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# Understanding Put-Call Parity

Alan Grigoletto
August 29, 2016
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Investors will often hear the term put-call parity without fully comprehending its meaning or how it keeps options prices in line. Put-Call Parity states that for a given underlying price with the same strike prices and the same expiry for both puts and calls, the value of a call at a given price implies a value for the put, and the relationship of the two holds in the reverse. It is often stated that put-call parity only holds for European-style options as there is no early exercise. However, in practice, market makers on the Canadian Derivatives Exchange keep prices in line using the same parity principles for all American-style options. It is arbitrage* that maintains the parity between call, put, and the underlying stock.

One way to visualize this concept is to imagine a teeter-totter with call on one side, put on the other, and the base or fulcrum being the underlying security.

As the call goes up in value, the put goes down in value and vice versa.

The easiest way to understand this concept is to look at the choice between purchasing the stock outright or buying a call and selling a put in combination (combo). The long exposure to the market is the same for each choice. The basic formula states that the call – put = stock price – strike price + (Interest – Dividend).

When purchasing the stock alone we have to finance the position, pay interest, and the receipt of any dividends acts to reduce that borrowing cost. Should we instead choose to purchase the call and sell the put, then we do not receive any dividends and our financial costs are negligible.

Put-Call Parity allows for the creation of many synthetic positions.

We just examined the combo above which is also known as Synthetic Long Stock. Flipping the position around, selling the call and purchasing the put creates Synthetic Short Stock. The combination of the calls and puts at the same strike should always equal 100 deltas. 100 deltas will be equivalent to ±100 shares of the underlying.

A Canadian investor may also wish to create any of the following synthetic positions:

Just as we defined above, market makers use the three-sided relationship of stock, call, and put to maintain quotes and prices.

One method for maintaining this relationship and a favorite of market makers is the Conversion.

The Conversion is long stock combined with a long put and a short call. We observe immediately that the trade is both long and synthetically short 100 shares, with the same strike calls and puts equal to -100 shares, while simultaneously being long 100 physical shares. As identified above this position can also be viewed as:

• Short call+synthetic long call (long put and long stock)
• Long put+synthetic short put (short call and long stock)

We calculate the cost to carry the Conversion until expiration as Strike Price x Interest Rate x Time to Expiration – Dividend Received. Note the carrying cost is based on the strike price, not the stock purchase price!

Let’s look at the following example with At-The-Money strikes:

XYZ Stock is trading at C\$50

Interest rate 3%

60 days until expiration

Dividend of 20 cents

50 strike put priced at \$1.18

Cost to carry position until expiration \$50.00 strike x .03 x 60/365 – \$0.20 = \$0.05

• What should 50.00 call trade for?
• Call = put cost + carrying cost
• \$1.18 + \$0.05 = \$1.23

The market maker has calculated that these are the fair values of the calls and puts. Should the call be trading at \$1.30, he would use the Conversion to buy the stock, buy the put, and sell the call. No matter what the underlying does, the market maker makes a profit at expiration of 7 cents for as many times as the arbitrage is available. Although the Conversion is often considered a riskless trade, there are potential risks involved should there be a change in dividend dates, the amount of the dividend, or in interest rates.

At expiration, a Conversion with European-style options will be closed at the strike price with either the 50 call or the 50 put being In-The-Money.

At expiration – XYZ above \$50.00

• 00 short call assigned – XYZ sold at \$50.00
• 00 long put out-of-the-money – no value
• Position liquidated @\$50.00 (received for stock)
• No position remaining

At expiration – XYZ below \$50.00

• 00 short call out-of-the-money – no value
• 00 long put exercised – XYZ sold at \$60.00
• Position liquidated @\$50.00 (received for stock)
• No position remaining

Final thoughts: Canadian investors can use these formulas in the option calculator to better understand the concepts of put-call parity to access how closely displayed prices are to the calculations.

https://www.m-x.ca/marc_options_calc_en.php

Also, note that the Synthetic Short Put described above is the same as the widely used Covered Call!

*Arbitrage is the opportunity to profit from price differences in identical or similar instruments. It holds the prices of these various instruments in line.

In today’s electronic and highly efficient markets, arbitrage is primarily the role of market makers because they have the combination of speed and low cost to profit from small discrepancies in pricing.

Alan Grigoletto

CEO

Grigoletto Financial Consulting