Put/Call Parity

 

Put/call parity is a captivating, noticeable reality arising from the options markets. By gaining an understanding of put/call parity, one can begin to better understand some mechanics that professional traders may use to value options, how supply and demand impacts option prices and how all option values (at all the available strikes and expirations) on the same underlying security are related. Prior to learning the relationships between call and put values, we’ll review a couple of items.

Arbitrage

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 selling pressure in the higher priced market will drive XYZ’s price down. Conversely, the buying of XYZ in the lower price market will drive XYZ’s price higher. 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. The “no-arbitrage principle” indicates that any rational price for a financial instrument must exclude arbitrage opportunities. 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.

Defining Derivatives

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.

For example, a $50 call option on XYZ expiring June of the current year must be priced at the same or lower price than the September XYZ $50 call option for the current year. 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.

Transaction 1: Buy September call for $3.00
Transaction 2: Sell June call for $3.50
Transaction 3: Assigned on June call, receive $50/share, short 100 XYZ
Transaction 4: Exercise September call, pay $50/share, flatten existing short position
Result: $0.50 per share profit
*note XYZ is a non-dividend paying stock*

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. Note too that if XYZ falls below the $50 strike price, it does not impact the trade as a result of the $0.50 credit received when the positions were opened. If both options expire worthless, the net result is still a profit of $0.50.

This example shows why a $50 XYZ call option expiring this June, must trade at the same or lower premium than a $50 call option expiring the following September. 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. For example, if an XYZ June $50 call was trading at $4.00 and the June $45 call was trading at $3.00, a rational investor would sell the $50 call, buy the $45 call, generating a $1 per share credit and pocket a profit.

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:

  1. Long Stock
  2. Short Stock
  3. Long Call
  4. Short Call
  5. Long Put
  6. Short Put
Original Position = Synthetic   Equivalent
Long Stock = Long Call + Short Put
Short Stock = Short Call + Long Put
Long Call = Long Stock + Long Put
Short Call = Short Stock + Short Put
Long Put = Short Stock + Long Call
Short Put = Long Stock + Short Call

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.

To replicate the gain/loss characteristics of a long stock position, one would purchase a call and write a put simultaneously. 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). If we were to look at the gain/loss characteristics of a long stock position, the gain/loss characteristics of a combined short put/long call position would be identical. 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.

Let’s take a closer look at a synthetic long stock position. ABC is trading at $49 per share. The $50 put is trading at $2.00 and the $50 call is trading at $1.00 – the call and the put have the same expiration - for purposes of this example the actual expiration does not matter. An investor can purchase the call and write the put. In doing so, the investor generated a $1.00 credit per share.

If assigned on the short put, the put writer pays the strike price of $50 (a total of $5,000 for one put) and receives 100 shares of ABC. If the investor elects to exercise the call, they would pay $50 per share and (similar to the assigned put) receives 100 shares. But remember the investor took in a credit of $1 when they entered the synthetic position, thus the “effective” purchase price of the stock is $50 (paid when assignment or exercise occurs) less $1 credit from initial trade equals $49/share – the price of ABC in the market.

Example:
ABC = $49/share
ABC $50 put = $2.00
ABC $50 call = $1.00

The relationship of put/call parity can now be seen. 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.

Change the ABC price to $49.50 and leave the call and put premiums the same. The synthetic long stock position can be established for $49/share - $0.50 less than the market price of ABC. Rational investors would buy calls and sell puts instead of purchasing stock (and maybe even short the stock to offset the position completely and lock in a $.50 profit – technically called a “reversal”). 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). This would occur until the put/call parity relationship falls back in line, thus diminishing the opportunity for arbitrage.

Bid/ask spreads and other transaction costs impact the ability of investors to implement the above trades. Other factors too will change the relationship – notably dividends and interest rates. Options are priced using the “no arbitrage principle”. The previous examples show how the markets participants would react to a potential arbitrage opportunity and what the impact may be on prices.

All this leads us to the final put/call parity equation-assuming interest rates and dividends equal zero: +stock = +call – put where “+” is long and “-“ is short; or stated as written: stock price equals long call premium less the put premium; any credit received or debit paid is added to or subtracted from the strike price of the options. 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.

Impact of Dividends & Interest Rates

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.

For a professional trader looking to remain “delta neutral” and not be impacted by market movements the offset to a short call is long stock. 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.

For example, an investor is looking to sell a one-year call option on a $75 stock at the $75 strike price. If the one-year interest rate is 5%, the cost of borrowing $7,500 for one year is: $7,500 x 5% = $375. Therefore, the call option on this non-dividend paying stock would have to be sold (at a minimum) for $3.75 just to cover the cost of carrying the position for one year.

Dividends reduce the cost of borrowing – if an investor borrows $7,500 (or some percentage thereof) to purchase 100 shares of a $75 stock and receives a $1/share dividend, he pays less interest on the money borrowed (assuming the $100 from the dividend is applied to the loan). 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.

Our position simulator and pricing calculators can help evaluate these relationships:

For additional insight into options pricing, our Pricing course may help.

Put/Call Parity Formula - Non-Dividend Paying Security

c = S + p – Xe–r(T– t)
p = c - S + Xe–r(T– t)


c = call value
S = current stock price
p = put price
X = exercise price of option
e = Euler’s constant – approximately 2.71828 (exponential function on a financial calculator)
r = continuously compounded risk free interest rate
T-t = term to expiration measured in years
    T = Expiration date
    t = Current value date

Put-call parity: The relationship that exists between call and put prices of the same underlying, strike price and expiration month.

Conversion: 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.

Reverse Conversion: 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.

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.

Synthetic positions

- Interest rates and dividends equal zero
- Strike prices and expirations the same for call and puts

Synthetic Long Stock = Long Call and Short Put (same expiration & strike)
*Synthetic Long Call = Long Put and Long Stock
Synthetic Long Put = Long Call and Short Stock

Synthetic Short Stock = Long Put and Short Call (same expiration & strike)
Synthetic Short Call = Short Put and Short Stock
*Synthetic Short Put = Short Call and Long Stock

*Most covered call writers (and protective put buyers) don't realize they are trading synthetic positions!

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