Use arbitrage arguments to prove the following simple bounds on European call options on an asset that pays no dividends: (a) \(C \leq S\); (b) \(C \geq S-E e^{-r(T-t)} ;\) (c) If two otherwise identical calls have exercise prices \(E_{1}\) and \(E_{2}\) with \(E_{1}

Arbitrage arguments are a fundamental concept in financial theory, particularly useful when assessing the value of derivatives like European call options. An arbitrage opportunity is a chance to make a risk-free profit with no initial investment, something that should not exist in an efficient and balanced market. In the context of option pricing, such arguments help in establishing boundaries for the option's value.

For example, if the price of a call option (\(C\)) were to exceed the current stock price (\(S\)), a savvy investor could exploit this by purchasing the stock and selling the call option. This instant profit without risk contradicts the no-arbitrage principle, therefore, we surmise that the call option's value must always be less than or equal to the stock price (\(C \leq S\)).

The no-arbitrage condition also sets a floor to the call option's price. Since an option allows the holder to buy stock at a fixed price in the future, it has inherent value. The lower boundary is established by taking into account that at expiration, the payoff is the excess of stock price over the exercise price, only if positive; in essence, looking at the present value of the exercise price gives us the conclusive inequality (\(C \geq S - E e^{-r(T-t)}\) indicating the minimal worth of the call.

The valuation of call options is an intricate process that relies on various factors including the current stock price, the strike price, the time to expiration, interest rates, and the underlying volatility of the stock. The European call option, a contract that gives you the right but not the obligation to buy a stock at a predetermined price on a specific date, is particularly interesting as it can be valued analytically using models like the Black-Scholes formula.

In concentrating on the bounds, call option valuation maintains that the value of a call must reflect the potential for future gain. For instance, when comparing two call options with different strike prices (\(E_1\) and \(E_2\) where \(E_1 < E_2\)), we deduce that the one with the lower strike price (\(E_1\)) should not exceed the value of the option with the higher one (\(E_2\)) by more than the difference in these strike prices (\(E_2 - E_1\) as presented in the inequality \(0 \leq C(S, t ; E_1) - C(S, t ; E_2) \leq E_2 - E_1\)).

Moreover, evaluating call options with varying expiration times requires acknowledgment of the time value of money. An option with a longer time to expiry (\(T_2\) should not be worth less than one with an earlier expiry date (\(T_1\) as the former carries more opportunities for profit, leading to the inequality \(C(S, t ; T_1) \leq C(S, t ; T_2)\)).

The no-arbitrage market principle is a backbone of modern financial theory, dictating that if a market is efficient, there should be no possibility to achieve guaranteed profits without exposing oneself to any risk or capital. This principle aids in the establishment of a fair valuation of financial securities, such as options.

Applying this principle to our earlier arguments, the very notion that an option's value can breach certain boundaries suggests an arbitrage opportunity, which should be impossible in a no-arbitrage market. For example, if an investor observes a price discrepancy suggested by any of the inequalities not holding (\(C > S\) or \(C < S - E e^{-r(T-t)}\)), they could exploit this to achieve a risk-free profit, which in turn adjusts the pricing anomaly.

The no-arbitrage principle doesn't only assist in deducing upper and lower limits for call or put options, but it also underpins complex pricing models and strategies. It further implies that market prices for options will naturally tend to levels where no arbitrage can be made, ensuring that the prices of derivatives are consistently a reflection of their intrinsic and time value.