Derive the closed-form formula for an option with the following payoff function:
• Basic requirement
(i) Utilize the martingale pricing method to derive the closed-form formula by hands. (ii) Based on the formula you derive, implement a program to price this option.
(Inputs: S0, r, q, σ, T, K1, K2, K3, K4. Output: Option value.)
• Bonus
Employ the Monte Carlo simulation to price this option.
Based on lnST ∼ NDQ(lnS0 + (r − q − σ2/2)T,σ2T), draw 10,000 random samples for
ST to compute an option price. Repeat the above step 20 times to obtain the 95% confidence interval for the option value:
[mean of 20 repetitions − 2×(s.d. of 20 repetitions), mean of 20 repetitions + 2×(s.d. of 20 repetitions)].
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