The price of a call option is inversely related to the exercise price. This is because the exercise price is the price at which the holder of the option can buy the underlying asset. If the exercise price decreases, the likelihood of the option being in the money (i.e., profitable) increases, which in turn increases the price of the call option. Similarly, the price of a put option is directly related to the exercise price. A put option gives the holder the right to sell the underlying asset at the exercise price. Therefore, if the exercise price increases, the potential profit from exercising the option also increases, which increases the price of the put option. Lastly, the value of both American call and put options increases as the time to maturity increases. This is because the longer the time to maturity, the greater the chance that the option will end up in the money, which increases the value of the option.

Choice A is incorrect. While it correctly states that the price of a call option increases as the exercise price decreases (Property I), and the price of a put option increases as the exercise price increases (Property II), it fails to acknowledge that the values of both American call and put options increase as time to maturity increases (Property III). This is an important aspect of options pricing, known as time value, which suggests that an option with more time until expiration has greater potential for profit, thus increasing its value.

Choice B is incorrect. Although it correctly identifies Properties II and III, it incorrectly omits Property I.