Put/Call Parity - Two Definitions

When traders use different options with the same net price (whether long or short and involving calls or puts), the degree of net debit or credit differences often defines potential profit or loss. If the two sides are at net zero difference, the position is at parity.
 

The bid/ask spread defines a series of options in terms of how popularly they are being traded, and how expensive or cheap they are. A very narrow spread is the result of a high volume of trading, but everyone has seen examples of very wide spreads. Invariably, this is found when little or no active trading is going on. This reveals that traders know the current pricing of the option is not a good deal. That usually results from a combination of moneyness and timing to expiration. The more remote the possibility of profit, the less reason there is to make a trade at all. And this is reflected in the bid/ask spread.

When the net spread is identical (at parity), what does it mean? This is seen when a bid price is reduced by the trading fee, and the offsetting ask price is increased. An example of parity:

•  Bid price 1.15 and trading fee is 0.05. Net is 1.10 ($110).
•  Ask price is 1.05 and trading fee is 0.05. Total is 1.19 ($110).

The net of both sides is identical; under this simplified definition of parity, the net debit or credit is zero.

This applies, for example, in a synthetic long stock trade. With a 60 strike, the following positions are possible:

• Buy 60 call, ask 1.25 plus trading fee = 1.30
• Sell 60 put, bid 1.35 less trading fee = 1.30

For the trader of the long synthetic, the positions starts out at parity, so there is no debit or credit. As the underlying price rises, the long call matches point for point and the short put loses time value, eventually expiring worthless. If the underlying declines, the long call loses value and the short put gains value. As this occurs, the appreciated short put represents a point-for-point net loss identical to the decline in stock. The advantage here is that the trader can roll out of the put and delay exercise.
 

The same comparison works for a synthetic short stock trade. Parity occurs when the net of a long put and a short call is zero. Parity as a trading ides matters because setting up a trade with a net credit provides flexibility for profits, but a net debit means more points must be earned before breakeven and profit become profitable. This argument works not only in synthetic trades but also in any straddle or other offsetting combination trade.
 

The more complex definition of parity considers the present value based on European style options. Exercise is allowed only at or close to expiration date. The purpose of this is to determine whether the calculated parity justifies the trade. The farther away the expiration date, the greater the impact of present value.
 

The formula for the complex version of parity is:

                ( C + X ) ÷ ( 1 + r )^t =P + U

            C = premium of the call

            X = strike

            R = interest rate

            t = time to expiration

            P = premium of the put

            U = underlying price

This formula might seem complex for what could be a small difference between he two sides. But when working with many contracts on each side of the trade, and when expiration is distant, calculated parity becomes a determining factor. It applies in arbitrage trading with options and with any strategies designed to set up riskless profits. For most options traders, especially those working with relatively short-term expiration cycles, reliance is on rapid time decay (for short trades) or time management (for long trades).

Parity defines the practicality of such combinations for calculating the likelihood of profit, a form of arbitrage in the extreme short term. The advanced method is appropriate for larger block trades and in cases where long-term options are used as part of contingent buy or sell forms of risk management. It is inaccurate when applied to American style options unless those options are kept open to last trading day, which happens only in a minority of cases. The present value calculation also fails to take dividends into account, which will distort the outcome when traders hold the underlying in addition to trading options.
 

This distinction between a spread and parity is important in another way as well. Many pricing calculations of options are based on the midpoint between bid and ask. This makes no sense. No traders – buyers or sellers – ever trader at the  midpoint. They either buy at the ask or sell at the bid. Thus, the midpoint is always an inaccurate version of “price.” Premium value should involve separate calculations, one based on big (for sellers) and another based on ask (for buyers). This misguided application of midpoint often is used in calculating probability. As a result, probability is always misrepresented as a detriment to one or both sides of a trade. A fully accurate report of probability would be based on two calculations, and only when both sides are at parity would the result be realistic.
 

Calculation of parity in its simple form is good enough for most people. When variations such as time positions are closed, whether dividends are earned, and the degree bid/ask spread and its effect on probability, are also considered, parity becomes a more complex issue. In its most simplified form, it defines the net difference or similarity between net bid and ask. For calculations of probability, the midpoint is a complete inaccurate measure in every case, whether options are at parity or widely divergent.

 
Michael C. Thomsett is a widely published author with over 80 business and investing books, including the best-selling Getting Started in Options, coming out in its 10th edition later this year. He also wrote the recently released The Mathematics of Options. Thomsett is a frequent speaker at trade shows and blogs on his websiteat Thomsett Guide as well as on Seeking Alpha, LinkedIn, Twitter and Facebook.