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## What Is Put-call Parity?

The term "put-call" parity refers to lớn aprinciple that defines the relationship between the price of European put & điện thoại tư vấn options of the same class. Put simply, this concept highlights the consistencies of these same classes. Put and call options must have the same underlying asmix, strike price, and expiration date in order to be in the same class. The put-gọi parity, which only applies lớn European options, can be determined by a phối equation.

Put-call parity shows the relationship that has to exist between European put và điện thoại tư vấn options that have the same underlying asphối, expiration, và strike prices.This concept says the price of a Gọi option implies a certain fair price for the corresponding put option with the same strike price & expiration and vice versa. Put-điện thoại tư vấn parity doesn't apply khổng lồ American options because you can exercise them before the expiry date. If the put-Hotline parity is violated, then arbitrage opportunities arise. You can determine the put-Call các buổi tiệc nhỏ by using the formula C + PV(x) = P + S.

## Understanding Put-hotline Parity

As noted above, the put-Gọi parity is a concept that applies to European options. These options are of the same class, meaning they have sầu the underlying asphối, strike price, và expiration date. As such, the principle doesn"t apply to American options, which can be exercised at any time before the expiration date.

Put-điện thoại tư vấn parity states that simultaneously holding a short European put and long European Hotline of the same class will deliver the same return as holding one forward contract on the same underlying asphối, with the same expiration, and a forward price equal khổng lồ the option"s strike price.

If the prices of the put and Gọi options diverge so that this relationship does not hold, an arbitrage opportunity exists. This means that sophisticated traders can theoretically earn a risk-không tính phí profit. Such opportunities are uncommon and short-lived in liquid markets.

PV(x) = the present value of the strike price (x), discountedfrom the value on the expiration date at the risk-free rate

The put-Gọi parity concept was introduced by economist Hans R. Stoll in his December 1969 paper "The Relationship Between Put & call Option Prices," which was published in *The Journal of Finance*.

## Special Considerations

When one side of the put-hotline parity equation is greater than the other, this representsan arbitrage opportunity. You can sell the more expensive side of the equation và buy the cheaper side to lớn make, for all intents và purposes, a risk-không lấy phí profit.

In practice, this means selling a put, shorting the stochồng, buying a Gọi, & buying the risk-không tính tiền asphối (TIPS, for example). In reality, opportunities for arbitrage are short-lived & difficult lớn find. In addition, the margins they offer may be so thin that an enormous amount of capital is required to take advantage of them.

### Put-call Parity & Arbitrage

In the two graphs above sầu, the*y-*axis represents the value of the portfolio, not the profit or loss, because we assume that traders give options away. But they don"t and the prices of European put and gọi options are ultimately governed by put-Hotline parity. In a theoretical, perfectly efficient market, the prices for European put and điện thoại tư vấn options would be governed by the equation that we noted above:

Let"s say that the risk-miễn phí rate is 4% & that TCKR stochồng trades at $10. Let"s continue khổng lồ ignore transaction fees and assume that TCKR does not pay a dividend. For TCKR options expiring in one year with a strike price of $15 we have:

In this hypothetical market, TCKR puts should trade at a $4.42 premium khổng lồ their corresponding calls. With TCKR trading at just 67% of the strike price, the bullish Gọi seems to have sầu the longer odds, which makes intuitive sense. Let"s say this is not the case, though, for whatever reason, the puts are trading at $12, the calls at $7.

Say that you purchase a European gọi option for TCKR stoông chồng. The expiration date is one year from now, the strike price is $15, và purchasing the hotline costs you $5. This contract gives you the right but not the obligation khổng lồ purchase TCKR stoông xã on the expiration date for $15, whatever the market price might be.

If one year from now, TCKR trades at $10, you will not exercise the option. If, on the other h&, TCKR is trading at $đôi mươi per nói qua, you will exercise the option, buy TCKR at $15 & break-even, since you paid $5 for the option initially. Any amount TCKR rises above $đôi mươi is pure profit, assuming zero transaction fees.

