The current price of a stock is $100. Consider the Black-Scholes model price of a six-month call option at strike $101, given an interest rate of 2% and a dividend rate of 1%? The volatility is 25%. What is the real-world (physical) probability of the option ending up in the money if the growth rate of the stock is expected to be 5% per year?
A) 0.45
B) 0.48
C) 0.49
D) 0.50

Question

Quizplus · Accepted Answer

Using the Black-Scholes formula, we get the option price to be $3.206. Since the option strike price is greater than the current stock price, it is an out-of-the-money option. Using the Black-Scholes model to calculate the implied probability of the option ending up in the money, we get 0.49. However, the Black-Scholes model assumes a constant stock growth rate, which is not realistic in the physical world. If we assume a constant stock growth rate of 5%, the real-world probability of the option ending up in the money can be calculated using the following formula:

d1 = [ln(S/K) + (r - q + 0.5*sigma^2)*T] / (sigma*sqrt(T)) 
= [ln(100/101) + (0.02 - 0.01 + 0.5*0.25^2)*0.5] / (0.25*sqrt(0.5)) 
= -0.191
d2 = d1 - sigma*sqrt(T) 
= -0.191 - 0.25*sqrt(0.5) 
= -0.424
N(d2) = 0.33 (from standard normal table)
N(d1) = 0.42 (from standard normal table)

real-world probability = (S*e^(r-q)*N(d1) - K*e^(-r+q)*N(d2)) / S*e^(r-q) 
= (100*e^(0.02-0.01)*0.42 - 101*e^(-0.02+0.01)*0.33) / 100*e^(0.02-0.01) 
= 0.49

Therefore, the answer is C.

The Current Price of a Stock Is $100