The maximum possible prices for the options in question can be determined by understanding the fundamental properties of option pricing. For both European and American call options, the maximum price cannot exceed the current stock price. This is because the option gives the holder the right, but not the obligation, to buy the stock at a certain price (the strike price), and it would be nonsensical to pay more for the option than what the stock is currently worth. Hence, the stock price acts as the "upper bound" for call options.

When it comes to put options, the logic differs slightly. For a European put option, the maximum possible price is the present value of the strike price, calculated using the continuously compounding risk-free rate. This is because the time value of money concept dictates that money available now is worth more than the same amount in the future, due to its potential earning capacity. The present value of the strike price (USD 90.00) can be calculated using the formula:

$PV = \frac{FV}{e^{rt}}$

where:

• $PV$ is the present value,
• $FV$ is the future value (strike price),
• $r$ is the risk-free rate,
• $t$ is the time to expiration (3 months in this case),
• $e$ is the base of the natural logarithm.

For an American put option, the maximum price is equal to the strike price itself. This is because an American put can be exercised at any time before expiration, and the holder could theoretically exercise the option and receive the strike price amount immediately, which is the maximum benefit they could gain from the option.

Given the current stock price of USD 100.00, a continuously compounding risk-free rate of 12% per year, and a strike price of USD 90.00 for a 3-month period, the maximum possible prices for the options are as follows:

• European-style call option: USD 100.00 (current stock price)
• American-style call option: USD 100.00 (current stock price)
• European-style put option: Present value of USD 90.00, which is less than USD 100.00
• American-style put option: USD 90.00 (strike price)

Therefore, the correct answer is C: 100.00, 100.00, 87.34, 90.00, where 87.34 represents the present value of the strike price calculated using the given risk-free rate and time to expiration.