The relationship between the upper and lower limits of option prices and option parity

Table of contents

2. The upper and lower limits of options

1. The basic concept of options

Option: It is a kind of option. After the option buyer pays a certain amount of option premium , he can obtain the right to buy or sell a certain amount of underlying objects at a certain price (execution price) within a certain period of time (expiration date) .

1. Expiration date Expire: The specific time specified in advance by the option contract to perform the contract in the future, also known as the repayment date.

2. Strike: The delivery price of the underlying asset determined by the option contract, also known as the strike price.

3. Option fee: Option is the right to spend money to buy/sell the underlying object in the future, and the option contract represents a kind of initiative. Therefore, the option itself has its value, and this value is the option premium, that is, how much the right is worth.

4. Option Buyer Long: The party who buys the option is long.

5. Option seller Short: The party who sells the option is short of the option.

6. Call option: The holder has the right to buy the underlying object at the strike price in the future, also called a call option.

7. Put option: The holder has the right to sell the underlying object at the strike price in the future, also called a put option.

8. European option: The holder can only perform the contract on the expiration date.

9. American option: The holder can perform the contract at any time before the expiration date (including the expiration date).

10. In-the-money option: the option that the holder will generate positive cash flow by exercising the option, in-the-money option.

11. At-the-money options: Options with cash flow close to zero are at-the-money options, at-the-money options.

12. Out-of-the-money options: options with negative cash flow, out-of-the-money options.

Whether it is real value or imaginary value only depends on whether the cash flow income of exercising option rights is positive or negative, and has nothing to do with whether the total income is positive or negative, that is, option premiums are not considered.

13. Intrinsic value: If the option is executed now (if possible), the profit or loss brought by the exercise of the option to the holder is compared with 0, whichever is greater.

Intrinsic value of call option: max( S - X , 0)
Intrinsic value of the put option: max( X - S , 0)

14. Time value: The premium minus the intrinsic value.

The value of an option comes from uncertainty, and the longer the time to expiration, the greater the uncertainty. The value of this part of uncertainty is reflected in the compensation for the option seller to bear unlimited risks, or in the possibility of the option holder gaining income, which is the part of the option premium minus the intrinsic value of the option.

2. The upper and lower limits of options

(upper limit)

Holders of European or American call options have the right to buy an underlying object at a certain price, and C of the call option price will not exceed the underlying object price S: C ≤ S

C is a right to buy something. If the price of this right is more expensive than the target item, then go directly to the market to buy the target item and you’re done. For example, if you buy the subject matter with a value of S, and then sell the right with a value of C, the current income is C - S. If the underlying object falls at that time, and the other party does not perform the contract, I don’t have to do anything, and the final income is C - S; Giving the subject matter in TA to TA can also get X, and the final income is C - S + X.

The holder of a European or American put option has the right to sell an underlying object at price X, no matter how much the market price of the underlying object falls, the price P of the put option will not exceed the strike price X: P ≤ X

P is a right to sell something. If the price of this right is higher than the execution price, then it’s over if you sell this right directly, and I can still keep my subject matter.

For European options, the current option price p: $p\leq Xe^{-rT}$

Because European options cannot be exercised in advance, the exercise price X needs to be discounted according to continuous compound interest.

(lower limit)

Without paying a dividend, the European call option has a lower bound of:

$c\geq max(S_{t}-Xe^{-rT}, 0)$

Assuming that you borrow a stock and sell it now $S_{t}$ , and then spend $S_{t}-Xe^{-rT}$ to buy a call option, $Vehicle^{-rT}$ the remaining money will be used for risk-free investment. At time T, the money becomes X, which happens to be able to buy a stock at X to close the position. If c is less than $S_{t}-Xe^{-rT}$ , then at time T, the money will be greater than X, and there will still be money left after the position is closed.

With the dividend paid, the lower bound for the European call option is:

$c\geq max((S_{t}-D)-Xe^{-rT}, 0)$

Where $S_{t}$ is the price of the underlying asset at time t, and D is the discounted return on the underlying asset during the option validity period.

As for why it is -D, this is mentioned in the pricing of forward contracts.

Without paying a dividend, the lower bound on the European put option is:

$p\geq max(Xe^{-rT}-S_{t}, 0)$

With the dividend paid, the lower bound on the European put is:

$p\geq max(Xe^{-rT}-(S_{t}-D), 0)$

Where $S_{t}$ is the price of the underlying asset at time t, and D is the discounted return on the underlying asset during the option validity period.

3. The parity relationship of options

Assume holding a European put option long P with strike price X, a European call option short C, plus a stock , that is, the total value is a portfolio of S+PC.

Why is shorting a minus sign? Because C corresponds to the value of the call option, if the stock price on the expiration date is higher than the strike price, then the option holder will execute the option, and TA has obtained a profit worth C, but you will lose C as a short position, so The minus sign used here.

Organized to get:

$P=C+Xe^{-rT}-S$

or:

$C=P+S-Xe^{-rT}$

This is the parity relationship of options , that is, the prices of call options and put options will remain at an equilibrium level. (If you know one option price, you can use the parity relationship to find another option price)

But in reality, considering factors such as transaction costs, the left and right sides of the option parity relationship are not equal, but have certain errors. As long as the error is within the acceptable range, the option parity relationship is established.