Question

1. Black-Scholes option pricing

Suppose the stock price is 50 and we need to price a call option with a strike of 55 maturing in 2 months. The stock is not expected to pay dividends. The continuously compounded risk-free rate is 3%/year, the mean return on the stock is 7%/year, and the standard deviation of the stock return is 30%/year.

1. What is the N(d1) and N(d2)?
2. What is the price of the call option?
3. What is the price of a put option with the same features?

Answer 1

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

The value of a call and put option are:

C = (S0 * N(d1)) - (Ke-rT * N(d2))

P = (K * e-rT)*N(-d2) - (S0)*N(-d1)

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(50 / 55). We input the same formula into Excel, i.e. =LN(50/55)

· (r + σ2/2)*T = (0.03 + (0.302/2)*(2/12)

· σ√T = 0.30 * √(2/12)

d1 = -0.6761

d2 = -0.7986

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

N(d1) = 0.2495

N(d2) = 0.2123

N(-d1) = 0.7505

N(-d2) = 0.7877

Now, we calculate the values of the call and put options as below:

C = (S0 * N(d1))   - (Ke-rT * N(d2)), which is (50 * 0.2495) - (55 * e(-0.03 * (2/12)))*(0.2123)    ==> $0.8579

P = (K * e-rT)*N(-d2) - (S0)*N(-d1), which is (55 * e(-0.03 * (2/12)))*(0.7877) - (50 * (0.7505) ==> $5.5836

Value of call option is $0.8579

Value of put option is $5.5836