### Protective Put

Another way to imagine put-gọi parity is khổng lồ compare the performance of a protective put and a fiduciary call of the same class. A protective put is a long stoông chồng position combined with a long put, which acts lớn limit the downside of holding the stochồng.

### Fiduciary Hotline

A fiduciary Hotline is a long Call combined with cash equal khổng lồ the present value (adjusted for the discount rate) of the strike price; this ensures that the investor has enough cash lớn exercise the option on the expiration date. Before, we said that TCKR puts và calls with a strike price of $15 expiring in one year both traded at $5, but let"s assume for a second that they trade for không tính tiền.

## Put-Điện thoại tư vấn Parity Example

Say you also sell (or "write" or "short") a European put option for TCKR stoông xã. The expiration date, strike price, and cost of the option are the same. You receive $5 from writing the option, & it is not up to lớn you whether or not lớn exercise the option since you don"t own it. The buyer purchases the right, but not the obligation, lớn sell you TCKR stoông xã at the strike price. This means you are obligated khổng lồ take that giảm giá khuyến mãi, whatever TCKR"s market share price.

So if TCKR trades at $10 a year from now, the buyer sells you the stoông xã at $15. You both break even—you already made $5 from selling the put, making up your shortfall, while the buyer already spent $5 to buy it, eating up their gain. If TCKR trades at $15 or above, you make $5 & only $5, since the other party doesn"t exercise the option. If TCKR trades below $10, you thua kém money—up khổng lồ $10, if TCKR goes to lớn zero.

The profit or loss on these positions for different TCKR stoông xã prices is highlighted in the graph directly above this section. Notice that if you add the profit or loss on the long gọi to that of the short put, you make or thua thảm exactly what you would have if you had simply signed a forward contract for TCKR stoông chồng at $15, expiring in one year. If shares go for less than $15, you thất bại money. If they go for more, you gain. Again, this scenario ignores all transaction fees.

Another way to lớn imagine put-điện thoại tư vấn parity is khổng lồ compare the performance of a protective put và a fiduciary gọi of the same class. A protective sầu put is a long stock position combined with a long put, which acts to lớn limit the downside of holding the stoông chồng.

A fiduciary Call is a long call combined with cash equal to the present value (adjusted for the discount rate) of the strike price; this ensures that the investor has enough cash to lớn exercise the option on the expiration date. Before, we said that TCKR puts and calls with a strike price of $15 expiring in one year both traded at $5, but let"s assume for a second that they trade for free.

Put-Gọi parity allows you to lớn calculate the approximate value of a put or a Gọi relative sầu to lớn its other components. If the put-điện thoại tư vấn parity is violated, meaning that the prices of the put & gọi options diverge so that this relationship does not hold, an arbitrage opportunity exists. Although such opportunities are uncommon and short-lived in liquid markets, sophisticated traders can theoretically earn a risk-không tính tiền profit. Furthermore, it offers the flexibility khổng lồ create synthetic positions.

Put-hotline parity states that the simultaneous purchase & sale of a European Hotline & put option of the same class (same underlying asphối, strike price, & expiration date) is identical khổng lồ buying the underlying asmix right now. The inverse of this relationship would also be true.

Call Option Price + PV(x) = Put Option Price + Current Price of Underlying Asset-or-

Current Price of Underlying Asset = Điện thoại tư vấn Option Price - Put Option Price + PV(x)where: PV(x) = the present value of the strike price (x), discounted from the value on the expiration date at the risk-không lấy phí rate

An option's price is the sum of its intrinsic value, which is the difference between the current price of the underlying asmix and the option's strike price, và time value, which is directly related lớn the time left until that option's expiry.

Nowadays, an option's price is determined by using mathematical models, lượt thích the well-known Black-Scholes-Merton (BSM). After inputting the strike price of an option, the current price of the underlying instrument, time to expiration, risk-miễn phí rate, & volatility, this mã sản phẩm will spit out the option's fair market value.

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