Put-Call Parity Definition

What Is Put-Name Parity?

The time period “put-call” parity refers to a precept that defines the connection between the worth of European put and name choices of the identical class. Put merely, this idea highlights the consistencies of those similar lessons. Put and name choices should have the identical underlying asset, strike value, and expiration date with a purpose to be in the identical class. The put-call parity, which solely applies to European choices, could be decided by a set equation.

Understanding Put-Name Parity

As famous above, the put-call parity is an idea that applies to European choices. These choices are of the identical class, which means they’ve the underlying asset, strike value, and expiration date. As such, the precept would not apply to American choices, which could be exercised at any time earlier than the expiration date.

Put-call parity states that concurrently holding a brief European put and lengthy European name of the identical class will ship the identical return as holding one ahead contract on the identical underlying asset, with the identical expiration, and a ahead value equal to the choice’s strike value.

If the costs of the put and name choices diverge in order that this relationship doesn’t maintain, an arbitrage alternative exists. Which means subtle merchants can theoretically earn a risk-free revenue. Such alternatives are unusual and short-lived in liquid markets.

The equation that expresses put-call parity is:

$beginaligned&C + PV(x) = P + S &textbfwhere: &C = textPrice of the European name choice &PV(x) = textPresent worth of the strike value (x), &textdiscounted from the worth on the expiration &textdate on the risk-free charge &P = textPrice of the European put &S = textSpot value or the present market worth &textof the underlying asset endaligned$​C+PV(x)=P+Sthe place:C=Value of the European name choicePV(x)=Current worth of the strike value (x),discounted from the worth on the expirationdate at the risk-free chargeP=Value of the European putS=Spot value or the present market worthof the underlying asset​

Particular Concerns

When one facet of the put-call parity equation is bigger than the opposite, this represents an arbitrage alternative. You possibly can promote the dearer facet of the equation and purchase the cheaper facet to make, for all intents and functions, a risk-free revenue.

In observe, this implies promoting a put, shorting the inventory, shopping for a name, and shopping for the risk-free asset (TIPS, for instance). In actuality, alternatives for arbitrage are short-lived and tough to seek out. As well as, the margins they provide could also be so skinny that an unlimited quantity of capital is required to benefit from them.

Put-Name Parity and Arbitrage

Within the two graphs above, the y-axis represents the worth of the portfolio, not the revenue or loss, as a result of we assume that merchants give choices away. However they do not and the costs of European put and name choices are in the end ruled by put-call parity. In a theoretical, completely environment friendly market, the costs for European put and name choices could be ruled by the equation that we famous above:

$beginaligned&C + PV(x) = P + S endaligned$​C+PV(x)=P+S​

As an instance that the risk-free charge is 4% and that TCKR inventory trades at $10. Let’s proceed to disregard transaction charges and assume that TCKR doesn’t pay a dividend. For TCKR choices expiring in a single 12 months with a strike value of $15 we’ve got:

$beginaligned&C + ( 15 div 1.04 ) = P + 10 &4.42 = P – C endaligned$​C+(15÷1.04)=P+104.42=P−C​

On this hypothetical market, TCKR places ought to commerce at a $4.42 premium to their corresponding calls. With TCKR buying and selling at simply 67% of the strike value, the bullish name appears to have the longer odds, which makes intuitive sense. As an instance this isn’t the case, although, for no matter purpose, the places are buying and selling at $12, the calls at $7.

Say that you simply buy a European name choice for TCKR inventory. The expiration date is one 12 months from now, the strike value is $15, and buying the decision prices you $5. This contract offers you the appropriate however not the duty to buy TCKR inventory on the expiration date for $15, regardless of the market value is likely to be.

If one 12 months from now, TCKR trades at $10, you’ll not train the choice. If, however, TCKR is buying and selling at $20 per share, you’ll train the choice, purchase TCKR at $15 and break-even, because you paid $5 for the choice initially. Any quantity TCKR rises above $20 is pure revenue, assuming zero transaction charges. 

$beginaligned&7 + 14.42 < 12 + 10 &21.42 textfiduciary name < 22 textprotected put endaligned$​7+14.42<12+1021.42 fiduciary name<22 protected put​

Protecting Put

One other option to think about put-call parity is to check the efficiency of a protecting put and a fiduciary name of the identical class. A protecting put is an extended inventory place mixed with an extended put, which acts to restrict the draw back of holding the inventory.

Fiduciary Name

A fiduciary name is an extended name mixed with money equal to the current worth (adjusted for the low cost charge) of the strike value; this ensures that the investor has sufficient money to train the choice on the expiration date. Earlier than, we mentioned that TCKR places and calls with a strike value of $15 expiring in a single 12 months each traded at $5, however let’s assume for a second that they commerce without cost.

Put-Name Parity Instance

Say you additionally promote (or “write” or “quick”) a European put choice for TCKR inventory. The expiration date, strike value, and price of the choice are the identical. You obtain $5 from writing the choice, and it’s not as much as you whether or not or to not train the choice since you do not personal it. The customer purchases the appropriate, however not the duty, to promote you TCKR inventory on the strike value. This implies you’re obligated to take that deal, no matter TCKR’s market share value.

So if TCKR trades at $10 a 12 months from now, the customer sells you the inventory at $15. You each break even—you already made $5 from promoting the put, making up your shortfall, whereas the customer already spent $5 to purchase it, consuming up their achieve. If TCKR trades at $15 or above, you make $5 and solely $5, for the reason that different social gathering would not train the choice. If TCKR trades under $10, you lose cash—as much as $10, if TCKR goes to zero.

The revenue or loss on these positions for various TCKR inventory costs is highlighted within the graph straight above this part. Discover that in case you add the revenue or loss on the lengthy name to that of the quick put, you make or lose precisely what you’ll have in case you had merely signed a ahead contract for TCKR inventory at $15, expiring in a single 12 months. If shares go for lower than $15, you lose cash. In the event that they go for extra, you achieve. Once more, this situation ignores all transaction charges.

One other option to think about put-call parity is to check the efficiency of a protecting put and a fiduciary name of the identical class. A protecting put is an extended inventory place mixed with an extended put, which acts to restrict the draw back of holding the inventory.

A fiduciary name is an extended name mixed with money equal to the current worth (adjusted for the low cost charge) of the strike value; this ensures that the investor has sufficient money to train the choice on the expiration date. Earlier than, we mentioned that TCKR places and calls with a strike value of $15 expiring in a single 12 months each traded at $5, however let’s assume for a second that they commerce without cost.