Binomial Option Pricing (3-period model)

Step 1: Up and Down Factors

• $u = e^{\sigma \sqrt{\Delta t}} = e^{0.1825 \times \sqrt{1}} = 1.20$
• $d = e^{-\sigma \sqrt{\Delta t}} = 0.83$

Step 2: Possible Stock Prices After 3 Periods

• $S_{uuu} = 75 \times 1.2^3 = 129.60$
• $S_{uud} = S_{udu} = S_{duu} = 75 \times 1.2^2 \times 0.83 = 89.64$
• $S_{udd} = S_{dud} = S_{ddu} = 75 \times 1.2 \times 0.83^2 = 62.00$
• $S_{ddd} = 75 \times 0.83^3 = 42.89$

Step 3: Option Payoff

• Strike price = 90
• Only $S_{uuu} = 129.60$ is in-the-money
• Payoff = $129.60 - 90 = 39.60$

Step 4: Probability of 3 Up Moves

• $p = 60\%$
• Probability = $0.6^3 = 0.216$ (21.6%)

Step 5: Present Value of Option

• Discount factor = $e^{-0.05 \times 3} = e^{-0.15}$
• Option value = $e^{-0.15} \times 39.60 \times 0.216 = 7.36$

Final Result:
The option’s value today = 7.36