# Binomial Option Pricing Model: Real Probability

Consider a stock with current stock price of \$20 and a call option on the stock with strike price of \$21 and 50 days to expire (T = 50/365). The stock is not paying dividend and its volatility is 50% (ó = 50%), the expected annual rate of return (continuously compounding) on the stock is 20%. Assuming that the annual continuously compounding risk-free interest rate is 5% (r = 5%).

a. Using n = 1 in binomial option pricing model to find the option value.
b. Use real probability to price the above call option.

For part a,
u = 1.2116, uS = 24.2320, Cu = 3.2320
d = 0.8368, dS = 16.7354, Cd = 0

Delta = 0.4311
B = -7.1667
C = Delta(S) + B = 1.4553

For part b, suppose p is the real probability of the stock price moving up, y is the expected rate of return on the option. Then:

20e^(0.2 * 50/365) = 24.2320 * p + 16.7354 *(1 + p) , p = 0.5096
Ce^(y * 50/365) = 3.2320 * p + 0 * (1 + p) = 3.2320 * 0.5096

From synthetic call ,

e^(y * 50/365)
= e^(0.2 * 50/365) * (0.4311 * 20 / 0.4311 * 20 - 7.1667) +
e^(0.05 * 50/365) * (-7.1667 / 0.4311 * 20 - 7.1667)
= 1.1307

Therefore, c = 3.2320 * 0.5096 / 1.1307 = 1.4566.
I have the answers but I don’t understand the b part. Can someone help???