Time Value

Q. Is it true that option buyers typically lose money? Is it better to be the seller?

A. It is widely believed that purchasing options is a losing proposition. The primary reason for that belief is that an option’s price contains a time premium, which means the buyer pays for the time remaining on the option. Because options expire, as time passes, so does part of the option’s value.

In fact, in a recent article, the following quote was published by an option “expert”:

“While purchasing call and put options can be profitable, they are actually losing investments 90% of the time. The primary reason why so many of these option investments fail is because of the time depreciation element of options. This means that options lose value simply with the passage of time.”

The author then goes on to say that you must set up your trades properly – you must be the seller of time premium. Is it true that option buyers are destined to lose because of the time premium? Is option selling a financial road to riches? Let’s find out.

Intrinsic Value and Time Value
To understand the answer to the question, you must first realize that an option’s price can be broken down into two components: the intrinsic value and time value. The intrinsic value shows how much of an “immediate benefit” is conveyed in the option. Any value over the intrinsic value is called the time value.

For instance, assume that stock is trading for $103 with the $100 call trading for $5. By holding the $100 call, the investor has the right to purchase shares for $100 per share, which is a $3 advantage over the person who purchases the stock at the current $103 market price. Therefore, the $100 call must be worth at least $3 in the open market. If not, arbitrage is possible. But because the call is trading for $5, there is $2 of value that is unaccounted for by the intrinsic value. The $2 value is due to time.

It is the only the time value that erodes with the passage of time. If the stock’s price remains at $103 at expiration, the call’s price falls from $5 to $3, which is 40% from nothing more than the passage of time.

The time premium creates a higher breakeven price for the option trader. In this example, the call buyer who pays $5 for the $100 call must have the stock’s price rise to $100 strike + $5 premium = $105 at expiration in order for the option to break even.

Two Time Values
We’ve just shown how to break an option’s price into the two components of intrinsic value time value. However, we can do something similar to time value; it can be broken down into two components as well. One part of the time value is due to volatility, which is the portion that most investors are familiar with. However, there is another source of time value that must be present. That source is due to the cost-of-carry on the exercise price. This cost-of-carry portion carries no adverse effects for the investor. To understand why, let’s find out more about the cost-of-carry.

The Cost-of Carry
The cost-of-carry is simply the interest that is lost from “carrying” a position over time. For instance, if the risk-free interest rate is 5% and you deposit $100 into a checking account, you will have $105 at the end of one year. However, if you buy $100 worth of stock rather than depositing the cash into an interest-bearing checking account, you will not have $105 at the end of the year. The for-sure $5 that you miss out on from buying the stock is the cost-of-carry. It is the “cost” to you from carrying $100 worth of stock for a one-year period. Had you not purchased the stock, you’d be ahead by $5.

This obviously does not mean you cannot have more money at the year’s end from purchasing the stock. But in order to take that chance, you must give up the $5 interest for sure.

Any call option’s time premium is comprised of a volatility component plus a cost-of-carry component. Let’s look at a hypothetical call option whose time premium is nothing but the cost-of-carry on the exercise price. Assume a one-year, $100 call is trading for today’s value of the cost-of-carry. In other words, if interest rates are 5% then the cost-of-carry is $5 for one year on a $100 exercise price. Therefore, that value today is $5/1.05 = $4.76. A $4.76 deposit in a checking account will grow to $4.76 * 1.05 = $5 in one year. So “today’s” value of that interest payment is $4.76.

Let’s compare two investors – Investor A and Investor B – who both start with $100 cash in their accounts. Investor A purchases $100 worth of stock. Investor B purchases a one-year call option for only $4.76. After buying the call, Investor B has $100 - $4.76 = $95.24 cash which will earn interest over the year. Now let’s compare different stock prices and see how the two investors perform.

First, we’ll assume the stock rises to $120 at the end of the year. With the stock at $120, Investor A makes a $20 gain on his $100 stock purchase. Investor Bs call is worth $20 of intrinsic value. After subtracting the cost, Investor B makes $20 - $4.76 = $15.24. However – and this is important to understand – Investor B also had $95.24 cash which has now grown to a value of $95.24 * 1.05 = $100 at the end of the year. So Investor B earns $4.76 worth of interest that he would not have earned had he elected to buy the stock. Investor Bs total profit is therefore $15.24 + $4.76 = $20, which is exactly the same as Investor A.

