Question

What is the price of a European call option with the following parameters?

s0 = $41
k = $40
r = 10%
sigma = 20%
T = 0.5 years

(required precision 0.01 +/- 0.01)

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 value of a call option is:

C = (S₀ * N(d₁)) - (Ke^-rT * N(d₂))

where :

S₀ = current spot price

K = strike price

N(x) is the cumulative normal distribution function

r = risk-free interest rate

T is the time to expiry in years

d₁ = (ln(S₀ / K) + (r + σ²/2)*T) / σ√T

d₂ = d₁ - σ√T

σ = standard deviation of underlying stock returns

First, we calculate d₁ and d₂ as below :

• ln(S₀ / K) = ln(41 / 40). We input the same formula into Excel, i.e. =LN(41 /40)
• (r + σ²/2)*T = (0.10 + (0.20²/2)*0.50
• σ√T = 0.20 * √0.50

d₁ = 0.5989

d₂ = 0.4574

N(d₁) and N(d₂) are calculated in Excel using the NORM.S.DIST function and inputting the value of d₁ and d₂ into the function.

N(d₁) = 0.7254

N(d₂) = 0.6763

Now, we calculate the values of the call option as below:

C = (S₀ * N(d₁))   - (Ke^-rT * N(d₂)), which is (41 * 0.7254) - (40 * e^{(-0.10 * 0.50)})*(0.6763)    ==> $4.0065

Value of call option is $4.0065