A compound option is an option on another option.

Let the value of the underlying option move with the stock price:

• A call value increases when the underlying asset price increases.
• A put value decreases when the underlying asset price increases.

Now evaluate each compound type:

1. Call on a call: call value rises as the stock rises, so this compound option is increasing.
2. Call on a put: put value falls as the stock rises, so the call-on-put value is decreasing.
3. Put on a call: call value rises as the stock rises, so the put-on-call value is decreasing.
4. Put on a put: put value falls as the stock rises, so the put-on-put payoff rises, making it increasing.

Therefore, the compound option values that are decreasing in the underlying asset price are II and III.

So the correct answer is C.