No matter which stock price you choose that is greater than or equal to $100, Investors A and Investor B have identical profits at the end of the year. Their selections perform the same once we account for the interest that is earned on Investor Bs cash balance.

What Investor B loses in time value is exactly made up for with interest. This can perhaps be better understood by considering what happens if the stock price is exactly $100 at expiration. In this case, having purchased the stock for $100, Investor A makes no profit and takes no loss. Investor B loses the entire $4.76 worth of time value. However, his cash balance has grown from $95.24 to $100 thus capturing his lost time value. So after accounting for the interest on the cash balance, Investor B hasn’t lost one cent either.

What if the Stock Price Falls?
Now let’s see what happens if the stock price falls by assuming the stock is $90 at expiration. Investor A is down $10 from his $100 purchase price. Investor Bs call expires worthless so he loses the entire $4.76 time value. But as before, his cash balance has grown to $100 thus earning $4.76 worth of interest. After accounting for the interest, Investor B loses $4.76 time value on the option and gains $4.76 in interest – he loses nothing. No matter how low the stock’s price may fall, Investor B always ends p with his $100 cash that he started with at the beginning of the year. Investor A, on the other hand, takes dollar-for-dollar losses for all stock prices below $100 at expiration.

So no matter which stock price you choose below $100, you’ll find that Investor Bs call option expires worthless but he exactly makes up that loss with the interest earned on his cash balance.

This shows that with the $100 call priced at $4.76 – the present value of the exercise price – Investor B might make a gain if the call ends up with intrinsic value. But he cannot lose. A call price of $4.76 time value provides for a possible profit with no risk.

This alone shows that it is incorrect to say that the presence of a time premium ensures that the investor will lose. It depends on the source of the time premium. If the time premium is entirely due to the cost-of-carry, the option investor cannot lose when compared to the stock investor as we have demonstrated here.

If you did find the call priced for less than $4.76, arbitrage is possible. Arbitrage is when a trader has an opportunity to make “free money” without any risk or any cash outlay. It’s like finding money on the street. Here’s how it would work. Assume the above call was trading for $2. Arbitrageurs would buy the call for $2 and short the stock for $100 thus leaving them with a $98 cash balance. Effectively, they are shorting the stock to generate the cash and then using $2 of that to buy the call. There’s no out-of-pocket-expense.

Would there be margin requirement? No, because the short stock is protected by the long $100 call. The arbitrageur always has the right to buy the shares for $100, which is exactly the price where he shorted them. Aside from commissions, there is no way to lose. The arbitrageur keeps the $98 cash, which grows to $98 * 1.05 = $102.90 in one year. If the stock’s price is above $100 at that time, he will exercise the call and purchase shares for $100 thus leaving him with a free profit of $2.90.

If the stock price is below $100 at expiration, the arbitrageur will let the call expire worthless and purchase the shares in the open market thus making more money. For instance, if the stock is $90, he can purchase the shares and be left with $102.90 - $90 = $12.90 profit. So the arbitrageur might make more money if the stock price falls but is guaranteed to make at least $2.90. He cannot lose; it is an arbitrage.

In fact, in the Alpha Trader Certificate Course, you’d find that the above combination of short stock and long call is a “synthetic” put option. That just means that from a profit and loss perspective that short stock and a long call behave exactly like a long put. Now it should be clearer why the arbitrageur might make money if the stock price falls but cannot lose. He was effectively paid $2.90 to hold the put.

The arbitrageur will therefore continue to buy the undervalued calls until their price rises to at least $4.76. We can see this is true by looking at the Black-Scholes Option Pricing Model and considering a one-year $100 call option with interest rates at 5%. What is the call worth even if there is no volatility? It has to be worth at least $4.76 otherwise arbitrage is possible, which can be verified in Figure 1:

Figure 1:

In Figure 1, we used 0.1 for volatility rather than zero as the calculator will not accept zero. But 0.1 is pretty close to zero and you can see the call’s price (red box) is $4.7619. Again, that is the minimum price that any one-year, $100 call can trade for with 5% interest rates to prevent arbitrage. Of course, if there’s no volatility then the put option is worthless, which is confirmed in Figure 1 as well.

