At any given point in time, the value of a call option or a put option cannot exceed a particular price. Option prices fluctuate between their upper and lower bounds. For example, a call option can never be worth more than the stock price; therefore, the value of a call option should be lower or equal to the stock price. If there is a violation of this rule, arbitrageurs will enter and make a riskless profit by buying the stock and selling the call option.

In the case of a put option, the upper bound is the strike price at which the contract has been entered. The value of a put will be lower than or (at most) equal to the strike price of the option. If this condition is violated, an investor can make use of the arbitrage opportunity by writing the option and investing the proceeds at the risk-free rate of interest.

As explained earlier, the lower bounds for an American call option are the same as the lower bounds for a European call option. Prices of American and European options differ mainly in whether they can be exercised before expiration, as is the case with American calls. As such, in most cases American calls and puts will be worth more than European calls and puts and their lower bounds will also differ.

• When the underlying asset does not make cash payments such as dividends or interest payments, the value of an American call option is equal to a European call option.
• When cash payments are involved, the value of an American call option tends to be higher than a European call option.
• American put options are almost always worth more than European put options.

To take into account cash flows on underlying assets, we must rewrite the maximum value formulas (for European options given above) as follows:

European Options:

Formula 15.12

Where: C = Call, S = Strike price, PV = Present value, CF = Cash Flow,
r = interest rate, T= Time to expiration of the option

Cash flows for underlying assets are as follows:

• Stocks pay dividends - in formula terms FV (D,O,T) or PV (D,O,T)
• Bonds pay interest - in formula terms FV (CI,O,T) or PV (CI,O,T)
• Currency pays interest
• Commodities have carrying costs

The underlying price is reduced by the PV of the cash flows of the underlying; therefore, the put-call parity relationship is calculated as:

Formula 15.13

This formula determines the reduction in the price of the underlying assets as related to the present value of the cash flows over the life of the trade.

Interest Rate Changes and Option Prices
Options are priced on a risk-neutral basis, so a long call (for example) would be paired with a short stock. A short-stock position generates interest revenue, which makes the call option more valuable. If interest rates go up, the interest revenue from the short stock position increases, which makes the call worth even more. For put options and dividends, it works in the opposite direction.

When interest rates are high, the prices of calls are higher and the prices of puts are lower. Why? When buying an option, one is essentially using leverage. When rates are high, the option itself is more attractive than the underlying asset. By purchasing an option instead of the underlying asset, an investor saves cash.

Puts are adversely affected by higher rates because investors lose interest while waiting to sell their underlying assets. This works for all underlying assets except when dealing with bonds or interest rates.

Interest rate volatility has a huge effect on option prices. When volatility increases, call and put prices both increase because of the increased possibility that a downside or upside event could occur concerning the option. The upside helps call price and has no effect on puts, while downside helps puts with no effect on calls and is especially true when options are out of the money. Downside does begin to matter when options become in the money.

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