Question

Consider an option on a non-dividend-paying stock when the stock price is $48, the exercise price is $46, the risk-free interest rate is 6% per annum, the volatility is 20% per annum, and time to maturity is four months. (a) What is the price of the option if it is a European call? (b) What is the price of the option if it is a European put? (c) What is the price of the option if it is an American call? (d) How would the result of a) change if a dividend of $1 is expected in two months? How would the result of a) change if a dividend of $2 is expected in six months?

Answer 1

Call option (C) and put option (P) prices are calculated using the following formulas:

… where N(x) is the standard normal cumulative distribution function.

The formulas for d1 and d2 are:

,

S₀ = Stock Price = $48

X = Exercise price = $46

σ = volatility (% p.a.) = 0.20

r = risk-free interest rate (% p.a.) =0.06

q = dividend yield (% p.a.) = 0

t = time to expiration= 4/12 year = 0.33 year

d1 = ln(48/46)+0.33*(0.06-0+.02^2/2)/(0.2*sqrt(0.33) = 0.6

d2 = d1-0.2*sqrt(0.33) =0.484

N(d1) = 0.725, N(-d1) = 0.274

N(d2) = 0.685, N(-d2) = 0.314

Putting these values in above equations to find call and put option price

European Call option Price(C) = $ 3.90

European Put option Price(P) = $ 0.99

c) American Non Dividend paying call option will have same price as European Call Option , i.e. $ 3.90

d) Dividend = $1 in 2 months

Dividend Yield = (1/48)*(12/2) =0.125

d1 = 0.253, d2 = 0.138

Call Option Price (C) = $ 2.61

Dividend = $2 in 6 months

Dividend Yield = (2/48)*(12/6) =0.833

d1 = 0.369, d2 = 0.253

Call Option Price (C) = $ 3.01