Question

Use the information below and the Black-Scholes Option Pricing Model to answer the following questions.

Stock price =	$90
Exercise price =	$86
Risk-free rate =	2.6% per year, compounded continuously
Maturity =	10 months
Standard deviation =	26% per year

a. What is the price of a call?
b. What is the exercise value of the call option?

c. What is the time value of the call option?

Answer 1

a]

We use Black-Scholes Model to calculate the value of the call and put option.

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(90 / 86). We input the same formula into Excel, i.e. = LN(90 /86)
• (r + σ²/2)*T = (0.026 + (0.26²/2)*(10/12)
• σ√T = 0.26 * √(10/12)

d₁ = 0.4015

d₂ = 0.1642

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

N(d₁) = 0.6560

N(d₂) = 0.5652

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

C = (S₀ * N(d₁))   - (Ke^-rt * N(d₂)), which is (90 * 6560) - (86 * e^{(-0.026 * (10/12))})*(0.5652)    ==> $11.4727

Price of call option is $11.4727

b]

Exercise value of in-the-money call option = stock price - exercise price

Exercise value of in-the-money call option = $90 - $86 = $4

c]

price of call option = intrinsic value + time value

$11.4727 = $4 + time value

time value = $7.4727