The price of an American put option should be equal to or lower than the strike price of the option on the underlying stock. This is because the value of a put option is derived from the right it provides to its holder to sell the underlying asset at the strike price. If the price of the put option were to exceed the strike price, it would mean that the holder is paying more for the option than they would receive from exercising it. This would not be rational, as the holder could instead sell the asset in the market at a higher price. Therefore, the price of a put option should never exceed the strike price of the option. If it does, an arbitrage opportunity would arise. An investor could sell the overpriced put option, invest the proceeds at the risk-free rate, and make a risk-free profit. This is why the price of an American put option should be equal to or lower than the strike price of the option on the underlying stock.

Choice A is incorrect. The price of an American put option should not be equal to or higher than the current price of the underlying stock. This would imply that the holder could sell the stock for its full market value and still retain a valuable put option, which contradicts no-arbitrage principles.

Choice B is incorrect. The price of an American put option should not be equal to or higher than the strike price of the option on the underlying stock. If this were true, it would mean that exercising the put option immediately would yield a profit greater than or equal to zero regardless of what happens in future, which again contradicts no-arbitrage principles.

Choice C is incorrect. While it's true that an American put option's value will generally decrease as its underlying asset's price increases (all else being equal), saying that its price "should" be lower than its underlying asset's current market value implies a stronger relationship between these two prices than actually exists in practice.