Understand Put-Call Parity

Call and Put Options

When it comes to market analysis, especially while choosing the timing of investment in an available stock option, there are certain things that must be kept in mind. These factors involve a detailed and empirical analysis of the relationships between market dynamics that tend to influence investment decisions. The PCP is a no-arbitrage relationship that must hold between the prices of any trading market as such. This is a mathematical tool that has been specifically used by firms on the basis of tests based on given stock and option market prices and aimed at checking the violations of no-arbitrage relationships among these prices. More specifically, any trading firm must restrict its attention to the PCP, which is the most famous condition of cross-market efficiency between the underlying and the option market.

Broadly speaking, put/call parity implies the underlying relation between the price of a call option and matches it with same strike price. It thus says that says the price of a call option implies a certain fair price for the corresponding put option with the same strike price and expiration. The inverse of this relation also holds true. In a simpler term, PCP can be understood as a relation between the long put (i.e. a long stock position with a long put) and the fiduciary call (i.e. a long stock position with a liquid cash value that is equal to the current value).

What is Put-Call Parity?

Before heading towards the mathematical relation involved here, it is important here to know exactly what is meant by call and put. Calls and puts represent the options i.e. the instruments for deriving the stock value or the index value, as may be applicable. In terms of buying, call is the option that gives a buyer the right to buy an underlying stock on a predetermined rate from a seller at a pre-decided rate. On the other hand, a put gives the right to a seller to sell an underlying stock to a buyer on a particular date at the present or the current price. 

The difference between the put of an underlying stock or index and the call of the underlying stock or index is what is called put-call parity. This parity indicates the difference or the variance in the rights to buy or sell the underlying stocks at predetermined rates. This can be understood from the following example:

Understanding Put-Call Parity

Take for example the expected or the anticipated range of any underlying stock or index results. Suppose that this expected range for BSE in any current term is 9,300 to 10,500. While trading on this index, any seller would definitely wish to trade around this range. So, he would sell at a put of 9,300 and a call of 10,500. This difference of call and put is what is called PCP or Put Call Parity. Needless to say that this range also forms the range for profits or losses that the seller may have in anticipation for any trade deal in the current index. As can be understood here, this parity often undergoes the same fluctuations as the market trends and this impacts the buying price range as well as the selling price range of any underlying stock.

Mathematically, this is represented by the following connotation or formula:

C+Xe = P+I,

C is the call price,
P is the put price,
X is the strike price, and
I is the current level of the underlying stock or index.

Put-Call Parity Arbitrage

There has to be utmost synchronization between the options and the indices which means that quality of data is or paramount importance. This has led the empirical research to improve on the data set quality, which has become possible given the availability of high-frequency prices that ensure a high level of synchronization between the option prices and the index. In order to implement an arbitrage strategy, a (long or short) position in the underlying has to be taken. However, given that the underlying of index options is a basket of stocks, the implementation of the arbitrage strategy is less straightforward than for single stocks and, in principle, it could be practically unfeasible. Therefore, for the effective implementation of arbitration strategies, it is very much essential that all the parameters involved are taken into account thoroughly.

It must be noted that transaction costs are basically represented by commissions on transactions and bid-ask spreads on the relevant prices. In case of option markets, where the majority of commodities are listed on the basis of speculative pricing strategies, these transaction costs affect the decisions of investment on a real-time basis. This is also because of the fact that these costs reflect the operational costs of a deal and therefore must be considered before understanding the net returns on an investment deal. Given that both commissions and the bid-ask spreads are very specific to the market investigated, the existing papers take different assumptions about them. As for the commissions, to make the arbitrage strategy realistic, one has to consider both the commissions in the option market and the cost of replicating the index.

Also, it is well known that the PCP condition has to be adjusted when the underlying stock pays a dividend during the lifetime of the option.

Put Call parity therefore is a factor that decides the eventual profitability of any trading venture and therefore, the market analysts focus keenly on it. It follows from this that in order to retain only reliable and informative data to test the PCP, the market analysts have to apply some filters to the original data set. Specifically, the data set has to satisfy the following requirements, which impact the overall bearings on the PCP:

 1. Prices synchronicity
 2. Maturity and strikes matching
 3. Index adjustments for dividends
 4. Estimation of the transaction costs.

These form the various dynamics that affect the PCP in detail and allow for a shifting in the PCP across trading markets in different domains.

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