Question

A stock currently trades at $50. It pays dividends at a rate of 2%, and the risk-free rate is also 2%. A 3-month call option with a strike price of $50 is trading at $2.

Calculate the implied volatility of the underlying stock.

Calculate the volatility of the call option.

(No Excel work. Please show step by step with formula)

Answer 1

Black Scholes Merton formula for valuing a European Call option is given by:

C₀= S₀*e^(-qt)*N(d₁) – K*e^(-rt)*N(d₂)

d₁ = (ln(S/K) + (r-q+²/2)*t)/ t

d₂ = d₁ – t

Where,

C₀ is current call option price

S₀ is the current stock price

K is the Strike price

r is the risk free rate

q is the dividend rate

t is time to expiry

is the volatility of the underlying stock

N(d₁) and N(d₂) are the standard cumulative normal distribution

In the given Problem,

C₀ = 2

S₀ = 50

K = 50

q = 2%

r = 2%

t = 0.25 years

= ??

d₁ = (ln (50/50) + (0.02-0.02+²/2)*0.25)/ 0.25

= 0.25

d₂ = 0.25 – 0.5 = -0.25

Hence, 2= (50*e^(-0.02*0.25)*N(0.25)) – (50*e^(-0.02*0.25)*N(-0.25))

Now, we shall need backward calculation or trial and error method. I suggest you to use Excel “Goal Seek” to find out the volatility. I am attaching the screenshot for the same.

Or You can use trial and error by assuming in any value for the volatility. If the call price is lower than 2, try increasing the volatility and vice-versa.