Derivatives - Effect of Cash Flows on Put-Call Parity and the Lower Bounds

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.

Swap Markets and Contracts


Related Articles
  1. Options & Futures

    Three Ways to Profit Using Call Options

    A brief overview of how to provide from using call options in your portfolio.
  2. Options & Futures

    Three Ways to Profit Using Put Options

    A brief overview of how to profit from using put options in your portfolio.
  3. Options & Futures

    The Basics of Options Profitability

    The adage "know thyself"--and thy risk tolerance, thy underlying, and thy markets--applies to options trading if you want it to do it profitably.
  4. Trading Strategies

    A Guide Of Option Trading Strategies For Beginners

    Options offer alternative strategies for investors to profit from trading underlying securities, provided the beginner understands the pros and cons.
  5. Professionals

    Exchange Traded Options

    CFA Level 1 - Exchange Traded Options. Contrasts over the counter options from those traded on the exchange. Discusses the characteristics of many exchange traded options.
  6. Professionals

    Minimum and Maximum Values for Options

    CFA Level 1 - Minimum and Maximum Values for Options. Learn how to calculate the maximum and minimum values of American or European put or call options. Includes sample calculations and formulas.
  7. Professionals

    European vs. American Options and Moneyness

    CFA Level 1 - European Vs. American Options and Moneyness. Differentiates European and American options and contrasts the concepts of instrinic and time value. Also discusses the moneyness of ...
  8. Options & Futures

    Getting Acquainted With Options Trading

    Learn more about stock options, including some basic terminology and the source of profits.
  9. Options & Futures

    Options Basics: What Are Options?

    An option is a contract that gives the buyer the right, but not the obligation, to buy or sell an underlying asset at a specific price on or before a certain date. An option, just like a stock ...
  10. Professionals

    Options

    Options
RELATED TERMS
  1. Put On A Call

    One of the four types of compound options, this is a "put" option ...
  2. Call On A Put

    One of the four types of compound options, this is a call option ...
  3. Compound Option

    An option for which the underlying is another option. Therefore, ...
  4. In The Money

    1. For a call option, when the option's strike price is below ...
  5. Call On A Call

    A type of compound option in which the investor has the right ...
  6. Bear Call Spread

    A type of options strategy used when a decline in the price of ...
RELATED FAQS
  1. How are call options priced?

    Learn how aspects of an underlying security such as stock price and potential for fluctuations in that price, affect the ... Read Answer >>
  2. Are put options more difficult to trade than call options?

    Learn about the difficulty of trading both call and put options. Explore how put options earn profits with underlying assets ... Read Answer >>
  3. When is a call option considered to be "in the money"?

    Learn about call options, their intrinsic values and why a call option is in the money when the underlying stock price is ... Read Answer >>
  4. Do options make more sense during bull or bear markets?

    Understand how options may be used in both bullish and bearish markets, and learn the basics of options pricing and certain ... Read Answer >>
  5. How do speculators profit from options?

    As a quick summary, options are financial derivatives that give their holders the right to buy or sell a specific asset by ... Read Answer >>
  6. How do I change my strike price once the trade has been placed already?

    Learn how the strike prices for call and put options work, and understand how different types of options can be exercised ... Read Answer >>
Hot Definitions
  1. Goodwill

    An account that can be found in the assets portion of a company's balance sheet. Goodwill can often arise when one company ...
  2. Return On Invested Capital - ROIC

    A calculation used to assess a company's efficiency at allocating the capital under its control to profitable investments. ...
  3. Law Of Demand

    A microeconomic law that states that, all other factors being equal, as the price of a good or service increases, consumer ...
  4. Cost Of Debt

    The effective rate that a company pays on its current debt. This can be measured in either before- or after-tax returns; ...
  5. Yield Curve

    A line that plots the interest rates, at a set point in time, of bonds having equal credit quality, but differing maturity ...
  6. Stop-Limit Order

    An order placed with a broker that combines the features of stop order with those of a limit order. A stop-limit order will ...
Trading Center