Focus on the reason behind trading and don't stick to options models based on ideal assumptions

A What is the value of time

The price of an option consists of the intrinsic value of the option and the time value of the option. Among them, the intrinsic value of an option defines the benefit that a buyer of a option can obtain if it exercises the right immediately. If C and P indicate the price of a call option and a put option contract respectively, S indicates the price of the underlying asset, and K is the execution price of the option. So for call options, there are:

The intrinsic value of a call option = max (S-K, 0)

Call option time value = C-max (S-K, 0)

For put options, there are:

The intrinsic value of a put option = max (K-S, 0)

Put option time value = p-max (K-S, 0)

Therefore, for options, the price of a real option consists of intrinsic value and time value. However, for fictitious options, their intrinsic value is zero, and the price of the option is entirely composed of time value.

Beginners generally think that the time value of an option is only related to time. In fact, the time value of an option is affected by many factors. Among them, the three main factors are the remaining time of the option, the implied volatility of the option, and changes in the price of the underlying asset. It also has the following properties:

1. The longer the remaining expiration time of the option, the greater the time value; the shorter the remaining expiration time of the option, the less time value.

2. If implied volatility increases, time value increases; if implied volatility decreases, time value decreases.

3. The closer the target price is to the exercise price, the greater the time value of the option; the farther the target price is from the exercise price, the smaller the time value of the option. In other words, equalization options have the greatest value in time, while deep imaginary and deep real options have less time value.

4. The rate at which the time value of an option passes will also accelerate as the expiration date of the option contract approaches. That is, the closer to the expiration date of the contract, the faster the time value of the option will decay.

The decline in the value of time B

Generally speaking, the more uncertain the buyer's future profit will be, the greater the time value of the option. Therefore, when an option expires for a longer period of time, the option's intrinsic value is also more likely to increase, so its time value also increases. Furthermore, an increase in implied volatility will also increase uncertainty about the profit of an option when it expires. The price of the option will be more expensive, and when the intrinsic value of the option does not change, the time value of the option will also increase. When the target price is closer to the exercise price, Delta is close to 0.5 at this point, indicating that the probability that the option will be profitable in the future is only 50%. Uncertainty is the highest, so the time value of options near parity is greatest. Meanwhile, Deep Imaginary Value and Deep Real Value Option Delta are close to 0 or 1, and it is relatively certain whether future profits will be made. So their time value is less.

Properties 1 and 2 can be better understood through the above analysis. Next, we use time value to calculate the price of the underlying asset and time separately to further explain why equalized options have the greatest time value, and why the time value of options declines at an accelerated pace. For the sake of simplicity, the following deductions all use call options as an example.

Let's say C is the price of a call option, S is the target price, K is the execution price, r is the risk-free interest rate, σ is the volatility, t is the current moment, and T is the expiration time of the option. Then from the Black-Scholes formula we get:

where S>0 follows a logarithmic normal distribution; the fluctuation rate sigma>0.

Then the time value of an option can be expressed as:

TVTimeValue=C-max (S-K, 0)

First, let's calculate the underlying asset price S:

When S ≤ K, max (S-K, 0) = 0

When S>K, max (S-K,0) = S-K

Furthermore, since Delta[ 0,1)

Therefore, when S=K, the option time value TV reaches its maximum.

Also, from the above derivation, it can also be obtained that when the option is a real value option, the time value of the option decreases as the price of the underlying asset increases. When an option is an imaginary option, the time value of the option increases as the price of the underlying asset increases.

Next, let's calculate t again:

That is, the time value of an option is a subtraction function of t.

Additionally, you'll also get:

In other words, the time value of an option is a convex function.

As a result, the time value of options passes faster as the expiration date of the option contract approaches.

Abnormal phenomenon of C time value

From the discussion above, we know that the time value of options should be positive. For buyers, time value reflects the possibility that the option's intrinsic value will increase in value in the future. If the buyer thinks the more likely it is, the higher the premium he is willing to pay. In other words, the price of an option contract is usually higher than the intrinsic value of the option. For option contracts of the same type and with the same execution price, the longer the remaining expiration period, the greater the time value of the option, and the higher the price of the option. In other words, the price of a long-month option contract is greater than that of a recent month's contract.

However, investors often find a strange phenomenon on T-type price lists for options. Many times when subscribed to real-value options will have a negative value. At the same time, the implied volatility of options is also 0. Why exactly is this?

The 50 ETF options currently on the market are European options, using the Black-Scholes pricing formula. In the assumptions of this model, one very important condition is that the option volatility is σ > 0. However, as we can see in the T-price chart above, many hidden subscription options were already 0 at this point, which did not meet the model's requirements and assumptions. Therefore, some of the results deduced above based on model conditions are no longer applicable. As a result, there have been some unexpected situations.

