Question

A 3 month call option is trading with an exercise price of US$50.The current price of the underlying stock is US$60.The risk free rate is 7% compounded continuously and the variance of the stock price return is 14.4%.

Required:

1.What is the intrinsic value of this call option?

2.Based on the Black Scholes model what is the total value of this call option?

3. what accounts for the difference between the total value and the intrinsic value?

.

Answer 1

1]

Intrinsic value of call option = underlying price - strike price = $60 - $50 = $10

2]

We use Black-Scholes Model to calculate the value of the call 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 maturity in years (time to maturity = 3 months = 0.25 years)

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

d₂ = d₁ - σ√T

σ = standard deviation of underlying stock returns. (standard deviation = variance. Standard deviation = 14.4% = 38%)

First, we calculate d₁ and d₂ as below :

• ln(S₀ / K) = ln(60 / 50). We input the same formula into Excel, i.e. =LN (60 / 50)
• (r + σ²/2)*T = (0.07 + (0.38²/2)*0.25
• σ√T = 0.38 * √0.25

d₁ = 1.1480

d₂ = 0.9583

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

N(d₁) = 0.8745

N(d₂) = 0.8310

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

C = (S₀ * N(d₁))   - (Ke^-rt * N(d₂)), which is (60 * 0.8745) - (50 * e^{(-0.07 * 0.25)})*(0.8310)    ==> $11.64

Value of call option is $11.64

3]

Difference between total value and intrinsic value is the time value of the option