The spot term structure for T-Bills (proxy for the risk free rate) is as follows

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. (if you can include the math, that would be great)

Explanation / Answer

d1=[ln(So/X)+(r-q+.5*sigma^2)*T]/[sigma*sqrt(T)]

d1=[ln(50/50)+(0.0765-0.04+.5*.35^2)*0.50]/[0.35*sqrt(0.50)]

d1=[ln(1)+(0.0365+.5*.1225)*0.50]/[0.35*0.70710678]

d1=[0+(0.0365+0.06125)*0.50]/0.247487373

d1=(0.09775*0.50)/0.247487373

d1=0.1975

d2=[ln(So/X)+(r-q-.5*sigma^2)*T]/[sigma*sqrt(T)]

d2=[ln(50/50)+(0.0765-0.04-5*.35^2)*0.50]/[0.35*sqrt(0.50)]

d2=-0.0500

European Put option price=p=X*exp(-rT)*N(-d2)-S0*exp(-qT)*N(-d1)

p=50*exp(-0.0765*0.5)*N(0.0500)-50*exp(-0.04*0.50)*N(-0.1975)

from normal cumulative table: N(0.0500)=0.5199

N(-0.1975)=1-N(0.1975)=1-0.5783=0.4217

p=50*exp(-0.0765*0.5)*0.5199-50*exp(-0.04*0.50)*0.4217

p=50*0.96247*0.5199-50*0.9802*0.4217

p=25.01940765-20.667517

p=4.35

Thus  price of a 6-month At-the-Money European Put option on this stock is $4.35.

current StockPrice S0 50 annual return volatility sigma 35.00% effective annual risk-free rate r(180 day T-billl rate as option expire after 180 days) 7.65% Exercise Price X(ATM,X=S0) 50 Time to Maturity(yrs) T=180/360 0.5 Dividend Yield q=(0.50*4)/50 4.00% d1 d1=[ln(So/X)+(r-q+.5*sigma^2)*T]/[sigma*sqrt(T)] 0.1975 d2 d2=[ln(So/X)+(r-q-.5*sigma^2)*T]/[sigma*sqrt(T)] -0.0500 N(-d1) N(-d1)=NORM.S.DIST(-d1,TRUE) 0.4217 N(-d2) N(-d2)=NORM.S.DIST(-d2,TRUE) 0.5199 put option value p=X*exp(-rT)*N(-d2)-S0*exp(-qT)*N(-d1) $        4.35

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.