Question

Consider a European call option on a stock. The stock price is $65, the time to maturity is 8 months, the risk-free interest rate is 10% p.a., the strike is $70, and the volatility is 32%. A dividend of $1 will be paid after 3 months and again after 6 months. What is price of the option?

Answer 1

First Dividend, D₁ = $ 1 at t₁ = 3 months = 3/12 = 0.25 year

Second Dividend, D₂ = $ 1 at t₂ = 6 months = 6/12 = 0.50 year

Present value of dividends = D₁e^-rt₁ + D₂e^-rt₂ = 1 x e^{-0.10 x 0.25} + 1 x e^{-0.10 x 0.50} = 1.9265

Stock Price, S = $ 65

Hence, equivalent stock price of a non dividend paying stock, S₀ = S - Present value of dividends = 65 - 1.9265 =  63.0735

We will use this stock price in all our calculation making use of Black Scholes formula for pricing a call option on a non dividend paying stock.

Time to maturity, T = 8 months = 8/12 years = 0.6667 years

Risk free rate, r = 10% per annum = 0.10

Strike Price, K = $ 70

Volatility, = 32% = 0.32

Let's use the Black Scholes model to value this European call option on a non dividend paying stock:

Where

and d2 = d1 -

On substituting the values,

Hence, N(d1) = 0.494816145

and N(d2) = 0.391937313

Hence, the value of the European Call Option

= $ 5.5436