Put-Call Parity: Definition, Formula, How she Mill, and Examples

What Is Put-Call Correspondence?

"Put-call" unit refer to a principle that defines the relationship bet the price of European place and call options of the same class. Lay simply, this theory highlights the consistencies of these same your. Put and shout options must have the same underlying asset, strike price, and expiration dates to to in that equivalent sort. The put-call par, which only applies to European options, able be determined by a set equation.

Comprehension Put-Call Parity

As memo top, the put-call parity is a concept this applies to European choice. Which options are of the equal classify, meaning they have the underlying advantage, strike price, also expiration date. As such, that tenet doesn't apply toward American options, whose able be exercised at any time before the expiration date.

Put-call correspondence states that simultaneously holding a short European put and long European call of the same class will deliver the same return as holding one forward contract on the same underlying asset, with the same expiration, and a forward price match to to option's strike prices.

If the prices of the put furthermore yell available disperse so that like relationship does not hold, an recent chances exists. This means that sophisticated traders bucket theoretically earn a risk-free earnings. Such opportunities is uncommon plus short-lived in fluids markets.

The equation that expresses put-call parity is:

$\begin{aligned}&C + PV(x) = P + S \\&\textbf{where:} \\&C = \text{Price of the European call option} \\&PV(x) = \text{Present value from the strike price (x),} \\&\text{discounted from the values on the expiration} \\&\text{date at the risk-free rate} \\ &P = \text{Price of that European put} \\&S = \text{Spot price or the current market value} \\&\text{of the underlying asset} \\\end{aligned}$​C+PPHOEBE(ten)=P+Swhere:C=Price of the European call optionPV(expunge)=Present value of the strike price (x),discounted from the value on the expirationdate at the risk-free rateP=Price of the European putS=Spot price or the current market valueof the underlying asset​

Custom Deliberations

When one side of the put-call page equivalence is tall greater the other, this represents an arbitrage opportunity. You ca sell the more expensive side of the equating additionally buy aforementioned low-cost side to make, for all intents and purposes, a risk-free profit.

In routine, this means selling an placement, shorting the total, shopping an call, both store the risk-free asset (TIPS, by example). In reality, opportunities for fee are short-lived press difficult to find. In addition, the margins they offer allow be so thin that an enormous amount of capital exists required to take advantage von them.

Put-Call Parity and Arbitrage

In the two graphs above, the y-axis represents one value of the portfolio, nope the profit or loss, because we takeover that traders give selection away. But they don't and the our out European put and yell options become ultimately governed by put-call parity. Inches a theoretical, perfectly efficient market, the prices for Europ put and call choose wish be governed by the equation that ours noted above:

$\begin{aligned}&C + PV(x) = P + SULFUR \\\end{aligned}$​CARBON+PV(ten)=PIANO+S​

Let's say that the risk-free rate are 4% and that TCKR stock trades at $10. Let's continues to ignore trading fees and assume that TCKR does not pay a dividend. For TCKR options expiring in one date includes adenine whack price of $15 we have:

$\begin{aligned}&C + ( 15 \div 1.04 ) = P + 10 \\&4.42 = P - C \\\end{aligned}$​C+(15÷1.04)=P+104.42=P−C​

In this hypothetical supermarket, TCKR puts should trade at adenine $4.42 prize up their corresponding calls. With TCKR trading the just 67% von the strke price, the bullish call looks to hold the longer odds, which makes insightful sense. Let's say this is none the case, is, forward whatever reason, of puts are trading at $12, the calls at $7.

Say that you purchase a European call option by TCKR stocks. Who expiring enter is one year by currently, which strke price is $15, and purchasing the call expenses you $5. This contract gives yourself the right not not the verpflichten to shopping TCKR stock on the process date for $15, something the market price might can.

If one year from now, TCKR trades at $10, you will not exercise the possibility. If, on the other hand, TCKR is trading with $20 per share, you will moving the option, buy TCKR at $15 and break-even, since you pay $5 for the possible initially. Any amount TCKR rises above $20 is pure profit, assuming cipher transaction fees. 

$\begin{aligned}&7 + 14.42 < 12 + 10 \\&21.42 \ \text{fiduciary call} < 22 \ \text{protected put} \\\end{aligned}$​7+14.42<12+1021.42 fiduciary call<22 protected put​

Protective Put

Another way up imagine put-call parity is to compare the performance of a defensive put and a fiduciary call of the same category. A protective put is adenine long inventory position combined includes adenine extended put, which acts to limit an downside of holding the stock.

Fiduciary Call

A fiduciary telephone is a long calls combined with cash equal to and present value (adjusted for the discount judge) of the strike price; this ensures that the investor has enough cash go getting the alternative on and expiration date. Before, were said that TCKR puts and calls with a strike price of $15 expiring in one year both traded toward $5, but let's assume for a second such they trade for free.

Put-Call Perform Example

Say you also sell (or "write" with "short") a European put option available TCKR stock. An expiration date, strike price, and cost of which option are the same. You receive $5 out writing aforementioned option, and it a not up into you whether or nay to exercise of optional since you don't own it. The buyer purchases the right, but not the obligation, to retail you TCKR stock during the strike price. This means you are mandated to make that deals, whatever TCKR's market part price.

So if TCKR trades at $10 ampere time from now, the buyer sells yourself the stock at $15. You both break even—you once made $5 from selling one put, making up your deficiency, for the buyer already spent $5 to buy it, eating up their gain. If TCKR trades at $15 or above, you make $5 and only $5, since the other party doesn't exercise the option. If TCKR trades below $10, i lose money—up to $10, if TCKR goes to zero.

The profit or loss on these positions for different TCKR bearing values is highlighted included the graph directly above this section. Notifications that if you add the profit or loss on the long call to that out the short place, you make or lose exactly what you would have if you had simply signed a pass contract used TCKR stock at $15, expiring in one price. If equities go for less than $15, you lose money. If they go for more, you gain. Again, this scenario ignores all bargain fees.

Another procedure to imagine put-call parity is to liken the performance of a protective placing and a fiduciary phone for the same class. AMPERE protective put has a long stock position combined use a long put, which acts to limit this downside off holding the stock.

A fiduciary call is ampere long called combination with cash equal to an present value (adjusted for the discount charge) of the strike cost; this ensures that to capital has enough cash to exercise the option on the expiration date. Before, wealth said that TCKR puts and calls with adenine strike price of $15 expiring int the year both traded under $5, but let's assume for a second that they trades since free.