Question

The spot term structure for T-Bills (proxy for the risk free rate) is as follows 30-Day T-Bill=7% per annum, 60-Day T-Bill=7.25% per annum, 90-Day T-Bill=7.5% per annum, 180-Day T-Bill=7.65% per annum and the 270-Day T-Bill=7.85% per annum all with continuous compounding. A stock pays $0.5 per quarter as dividends, the first dividend has just been paid and the current stock price is $50. What is the price of a 6-month At-the-Money European Put option on this stock if the volatility is 35%? Assume the year has 360 days.

Answer 1

The formula for European non-dividend paying put option on an underlying Black & Scholes model is:

Where S0 = underlying price = $50

For stock paying dividends, the spot price is modified as:

Modified stock price = Current stock price - Present value of expected dividends during the life of the option

The company pays a dividend of $0.5 quarterly. Hence, the investor will receive two dividends (one in 90 days and second in 180 days):

Present value of expected dividends = 0.5*e^-(0.075)*(0.25) + 0.5*e^-(0.0765)*(0.5)

Present value of expected dividends = 0.5*(0.981425 + 0.962472) = $0.971948

Modified stock price = $50 - $0.971948 = $49.02805

K = strike price = $50 (because the option is at-the-money, hence, S0 = X)

σ = volatility = 35%

r = continuously compounded risk-free interest rate

Since, 180-days stock option is to be valued, hence, 180-day T-bill is considered.

r = 7.65%

t = time to expiration = 6 months = 180 days

Since, 360 days year has been given so, t = 180/360 = 0.5

Putting these values in equation, we get:

d1 = {ln(49.02805/50) + (0.0765 + (0.35^2)/2)*0.5}/{0.35*sqrt(0.5)}

d1 = {ln(0.980561) + (0.0765 + 0.1225/2)*0.5}/{0.35*sqrt(0.5)}

d1 = {-0.01963 + (0.0765 + 0.06125)}*0.5/{0.35*sqrt(0.5)}

d1 = (-0.01963 + 0.13775)*0.5/0.35*sqrt(0.5)}

d1 = 0.11812*0.5/{0.35*sqrt(0.5)}

d1 = 0.05906/{0.35*sqrt(0.5)}

d1 = 0.05906/{0.35*0.707107}

d1 = 0.05906/0.247487

d1 = 0.238639

d2 = d1 - σ*sqrt(t)

d2 = 0.238639 - 0.35*sqrt(0.5)

d2 = -0.00885

N(-d1) = N(-0.238639)

Using the =NORMSDIST(z) function in Excel, the value of N(-d1) is calculated as: N(-0.238639) =NORMSDIST(-0.238639)

Hence, N(-d1) = 0.405693

Using the =NORMSDIST(z) function in Excel, the value of N(-d2) is calculated as: N(0.00885) =NORMSDIST(0.00885)

Hence, N(-d2) = 0.503531

The price of put option = -S*N(-d1) + Ke^rt*N(-d2)

The price of put option = -49.02805*0.405693 + 50*e^{(0.0765)*0.5}*0.503531

= -19.8903 + 50*1.038991*0.503531

= -19.8903 + 26.15821

= 6.26787

The price of put option = $6.26787