# Determine the upper bound of the option premium, Boundary Conditions Definition

Identify and compute upper and lower bounds for option prices on non-dividend and dividend-paying stocks. Explain put-call parity and apply it to the valuation of European and American stock options, with dividends and without dividends and express it in terms of forward prices.

### Properties of Stock Options

Explain and assess potential rationales for using the early exercise features of American call and put options. Explain the relationship between options and forward prices. The strike price of the option For call options, the value decreases increases as the strike price increases decreases.

For put options, the value increases decreases as the strike price increases decreases.

Time to expiration With American style options as the time to expiration increases, the value of the option increases. With more time, there are higher chances of the option moving in-the-money. As the time to expiration increases, the value of a call option increases.

As the time to expiration increases, the value of a put option also increases. However, the same does not apply to European-style options, precisely when the underlying has scheduled dividends.

### Minimum and Maximum Value of European/American Options

Assume further that a sizeable dividend is expected in three months. The ex-dividend stock price and call price will decrease. As such, the two-month call could actually be more valuable than the four-month call.

Risk-free Rate over the Lifespan of the Option Here, the simplest way to think about this is as a rate of return on a stock. Otherwise, why would you buy a share of stock instead of investing in a risk-free bond? Dividends Payments from an underlying may include dividends.

However, the benefits of these cash flows to the holders of the underlying security do not pass to the holder of a call option. Expected volatility of stock price over time Volatility is considered the most significant factor in the valuation of options. As volatility increases, the value of all options increases.

Since the maximum loss for the buyer of a call or put option is limited to the premium paid we can conclude that as volatility increases, there are higher chances of the option expiring in-the-money. A call option gives the holder the right to buy the stock at a specified price. The value of the call is always less than the value of the underlying stock.

### AnalystPrep

A put option gives the holder the right to sell the underlying stock at a specified price. The value of a put is always less than the strike price. European options can only be exercised at expiration. As such, the value of a European put is always less than the present value of the strike.

The lowest value of a call option has a price which is the maximum of zero and the underlying price less the present value of the exercise price.

A put option can never be worth less than zero as the option owner cannot be forced to exercise the option. The lowest value of a put option is the maximum of zero and the present value of the exercise price less the value of the underlying.

The below illustration will tell us if it is prudent to exercise American options before their maturity dates. Can an investor exercise this option before the three months maturity period? However, this option should not be exercised before maturity if interest rates are positive.

The option holder is faced with two scenarios. Had the investor waited until maturity then the option would not have been exercised.

An investor loses insurance against losses by exercising an American option earlier than its maturity date. This insurance against losses is provided by the choices offered by an option optionality. Scenario 2: The investor does not want a position in the stock Exercising the option now gives a profit equal to the intrinsic value of the option, i.

Example 2: Two portfolios An investor has two portfolios: Portfolio X comprises a call option plus cash equal to the present value of the strike price. Portfolio Y comprises the stock. The table below summarizes the possible outcomes when the stock price is i greater than and ii lesser than the strike price at maturity. Conclusion: American Options Exercising an American call option earlier gives a profit equal to the stock price minus the strike price.

The above proves that American call options should not be exercised earlier than their maturity dates.

### Boundary Conditions

Since a call option cannot be more worth than the stock price, the stock price is the upper bound of a call option. The lower bound is obtained by subtracting the present value of the strike price from the stock price.

Employee Stock Options Employee stock options cannot be sold — they have to be exercised by the employee to which the stock option has been given.

As with all other call options, employee call options on stocks that pay no dividends should not be exercised before maturity. Effects of Dividends Dividends reduce the price of a stock.

## Boundary Conditions Definition

For an American call dividend-paying option to be profitable, the option should be exercised just before the ex-dividend date. American Put Options Case of No Dividends Whereas call option holders pay the strike price, put option holders receive the strike price.

The decision to exercise an American put option, thus, is dependent on a trade-off between receiving the strike price early so as to reinvest it and benefitting from a very small probability that the stock price will be greater than the strike price at maturity.

Impact of Determine the upper bound of the option premium Dividends make it undesirable for a put option to be exercised before maturity. A holder of a put option is less likely to exercise the position earlier if the time to maturity determine the upper bound of the option premium, the stock price increases, the dividends to be received increases, and if the interest rate decreases.

Put-call Parity In European Options Put-call parity states that the price of a call option implicitly informs a certain price for the corresponding put option with the same strike and expiration and vice versa.

Thus, put-call parity demands that the value of the two portfolios today is the same.