In fact, we can understand this issue from a more fundamental perspective. You need to know that option prices are traded on the market. When the price of the underlying asset does not change, if the buyer's strength is superior to the seller's strength, the option contract price will rise. In contrast, the option price will be depressed, and there will even be situations where the option price is less than the intrinsic value of the option, which ultimately causes the time value of the option to become negative. When the time value of the option becomes negative, the Black-Scholes model will fail at this point, and we cannot reverse the implied volatility of the option through the option price. At this point, we set the implied volatility of the options contract to zero. This is also why in T-type quotation tables, the option's time value is negative and the option's hidden wave is 0 often in double pairs.

As can be seen from the above, when it comes to subscription options, when the price of the underlying asset does not change, since the strength of selling is stronger than buying, this will cause the option price to be depressed, and sometimes there are even situations where the option price is less than the intrinsic value. This shows that investors are pessimistic about the future market or are relatively cautious about future market conditions. In fact, if we use the option parity formula, we can obtain the composite target asset at this time that will be applied to the target. This is similar to the situation where stock indexes are discounted all year round.

Furthermore, since the Delta of deep real value options is relatively large, changes in the underlying price have a greater impact on the price of the option contract (changes in the absolute value of the contract), investors are often even less inclined to trade deep real value options. This also causes the liquidity of deep real options to be poor, so that changes in the price of deep real options may lag behind changes in the price of the underlying asset, resulting in a situation where the time value of the option is negative.

The time value of the D due date

In addition to the “strange” phenomenon described above, many investors are also very puzzled by the “change” in the time value of option contracts on the expiration date. The following is a T-type quotation table for 50 ETF options after a certain expiration date has closed.

As can be seen, after the expiration and closing of the 50 ETF options contract, there are still quite a few subscribed options with a negative time value, while the time value of many put options is positive. Since the contract has already expired, why doesn't the time value return to zero? From the above discussion, we can see that this also shows that investors are cautious about the future market, which causes subscription options to be discounted and put options to a premium. Another reason is the liquidity problem of real value options. Normally, the liquidity of deep real value options is relatively low, and there may even be situations where there are no counterparties on the expiration date. As a result, after the option contract expires, the time value of real value options does not return to zero.

Also, we can look at the above issues from the perspective of options contract delivery. Since ETF options are physical delivery, when the options enter the delivery process, the purchaser of the option needs to prepare enough cash to meet the delivery requirements, and this sometimes requires a large amount of capital. For example, if an investor holds 100 50 ETFs to buy the October 3.4 contract, then the investor needs to prepare 100*10000*3.4 = 3.4 million on the delivery date. Obviously, it is quite difficult to raise such a large amount of capital in a short period of time. Furthermore, subscription option buyers can only operate on T+2 trading days after delivery and receive ETF shares, which also increases uncertainty. Therefore, investors prefer to close positions directly to earn the price difference between closing and opening a position. First, there is no need to raise 3.4 million, thereby greatly improving the efficiency of using capital; second, there is no need to bear the risk of ETF “staying overnight”. This is also the reason why real-value options are often heavily discounted when they are close to the expiration date, so even after the contract expires, the time value is negative. In fact, in order to solve the above problems, the exchange has introduced measures to exercise combined powers, which have reduced the discounted price of subscription options on expiration dates to a certain extent.

So after the contract expires, the time value of put options is positive. How do you understand this? Similarly, in order not to enter the delivery process, sellers of put options also need to close their positions. Due to the liquidity relationship, a premium situation occurred, which led to a negative value of the option. Of course, if investors hold a large amount of 50 ETFs in stock, selling directly on the secondary market may have an impact on the price of 50 ETFs, while reducing 50 ETF holdings through the delivery process does not have the above problems. Therefore, investors can also buy put options at a premium to achieve the goal of reducing their holdings.

The time value of options is a very important concept in options trading. It is mainly influenced by the three factors remaining in the option, the implied volatility of the option, and changes in the price of the underlying asset. According to the Black-Scholes model, the time value of European options should be positive (European put options when the underlying asset is low are not taken into account) and will accelerate as the value of the option contract approaches its expiration date. However, in everyday trading, the time value of options is often negative. This is mainly influenced by factors such as investors' judgments on future market conditions, market liquidity, and trading rules, thereby “invalidating” the Black-Scholes model. Therefore, in everyday trading, investors should pay more attention to the reasons behind the transaction and not stick to an option model based on ideal assumptions.