Question

Use the BSM model to calculate the price of a 13-month European call option with a strike price of $40 on a stock that is currently $48 and is expected to pay a $5 dividend in 6 months. The risk-free interest rate is 4% (annualized, continuously compounded), and the volatility of the stock’s returns is 55% per annum. (Reminder: your answer can have N(.) terms in it.)

Answer 1

First of all lets see the formula for computing european call option price by using BSM Model

S₀$e^{-\delta }$^T N (d1) - e^{- r T}N (d2) S_t

d1 = [ Natural Log of (S₀ / S_t) + (r - $\delta$ + ((annualised volatility of stock)² / 2) T ] / Volatility x $\sqrt{T}$

d2 = d1 - Volatility x $\sqrt{T}$

Now lets understand what each term denotes in the above formula

S₀ = Current Stock Price = 48

$\delta$ = dividend yield (Current value of dividend / Price of stock. This is to be expressed in decimals) = current value of dividend = 5 x e ^{- 0.02} = 4.90, Price of stock = 48

Hence dividend yield = 4.90 / 48 = 0.1021

e ^{- 0.02} is taken since the annualized rate is 4% and we have to discount it for 6 months.

S_t = Strike price of stock = 40

r = Risk Free Interest rate = 0.04

T = Time to expiration expressed in years = 13 / 12 = 1.08

Annualised volatility of stock = 0.55

d1 = [ Natural log of ( 1.2) + ( 0.04 - 0.1021 + (0.55² / 2) 1.08 ] / 0.55 x 1.04 = 5.98

d2 = 5.98 - 0.572 = 5.41

Now we shall put these figures in the formula mentioned above:

= 48 e ^{- 0.1021 x 1.08} x N (5.98) - e ^{- 0.04 x 1.08} x N (5.41) x 40

= 42.99 x N (5.98) - 38.31 x N (5.41).