Question

What is the price of a European call option with the following parameters? s0 = $41 k = $42 r = 10% sigma = 20% T = 0.5 years (required precision 0.01 +/- 0.01) black scholes equation.PNG As a reminder, the cumulative probability function is calculated in Excel as follows: N(d1) = NORM.S.DIST(d1,TRUE) N(d2) = NORM.S.DIST(d2,TRUE) If the above equations don't load for whatever reason, here are the text versions of the equations as a back-up: c = So*N(d1) - K*e^(-rT)*N(d2) p = K*e^(-rT)*N(-d2) - So*N(-d1) d1 = [ln(So/K) + (r + 0.5*(sigma^2))*T] / [sigma * sqrt(T)] d2 = d1 - sigma*sqrt(T) To validate your equations, you may use the following information to ensure you have it coded correctly: s0 = 22 k = 25 r = 0.1 sigma = 0.2 T = 0.75 d1 = -0.2184 d2 = -0.3916 c = 1.03446 p = 2.22805

Answer 1

The formula for European call & Put option using Black Scholes model is given by

Call = S0 x N(d1) - K x e^(-r x t) x N(d2)

Put = K x e^(-r x t) x N(-d2)- (S0 x N(-d1))

Where,

S0 = Price of the underlying, eg stock price.

K = Exercise price.

r = Risk free interest rate

t = Time to expiry

σ = Standard deviation of the underlying asset, eg stock.

N(d1) =  standard normal cumulative distribution function using the value of d1

N(d2) =  standard normal cumulative distribution function using the value of d2

The value of d1 & d2 is calculated using the below formula

d1 = (Ln(So/X) + ((r + σ ^2)/2) x t)) / ((σ^2) x t)^1/2)

d2 = (Ln(So/X) + ((r - σ ^2)/2) x t)) / ((σ^2) x t)^1/2)

Here,

S0 = 41

K = 42

r = 10% = 0.10

σ = 20% =0.20

t = 0.5

We will first calculate the values of d1 & d2

d1 = [(Ln(So/X) + ((r + σ ^2)/2) x t))] / ((σ^2) x t)^1/2)

= [Ln (41/42) + (0.1 +(0.2^2)/ 2) x 0.50)] / (0.2^2 x 0.5)^(1/2)

= (-0.0241 + 0.06) / (0.1414)

d1 = 0.2539

d2 = [(Ln(So/X) + ((r - σ ^2)/2) x t))] / ((σ^2) x t)^1/2)

= [Ln (41/42) + (0.1 - (0.2^2)/ 2) x 0.50)] / (0.2^2 x 0.5)^(1/2)

= (-0.0241 + 0.04 ) / (0.1414)

d2= 0.1124

N(d1) = 0.6002

N(d2) = 0.5447

N(-d1) = 0.3998

N(-d2) = 0.4552

Value of Call C = S0 x N(d1) - K x e^(-r x t) x N(d2)

= 41 X 0.6002 - 42 X e^(-0.1 x 0.5) x 0.5447

= 24.6082 - 21.7642

C = 2.8440

The value of European call with the above data = 2.8440