The main answer to the question is that a European call option is bounded below by \( S - E e^{-r T} \) and above by \( S \).
Step-by-step explanation:
1. Let's start with the lower bound of the European call option, which is given by \( S - E e^{-r T} \).
- \( S \) represents the current price of the underlying asset.
- \( E \) represents the strike price of the option.
- \( r \) represents the risk-free interest rate.
- \( T \) represents the time to expiration of the option.
2. The lower bound of the call option is obtained by considering the scenario where the underlying asset becomes worthless at the expiration date.
- In this scenario, the option holder can exercise the call option and sell the underlying asset at the strike price (\( E \)).
- The present value of the strike price at the expiration date is given by \( E e^{-r T} \).
- Therefore, the lower bound of the call option is \( S - E e^{-r T} \), as the option holder can always sell the underlying asset for at least the current price (\( S \)) minus the present value of the strike price.
3. Moving on to the upper bound of the European call option, which is simply the current price of the underlying asset (\( S \)).
- This upper bound arises from the fact that the option holder has the right to buy the underlying asset at the strike price (\( E \)).
- If the current price of the underlying asset is higher than the strike price, it is more profitable for the option holder to exercise the option and buy the asset at the lower strike price (\( E \)).
- Therefore, the upper bound of the call option is \( S \), as the option holder can always buy the underlying asset at the current price.
In conclusion, the European call option is bounded below by \( S - E e^{-r T} \) and above by \( S \). The lower bound is determined by the present value of the strike price, while the upper bound is simply the current price of the underlying asset.