The following is an adapt question:
The answer key and formula sheet states that P(S>k)=N(d2-hat). I trust this is correct. However, I thought it should be N(d1-hat) because of the way All-or-Nothing Options are explained. Ie, you use N(d1) (+d1 if S>K and -d1 if S<K) for the S portion since S is not fixed, but N(d2) for the K portion since K is a constant. This formula seems inconsistent (or to conflict with) the formulas for all-or-nothing options.
Can anybody explain why d2 is appropriate here? Furthermore, what does N(d1) or N(d2) really represent?
Thanks for any input!
Assume the Black-Scholes framework.
The continuously compounded risk-free interest rate is r.
The price of the stock is S. The stock pays continuously compounded dividends at a rate of 4% per year. The stock’s volatility is \sigma. The continuously compounded expected return on the stock is r+\sigma^{2}.
A 3-month, K-strike European call option on the stock has a delta of 0.68.
Calculate the probability that the stock price is above K at the end of 3 months.
I am confused about why this probability would be equal to N(d2-hat).
The answer key and formula sheet states that P(S>k)=N(d2-hat). I trust this is correct. However, I thought it should be N(d1-hat) because of the way All-or-Nothing Options are explained. Ie, you use N(d1) (+d1 if S>K and -d1 if S<K) for the S portion since S is not fixed, but N(d2) for the K portion since K is a constant. This formula seems inconsistent (or to conflict with) the formulas for all-or-nothing options.
Can anybody explain why d2 is appropriate here? Furthermore, what does N(d1) or N(d2) really represent?
Thanks for any input!
ADAPT Probability Problem
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