Question

Consider a 1-year option with exercise price $115 on a stock with annual standard deviation 10%. The T-bill rate is 2% per year. Find N(d1) for stock prices (a) $110, (b) $115, and (c) $120. (Do not round intermediate calculations. Round your answers to 4 decimal places.)

S	N(d1)
$110
$115
$120

Answer 1

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

where :

S₀ = current spot price

K = strike price

r = risk-free interest rate

t is the time to maturity in years

N(x) is the cumulative normal distribution function for value x.

a]

We calculate d₁ as below :

• ln(S₀ / K) = ln(110 / 115). We input the same formula into Excel, i.e. = LN(110 /115)
• (r + σ²/2)*T = (0.02 + (0.10²/2)*1
• σ√T = 0.48 * √1

d₁ = -0.1945

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

N(d₁) = 0.4229.

b]

We calculate d₁ as below :

• ln(S₀ / K) = ln(115 / 115). We input the same formula into Excel, i.e. = LN(115 /115)
• (r + σ²/2)*T = (0.02 + (0.10²/2)*1
• σ√T = 0.48 * √1

d₁ = 0.2500

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

N(d₁) = 0.5987.

c]

We calculate d₁ as below :

• ln(S₀ / K) = ln(120 / 115). We input the same formula into Excel, i.e. = LN(120 /115)
• (r + σ²/2)*T = (0.02 + (0.10²/2)*1
• σ√T = 0.48 * √1

d₁ = 0.6756.

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

N(d₁) = 0.7504.