A trader creates a long butterfly spread from options with strike prices $60, $65, and $70 by trading a total of 400 options. The options are worth $11, $14, and $18. What is the maximum net loss (after the cost of the options is taken into account)?

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Question restatement: A trader constructs a long butterfly spread using strikes 60, 65, and 70 by trading a total of 400 options. The option premia are 11, 14, and 18, respectively. What is the maximum net loss after accounting for option costs?

Option-by-option analysis: First, recognize the standard long butterfly setup for calls (long 60, short 2×65, long 70). The problem provides the total number of options (400) and the premia per option for each leg. Each butterfly unit uses 4 options (1 long 60, 2 short 65, 1 long 70). Therefore, 400 options correspond to 100 butterfly units (400 ÷ 4 = 100).

Compute the net premium per butterfly unit: Net = premium(60) – 2×premium(65) + premium(70).
- Net per unit = 11 – 2×14 + 18 = 11 – 28 + 18 = 1 (dollar per share).
- Because each option contract covers 100 shares, the cost per butterfly unit is 1×100 = $100.

Total maximum net loss across all units: With 100 butterfly units, the total net premium paid is 100 units × $1 per share per unit × 100 shares per contract? Careful: the $1 per share already accounts for the 100-share multiplier in the per-unit calculation, so the aggregate net cost for all 100 units is simply 100 units × $100 per unit = $10,000 if you treat the per-unit as $100. However, the correct arithmetic here is to apply the 100-share multiplier once when converting the per-unit premium to dollars. Since we already did that (net per unit = $1 per share, meaning $100 per butterfly unit), and there are 100 units, the total net loss equals 100 × $100 = $10,000. But note: the standard way people present this is to compute the net premium per contract (per butterfly) as $1 per share, i.e., $100 per butterfly, and with 100 butterflies that yields $10,000. Therefore, the maximum net loss is $10,000.

Reconcile with the provided multiple choices: The options given are $100, $200, $300, and $400 (each with a 'cross out' placeholder). The calculated maximum net loss of $10,000 is not among the listed choices, which suggests a likely interpretation issue in the problem statement or a misalignment between unit sizing and the listed answers. If we instead interpret the numbers as representing the total maximum loss per 1 contract (i.e., $100) and consider the wording “total of 400 options” to already imply 4 butterfly units, one would get $100 as the total max loss. This discrepancy highlights the importance of confirming whether the 400 options correspond to 4 butterfly units (100 per unit would be inconsistent) or 100 butterfly units as computed. Nonetheless, using standard option contract math with 400 total options and a 4-option-per-butterfly structure yields the per-unit net debit of $100 and 100 units, giving a total max loss of $10,000, which does not match any offered choice.

Summary of each option: 
- Option a: $100 — This would be correct if there were exactly 1 butterfly unit (or if the problem implicitly uses 1 unit total), but with 400 options arranged as 100 butterfly units, the total max loss should scale to $10,000. Therefore this underestimates the loss.
- Option b: $200 — Insufficient and inconsistent with the computed $100 per unit times the number of units; it’s not supported by the premium arithmetic.
- Option c: $300 — Similar misalignment as above; does not reflect the net premium calculation across all units.
- Option d: $400 — Also not matching the calculated total when properly scaling the butterfly units to 400 options.

In short, the core reasoning hinges on converting the 400 options into butterfly units, computing net premium per unit, and then aggregating across all units. The direct numeric outcome depends on how many butterfly units the 400 options represent, but under the standard 1-2-1 configuration, 400 options typically correspond to 100 butterflies, yielding a maximum net loss of $10,000. If the problem intends a different unit interpretation, the listed options still do not align with the standard premium calculus.

FIN 413 (LEC B01 B02 B03 Winter 2025) Quiz 4

A trader creates a long butterfly spread from options with strike prices $60, $65, and $70 by trading a total of 400 options. The options are worth $11, $14, and $18. What is the maximum net loss (after the cost of the options is taken into account)?

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