By Yuriy Smirnov Ph.D.

A put option is a contract between two parties. It gives the buyer (also referred to as holder) the right but not the obligation to sell to the writer (also referred to as seller) the specified amount of underlying asset at the strike price. The American-style put option enables the buyer to exercise the contract at any time before the expiration date, but the European-style put option can be exercised only on the expiration date.

**Important!** The writer of a put option is liable to exercise the contract at the request of the buyer. In turn, the buyer must pay the premium to the writer when buying the option.

The value of any put option is composed of intrinsic value and extrinsic value (also referred as time value).

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The intrinsic value is the amount that a buyer would receive if the put option were exercised immediately. The intrinsic value of a put option is determined by the price difference between the strike price and the spot price of the underlying asset.

- If the underlying asset spot price is below the strike price, a put option is in the money. The buyer will purchase the underlying asset at the spot price and immediately sell it to the writer at the strike price, benefiting from the price difference.
- If the spot price is above the strike price, a put option is out of the money. There is no sense for the buyer to exercise the contract; therefore, its intrinsic value equals zero.
- If the spot price equals the strike price, a put option is at the money, and its intrinsic value equals zero.

**Important!** The intrinsic value can be either positive or equal to zero.

The time or extrinsic value is the difference between premium and intrinsic value. It mainly depends on price volatility of the underlying asset and the length of time remaining to the expiration date.

**Important!** The time value of a put option decreases as the expiration date approaches. A put option has zero time value at the expiration date. If other things are equal, a put option with the latest expiration date will have the highest time value.

If the premium of a put option at moment *t* is known, the time value can be calculated as follows:

Premium_{t} = Intrinsic Value_{t} + Time Value_{t}

Time Value_{t} = Premium_{t} – Intrinsic Value_{t}

Let’s consider a 6-month put option for 100 shares of Tesla Inc. at a strike price of $244.50 and a premium of $28.36 per share, assuming that the spot price of the underlying stock is $227.72.

As far as the spot price is below the strike price, a put option is in the money and has an intrinsic value of $1,678.

Intrinsic value = ($244.50 - $227.72) × 100 = $1,678

Let’s calculate its time value using the formula above.

Time value = $28.36 × 100 - $1,678 = $1,158

The essential feature of a put option contract is unequal rights and liabilities for the writer and buyer. Therefore, they have different payoff diagrams.

The buyer of a put option expects that the price of the underlying asset will decline below the strike price, but the objectives of the buyer can be different.

__Speculation__. The investor is going to benefit on the price difference when the price of the underlying asset will drop.__Hedging__. If an investor holds some shares, it is possible to hedge against a price decline by buying a put option.

In the first case, the buyer expects that the price of the underlying asset will drop below the strike price. If this expectation turns out to be wrong and the price of the underlying asset increases above the strike price, the losses of the buyer will be limited by the amount of the premium paid.

In the second case, the investor protects a long position in the underlying asset against price reductions. The purchase of a put option will allow it to limit the amount of potential losses in a falling market.

**Important!** Selling a futures contract is an alternative to buying a put option. If shares are the underlying asset, the short selling can also be used as an alternative.

There is a 6-month put option for 100 shares of Walmart Inc. at a strike price of $112.35. The premium is $8.71 per share. Let’s consider two alternative scenarios for the expiration date.

__Scenario A__: spot price drops to $96.38__Scenario B__: spot price rises to $123.51

Under Scenario A, the put option expires in the money; therefore, the buyer will earn a profit of $726.

[($112.35 - $96.38) × 100 - $8.71 × 100] = $726

If instead an investor would sell a futures contract, his or her profit would be $1,597.

($112.35 - $96.38) × 100 = $1,597

**Important!** The amount of premium paid reduces the payoff of the buyer.

Under Scenario B, the expectations of the buyer turn out to be wrong. As far as a put option expires out of the money, the buyer will incur losses in the amount of the premium paid ($8.71 × 100 = $871).

If futures were sold, the investor would suffer losses of $1,160.

($112.35 - $123.51) × 100 = -$1,160

Thus, buying a put option enables one to benefit if the spot price of the underlying asset drops below the strike price and limit the losses if the spot price surges above the strike price.

The buyer’s payoff diagram is illustrated in the figure below.

A put option is in the money when the spot price of an underlying asset declines below the strike price, but the buyer will make profit if the spot price drops below the breakeven point (strike price less premium per share). In the example above, the breakeven point equals $103.64 ($112.35 - $8.71).

**Important!** If the spot price is equal to the breakeven price, both the writer and the buyer have zero profit.

If the spot price rises above the strike price, the buyer will not use the right to exercise the contract. Therefore, at any price level above the strike price, his or her losses will be limited by the amount of premium paid.

**Important!** The losses of the buyer of a put option are always limited by the amount of premium paid, but his or her profit is not limited.

Please note that selling of futures shows better performance for an investor while the spot price of the underlying asset is less than the strike price and premium per share amount. In the example above, this price equals $121.06 ($112.35 + $8.71). At any price above this point, a futures contract has a worse performance.

The writer of a put option expects that the spot price of an underlying asset will be above the strike price at the expiration date. If these expectations turn out to be true, the put option will expire out of the money, and the writer will make a profit in the amount of the premium received.

The writer is also obligated to buy the specified amount of the underlying asset at the strike price. Therefore, if the spot price drops below the breakeven point, the writer will lose an amount that is theoretically unlimited.

The payoff diagram for the writer of the put option from the example above is illustrated in the figure below.

If the spot price is above the strike price at the expiration date, the contract will expire out of the money. Therefore, the writer will make a profit in the amount of the premium received.

If the spot price is captured in the range between the breakeven point and the strike price at the expiration date, the contract will expire in the money, but the writer will still make some profit. In the example above, this range is between $103.64 and $112.35.

If the spot price drops below the breakeven point at the expiration date, the writer will suffer losses.

**Important!** The loss of the put option of the writer is not theoretically limited.

The writer of a put option can pursue one of two objectives.

- Speculation
- Buying of underlying asset at a predetermined price

The speculation is the writing of a put option with the only hope that the price of the underlying asset will increase above the strike price at the expiration date. The key advantage of such strategy is that the time delay of a put option works in favor of the writer and against the buyer.

However, the writer of a put option can also benefit if the underlying price declines below the strike price. Strange, isn’t it? Let’s consider the example below to better understand the issue.

An investor is considering buying 100 stocks if their price drops below $62. However, the spot price is $68.37, which is not acceptable for the investor at the moment. Assume that the investor can write a put option for 100 shares at a strike price of $66.50 and a premium of $4.75 per share.

If the spot price is above the strike price at the expiration date, the contract will end up out of the money, and the investor will gain $475 ($4.75 × 100).

If the spot price drops below the strike price, the investor will be liable to buy 100 stocks at a strike price of $66.50, but his or her actual costs will be lower by the amount of the premium received, which is $61.75 ($66.50 - $4.75). In this case, the investor will reach the goal to.

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