Too Good to be True
The arbitrage argument shows that a call option priced for exactly the cost-of-carry is too good to be true. You will therefore not find it in the open market. However, as stated before, it clearly shows that it is incorrect to say that the presence of a time premium ensures that option buyers lose in the long run.

That’s why it’s so important to separate the two sources of time value, especially if you are dealing with longer dated options with high strike prices where the cost-of-carry can be sizeable.

Any value in the option over and above the cost-of-carry is due to the volatility of the underlying stock. But just because there is a price for the volatility does not ensure a loss either. After all, there is a reason that option traders are willing to pay for volatility; that’s because there is a perceived value. And if there is a value, there is a price.

The question we must answer is this: Is the price we’re paying worth the volatility we’re receiving? And to answer that question, you must understand the concept of fair value.

Fair Value
The fair value of any asset is the price that would make the buyer and seller break even over the long run if they were to buy and sell the asset thousands of times. For example, if I am willing to toss a coin thousands of times and pay you $1 each time it lands “heads,” how much should you be willing to pay me every time it lands “tails?”

First of all, you have to realize there is a value to this game; after all, you’d certainly be willing to play it for free. If I offered you the chance to play it for no payment when the coin lands “tails” you’d certainly accept. Therefore it has value and must have a price. But what is a fair price to pay?

Let’s try a few numbers and see if we can find a fair price. If you decide to pay $2 every time it lands “tails,” you’d quickly find that about half the time you’d win $1 and half the time you’d lose $2. You’d end up losing in the long run. Two dollars is just too high of a price to pay relative to the frequency or size of the wins. In other words, it is too much to pay relative to the “volatility” of the appearances of heads and tails.

On the other hand, if you were able to play this coin flipping game for less than $1, say 50 cents, you’d find that you’d win $1 half the time and only lose 50 cents half the time thus ensuring a nice profit in the long run. You may lose on any given flip but, on average, you will win far more money than you will lose.

In a similar way, an option’s price can be too high or too low. If the stock’s price turns out to be highly volatile during the life of the option but you spend relatively little for it, you will make money. If you spend too much, you’ll lose.

Of course, there’s no way to tell beforehand if the option’s price is too high or too low. Only at expiration will we know for sure whether the buyers or sellers got the better end of the deal. But prior to expiration, it’s up to the market though to decide what the expected fair price is. On average though, the market should not systematically over or under price options.

If options are consistently overpriced so that sellers always win then everybody would want to be sellers and there would be no buyers. There would be no options market. So how are buyers introduced to the market? Buyers appear once sellers lower their price. Sellers will continue to lower the price until every seller who wants to sell at that price can do so. And at the new lower price, sellers lose some of their initial advantage.

Once the option’s price stabilizes then it is considered fair by both the buyer and seller.

Using the same reasoning, buyers cannot always have an advantage otherwise all traders would want to purchase options and nobody would want to sell. Prices would have to rise in order to attract sellers.

The point to understand is that it is a fallacy to believe that one side or the other – the buyer or the seller – always has the advantage. That cannot be true over the long run.

In addition, any argument touting that one side wins more often than another is not proof that the option is overpriced. At the beginning of this Q&A, we quoted someone stating that option buyers lose 90% of the time. This is not a true statement but if we assume that it is, we can still show that the bet can be fairly valued. For instance, assume I offer you the following game. I will write down a number from 1 to 10 on a piece of paper. If you guess the number, you win $10. How much is this game worth?

You have a one-in-ten chance of winning, which is the same thing as saying you have a 90% chance of losing. If you pay $1 for this game, you can expect to lose 10 times for every one time you win. On average, it will cost you $10 to win $10 and the bet is fair. So even though you have a 90% chance of losing, it is reflected in the low $1 price of the game.

The most important point to take from this is that it is not true that option sellers are always better off. It is also not true that the presence of a time premium ensures that you will lose. Options are nothing more than packaged expected cash flows and they have value. On average, buyers and sellers should win equally often. Do not devise your option strategy around one of the biggest myths in all of options trading.

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