3month = 0.25 year

d1 = -0.5104

d2 = -0.6604

N(d1) = normsdist(d1) = 0.3049

N(d2) = normsdist(d2) = 0.2545

C: value of call option

e: natural exponent

c = 2.703

p: price of put option

Using put call parity

c + K*exp^(-r*t) = p + So

p = c + K*exp^(-r*t) - So = 11.881

(a) Replicating portfolio call option

Buy N(d1) stocks = 0.3049

Value of stock = 100*0.3049 = 30.49

Sell bond worth = 30.49 - 2.703 = 27.787

(b) Replicating portfolio put option

Sell N(-d1) stocks = 1- 0.3049 = 0.6951

Value of stock = 100*0.6951 = 69.51

Buy bond worth = 69.51 + 11.881 = 81.391