Put-Call Parity

What Is Put-Call Parity?

The term "put-call" parity alludes to a principle that characterizes the relationship between the price of European put and call options of a similar class. Put essentially, this concept features the textures of these equivalent classes. Put and call options must have the equivalent underlying asset, strike price, and expiration date to be in a similar class. The put-call parity, which just applies to European options, can be determined by a set equation.

Grasping Put-Call Parity

As indicated over, the put-call parity is a concept that applies to European options. These options are of similar class, meaning they have the underlying asset, strike price, and expiration date. In that capacity, the principle doesn't make a difference to American options, which can be exercised whenever before the expiration date.

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

Assuming the prices of the put and call options veer so this relationship doesn't hold, a arbitrage opportunity exists. This means that sophisticated traders can theoretically earn a risk-free profit. Such opportunities are exceptional and short-lived in liquid markets.

The equation that communicates put-call parity is:

C + PV(x) = P + S

where:

C = price of the European call option

PV(x) = the current value of the strike price (x), discounted from the value on the expiration date at the risk-free rate

P = price of the European put

S = spot price or the current market value of the underlying asset

The put-call parity concept was presented by economist Hans R. Stoll in his December 1969 paper "The Relationship Between Put and Call Option Prices," which was distributed in The Journal of Finance.

Special Considerations

At the point when one side of the put-call parity equation is greater than the other, this addresses a arbitrage opportunity. You can sell the more costly side of the equation and buy the less expensive side to make, in every way that really matters, a risk-free profit.

In practice, this means selling a put, shorting the stock, buying a call, and buying the risk-free asset (TIPS, for instance). In reality, opportunities for arbitrage are short-lived and hard to track down. Also, the margins they offer might be meager to such an extent that a tremendous amount of capital is required to exploit them.

Put-Call Parity and Arbitrage

In the two graphs over, the y-axis addresses the value of the portfolio, not the profit or loss, since we assume that traders part with options. Be that as it may, they don't and the prices of European put and call options are at last represented by put-call parity. In a hypothetical, impeccably efficient market, the prices for European put and call options would be represented by the equation that we noted previously:

C + PV(x) = P + S

Suppose that the risk-free rate is 4% and that TCKR stock trades at $10. We should keep on disregarding transaction fees and assume that TCKR doesn't pay a dividend. For TCKR options lapsing in one year with a strike price of $15 we have:

C + (15 \u00f7 1.04) = P + 10

4.42 = P - C

In this theoretical market, TCKR puts ought to trade at a $4.42 premium to their comparing calls. With TCKR trading at just 67% of the strike price, the bullish call appears to have the longer chances, which seems OK. Suppose this isn't the case, however, out of the blue, the puts are trading at $12, the calls at $7.

Say that you purchase an European call option for TCKR stock. The expiration date is one year from now, the strike price is $15, and purchasing the call costs you $5. This contract gives you the right however not the obligation to purchase TCKR stock on the expiration date for $15, regardless.

Assuming one year from now, TCKR trades at $10, you won't exercise the option. If, then again, TCKR is trading at $20 per share, you will exercise the option, buy TCKR at $15 and break-even, since you paid $5 for the option initially. Any amount TCKR transcends $20 is pure profit, assuming zero transaction fees.

7 + 14.42 < 12 + 10

21.42 fiduciary call < 22 protected put

Protective Put

One more method for envisioning put-call parity is to compare the performance of a protective put and a fiduciary call of a similar class. A protective put is a long stock position combined with a long put, which acts to limit the downside of holding the stock.

Fiduciary Call

A fiduciary call is a long call combined with cash equivalent to the current value (adjusted for the discount rate) of the strike price; this guarantees that the investor has sufficient cash to exercise the option on the expiration date. Before, we said that TCKR puts and calls with a strike price of $15 lapsing in one year both traded at $5, yet we should assume briefly that they trade for free.

Put-Call Parity Example

Let's assume you likewise sell (or "express" or "short") an European put option for TCKR stock. The expiration date, strike price, and cost of the option are something very similar. You receive $5 from composing the option, and it no longer has anything to do with you the decision about whether to exercise the option since you don't possess it. The buyer purchases the right, yet not the obligation, to sell you TCKR stock at the strike price. This means you are committed to take that deal, whatever TCKR's market share price.

So on the off chance that TCKR trades at $10 per year from now, the buyer sells you the stock at $15. You both break even — you previously made $5 from selling the put, making up your shortfall, while the buyer previously burned through $5 to buy it, gobbling up their gain. Assuming TCKR trades at $15 or above, you make $5 and just $5, since the other party doesn't exercise the option. Assuming TCKR trades below $10, you lose money — up to $10, in the event that TCKR goes to zero.

The profit or loss on these positions at various TCKR stock costs is featured in the graph straight over this section. Notice that on the off chance that you add the profit or loss on the long call to that of the short put, you make or lose precisely exact thing you would have assuming you had basically marked a forward contract for TCKR stock at $15, lapsing in one year. Assuming shares go for under $15, you lose money. On the off chance that they go for more, you gain. Again, this scenario overlooks all transaction fees.

Features

This concept says the price of a call option infers a certain fair price for the comparing put option with a similar strike price and expiration and vice versa.
Put-call parity doesn't make a difference to American options since you can exercise them before the expiry date.
In the event that the put-call parity is disregarded, arbitrage opportunities emerge.
Put-call parity shows the relationship that needs to exist between European put and call options that have a similar underlying asset, expiration, and strike prices.
You can determine the put-call party by utilizing the formula C + PV(x) = P + S.

FAQ

Why Is Put-Call Parity Important?

Put-call parity permits you to work out the estimated value of a put or a call relative to its different parts. Assuming the put-call parity is disregarded, implying that the prices of the put and call options wander so this relationship doesn't hold, an arbitrage opportunity exists. Albeit such opportunities are remarkable and short-lived in liquid markets, sophisticated traders can theoretically earn a risk-free profit. Besides, it offers the flexibility to make synthetic positions.

What's the Formula for Put-Call Parity?

Put-call parity states that the simultaneous purchase and sale of an European call and put option of a similar class (same underlying asset, strike price, and expiration date) is indistinguishable from buying the underlying asset right at this point. The inverse of this relationship would likewise be true.> Call Option Price + PV(x) = Put Option Price + Current Price of Underlying Asset-or-> Current Price of Underlying Asset = Call Option Price - Put Option Price + PV(x)where: PV(x) = the current value of the strike price (x), discounted from the value on the expiration date at the risk-free rate

How Are Options Priced?

An option's price is the sum of its intrinsic value, which is the difference between the current price of the underlying asset and the option's strike price, and time value, which is straightforwardly connected with the time left until that option's expiry.Nowadays, an option's price is determined by utilizing mathematical models, similar to the notable Black-Scholes-Merton (BSM). Subsequent to inputting the strike price of an option, the current price of the underlying instrument, time to expiration, risk-free rate, and volatility, this model will let out the option's fair market value.