Show that the value of a European call option on an asset that pays a constant continuous dividend yield lies below the payoff for large enough values of \(S\). Show also that the call on an asset with dividends is less valuable than the call on an asset without dividends.

When dealing with financial securities that provide dividends, understanding the concept of continuous dividend yield is crucial. Essentially, a continuous dividend yield, represented by 'q', is an assumption used in option pricing that accounts for the dividends paid by a stock over the period until option expiration. Unlike traditional dividends which are paid at regular intervals, the continuous yield models dividends as if they are paid at an infinitesimally small rate but continuously over time.

This is important because dividends directly impact the price of the underlying asset, which ultimately affects the value of options written on that asset. The equity's value decreases as dividends are paid, as investors receive payouts at the expense of the stock’s retained earnings. Hence, the present value of these expected dividends is subtracted from the stock price in option valuation models, lowering the call option’s value for the dividend-paying asset when compared to similar assets that do not pay dividends.

The Black-Scholes formula is a mathematical model used for option pricing. Developed by economists Fischer Black, Myron Scholes, and Robert Merton, this formula provided the foundation for the modern field of financial derivatives. It calculates the theoretical price of European options using several parameters: the current price of the stock (S), the strike price (K), the risk-free interest rate (r), the time to expiration (T), and the volatility (σ) of the stock’s returns.

In the context of a continuous dividend yield, the Black-Scholes formula gets adjusted so that the stock price (S) is reduced by the factor of the present value of the expected dividends, expressed as \( S \times e^{-qT} \). Although the formula assumes several ideal conditions, such as continuous trading and no transaction costs, it remains one of the cornerstones of option pricing. By utilizing this formula, traders and investors can estimate fair market prices for option contracts.

Option pricing refers to the process of determining the fair value of an option contract, which is essential for investors and traders to make informed decisions. The core principle involves assessing the likelihood of an option finishing 'in the money'—that is, having a positive exercise value at expiration. Various models exist for this purpose, but one of the most prominent is the Black-Scholes model mentioned earlier.

Option pricing is influenced by factors such as underlying asset price, strike price, time to expiration, interest rates, volatility, and dividends paid by the asset. Continuous dividend yield significantly impacts option pricing because it lowers the value of call options on dividend-paying stocks. Since dividends are assumed to be paid out continuously, the expected value of these payouts (in terms of current dollars) must be subtracted from the current value of the asset, thus reducing the price an investor should be willing to pay for the call option.

Financial derivatives are contracts whose value is derived from the performance of an underlying asset, index, or interest rate. They are mainly used for hedging risk or speculating on the future price of an asset. Common types of derivatives include options, futures, forwards, and swaps.

Options, as one kind of financial derivative, provide the buyer the right, but not the obligation, to buy or sell the underlying asset at a predetermined price on or before a specified date. The two principal types of options are calls and puts. Call options give the holder the right to purchase the underlying asset at a set price, while put options confer the right to sell. Understanding the valuation of options, including the impact of continuous dividend yield and the use of pricing models like Black-Scholes, is pivotal in the trading and risk management of financial derivatives.