# Put call parity formula continuous dividend. The above shows that the value of an option on a dividend-paying asset with current price. S equals the value of an option on a non-dividend paying asset having current price S* = Se-qτ. Hence, we can value a European call or put option on a dividend-paying asset using the. Black-Scholes formula, but replacing S with S*.

## Put call parity formula continuous dividend. The most simple formula for put/call parity is Call - Put = Stock - Strike. So, for example, if stock XYZ is trading at \$60 and you checked option prices at the \$55 strike, you might see the call at \$7 and the put at \$2 (\$7 - \$2 = \$60 - \$55). That's an example of put/call parity. If the call was trading higher, you could sell the call, buy.

Let us begin by defining arbitrage and how arbitrage opportunities serve the markets. Arbitrage is, generally speaking, the opportunity to profit arising from price variances on one security in different markets.

For example, if an investor can buy XYZ in one market and simultaneously sell XYZ on another market for a higher price, the trade would result in a profit with little risk. The buying and selling pressure in the two markets will move the price difference between the markets towards equilibrium, quickly eliminating any opportunity for arbitrage. That is, we can determine the value of a financial instrument if we assume arbitrage to be unavailable.

Using this principle, we can value options under the assumption that no arbitrage opportunities exist. When trying to understand arbitrage as it relates to stock and options markets, we often assume no restrictions on borrowing money, no restrictions on borrowing shares of stock, and no transactions costs.

In the real world, such restrictions do exist and, of course, transaction costs are present which may reduce or eliminate any perceived arbitrage opportunity for most individual investors. For investors with access to large amounts of capital, low fee structures and few restrictions on borrowing, arbitrage may be possible at times, although these opportunities are fairly rare.

Options are derivatives; they derive their value from other factors. In the case of stock options, the value is derived from the underlying stock, interest rates, dividends, anticipated volatility and time to expiration. There are certain factors that must hold true for options under the no arbitrage principle. If the September call is less expensive, investors would buy the September call, sell the June call and guarantee a profit. Note that XYZ is a non-dividend paying stock, the options are American exercise style and interest rates are expected to be constant over the life of both options.

Here is an example of why a longer term option premium must be equal to or greater than the premium of the short term option. In our interest free, commission free, hypothetical world, the timing of the assignment does not matter, however the exercise would only occur after an assignment.

If the June premium was higher like in the example , investors would sell the June call, causing the price to decline and buy the September, causing the price of that option to rise. These trades would continue until the price of the June option was equal to or below the price of the September option. A similar relationship can be seen between two different strike prices but the same expiration. With stock and options, there are six possible positions from three securities when dividends and interest rates are equal to zero — stock, calls and puts:.

Synthetic relationships with options occur by replicating a one part position, for example long stock, by taking a two part position in two other instruments. Similar to how synthetic oil is not extracted from the fossil fuels beneath the ground.

Rather synthetic oil is manufactured with chemicals and is man-made. Similarly, synthetic positions in stocks and options are generated from positions in other instruments. The call and put would have the same strike price and the same expiration. By taking these two combined positions long call and short put , we can replicate a third one long stock.

Remember the put premiums typically increase when the stock prices decline which negatively impacts the put writer; and of course the call premiums typically increase as the stock price increases, positively impacting the call holder. Therefore, as the stock rises, the synthetic position also increases in value; as the stock price falls, the synthetic position also falls. An investor can purchase the call and write the put.

In the previous example, if the relationship did not hold, rational investors would buy and sell the stock, calls and puts, driving the prices of the calls, puts and stock up or down until the relationship came back in line.

Eventually the buying of the calls would drive the price up and the selling of the puts would cause the put premiums to decline and any selling of the stock would cause the stock price to decline also. Other factors too will change the relationship — notably dividends and interest rates.

The previous examples show how the markets participants would react to a potential arbitrage opportunity and what the impact may be on prices. The strike price of the call and put are the same. This assumes the strike prices and the expirations are the same on the call and put with interest rates and dividends equal to zero.

The next logical question is how ordinary dividends and interest rates impact the put call relationship and option prices. Interest is a cost to an investor who borrows funds to purchase stock and a benefit to investors who receive and invests funds from shorting stock typically only large institutions receive interest on short credit balances.

Higher interest rates thus tend to increase call option premiums and decrease put option premiums. Long stock requires capital. The cost of these funds suggests the call seller must ask for higher premiums when selling calls to offset the cost of interest on money borrowed to purchase the stock. Conversely, the offset to a short put is short stock.

As a short stock position earns interest for some large investors at least , the put seller can ask for a lower premium as the interest earned decreases the cost of funds. This reduces the cost of carry — as the cost of carrying the stock position into the future is reduced from the dividend received by holding the stock. Opposite of interest rates, higher dividends tend to reduce call option prices and increase put option prices.

Professional traders understand the relationships among calls, puts, interest rates and dividends, among other factors. For individual investors, understanding the early exercise feature of American style options is essential. When writing options, intuition as to when assignment may occur and when holding options understanding when to exercise at an opportunistic time is very important.

For dividend paying stocks, exercise and assignment activity occurs more frequently just before call exercises and after put exercises an ex-dividend date. The relationship that exists between call and put prices of the same underlying, strike price and expiration month. An investment strategy in which a long put and short call with the same strike and expiration is combined with a long stock position.

This is also referred to as conversion arbitrage. An investment strategy in which a long call and short put with the same strike and expiration is combined with a short stock position. This is also referred to as reversal arbitrage. Purchase or sale of instruments in one market versus the purchase or sale of similar instruments in another market in an effort to profit from price differences.

Options arbitrage uses stock, cash and options to replicate other options. Synthetic options imitate the risk reward profile of "real" options using a combination of call and put options and the underlying stock. This web site discusses exchange-traded options issued by The Options Clearing Corporation. No statement in this web site is to be construed as a recommendation to purchase or sell a security, or to provide investment advice. Options involve risk and are not suitable for all investors.

Prior to buying or selling an option, a person must receive a copy of Characteristics and Risks of Standardized Options. Copies of this document may be obtained from your broker, from any exchange on which options are traded or by contacting The Options Clearing Corporation, One North Wacker Dr.

Please view our Privacy Policy and our User Agreement. What is an Index? Putting It All Together. Arbitrage Let us begin by defining arbitrage and how arbitrage opportunities serve the markets. Defining Derivatives Options are derivatives; they derive their value from other factors.

Synthetic Relationships With stock and options, there are six possible positions from three securities when dividends and interest rates are equal to zero — stock, calls and puts: Our position simulator and pricing calculators can help evaluate these relationships: Email an options professional now. Chat with Options Professionals Questions about anything options-related? Chat with an options professional now. More Info Register Now.

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