Explain Option Trading: The Value of Time
Why are options worth more than their intrinsic value? Why are options with a long time to maturity worth more than options with a short time to maturity? And why don’t futures have time value? And what about lottery tickets? It turns out that it all boils down to hope.
This page explains the reasons for and details of the time value of options.
Options have time value
Stating that sounds like saying ”water is wet”. With time value I mean that options are always priced above their intrinsic value. This is not the same a the ”time value of money” which is purely a matter of interest rates. Let’s take a closer look. We’ll use this chart of the theoretical value of a call option as a starting point (The values were calculated using the Black -76 formula).
Each line represents a call option with a strike price of 100$ (Underlying and everything else equal). The higher the curve, the longer time left to maturity. On the horizontal x-axis is the price of the underlying. On the vertical y-axis is the value of the option. You might point out that an option can’t possibly have many prices at the same time. Correct, in reality there is only one price, the price you can buy it for in the market. The explanation for the calls being represented by lines is a ”what if?”. What would be the value of a call option with a strike of 100$ and 90 days left to maturity if the underlying was 94$. If the underlying was at 96$ and so on. Then connect the dots and you have a curve for that specific time to maturity.
The top line in the chart represents the value of the option with 3 months left to maturity. The one under it 2 months, next 1 month and the bottom line 1 day left. More time left means a higher value and thus a higher line. Very useful to know when you read option quotes, or think about buying or writing options or have an option or....
Before getting to the time value an important thing to note is that there is one more assumption behind the lines in the graph. The volatility is assumed to be the same all the time for all the options in the chart. The volatility assumed here is 20% yearly. As you can read in the explanation of what an option is, volatility is one of two most important factors effecting the value of an option. The factor is time, also introduced in the explanation of what an option is and further explained on the tutorial page you’re reading right now.
Explaining time
Why do options have time value? It’s partially because more time gives volatility a better chance to move the stock price so that the option goes very deeply into the money. That is not the whole story though. Consider futures. Futures on stocks also have a volatile underlying asset. But they don’t have any time value, except for that caused by the interest rate. The other important part of the story is that the minimum value of an option at expiry is zero. Imagine if you had a call and the underlying price went very low so that the call was very far out of the money. Would you end up with a huge loss and owe the option writer loads of money? No, the option would expire worthless and you’d lose only what we paid for it. Like a lottery ticket. Now, imagine the opposite, that the minimum value of the option was negative. Imagine that it would be possible for us to end up having to pay the option writer lots of extra money at expiry. That would give time a different meaning. Not only would time give volatility the opportunity to move the option deeply into the money and make us a big profit. It would also give volatility chance to do the opposite. Move the price of the underlying down so that we’d owe the writer more money. In that case, long time to maturity would not necessarily be a good thing. Time would work both ways and not be worth much. Just look at futures and forwards. They have huge maximum losses in dollar terms speaking. You could be forced to put more and more money into a margin account with your broker and then lose all that money. Thus futures and forwards don’t have much time value. What little time value you can find in a future is because of the interest rate, cost-of-carry and some other factors that are, in most cases, very insignificant.
Let me try to clarify the comparison between options and futures. You could say that an option that is far away from the money (deep in or deep out) is one of two things: If it’s very deeply in the money it’s like a futures contract, if it’s deeply out of the money it’s like a lottery ticket. Explanation of each below.
Options as futures
When the option is deeply in the money it changes value in tandem with the underlying asset, much like a futures contract does. Dollar for dollar they move almost the same as the underlying asset.
Calls move like long futures and puts move like short futures. Being deeply into the money the option is almost certain to have an intrinsic value when it expires. Above I said that the time value comes from the fact that the option, as opposed to the future, has a maximum loss at maturity of zero. If it’s deeply in the money, the probability that it will expire worthless is very low. In such a situation the difference between a future and an option becomes unimportant. Imagine things go really badly and the market makes a huge move in the wrong direction. This would be bad for both a person who is long a future and a person who is long a call option. However, the owner of a call option is protected on the down side by the fact that the option has a maximum loss of zero once it is purchased. The person long the future has no such protection. The difference between the long future and the long option is the option’s limited downside. But if the option is very deeply into the money, the downside protection is highly unlikely to come into play. So then the difference in price moves between a future and an option becomes very small in dollar for dollar terms.
Options as lottery tickets
The opposite case of the options-as-futures is the options-as-lottery-tickets. By this I mean the scenario when the option is very deeply out of the money. The option then has very little chance of expiring with an intrinsic value. With all probability it will it will expire worthless. It’s especially unlikely that it will every achieving the ”almost a future” status described above. In this situation the option is worth so little it simply doesn’t have much value to lose over time.
Put those two extreme cases together, options as futures and options as lottery tickets, in one contract. What you’ve got is the actual, real world option. So how was reading all that useful? The time value comes from the fact that we don’t know which situation we’ll end up in.
What effects time value?
Now that you understand why an option has time value it’s possible to understand how time value changes under different conditions. If you look at the graph you see that the distance between the bottom line and the line above it is not the same everywhere. Apparently the amount and loss of time value depends how far under or over the option’s strike the price of the underlying is. Observe that as a month goes by we move from the top line to the line below it. If the price of the underlying is 100$ when we start and still is 100$ after the month passes, the option loses 0.71$ in value.
In another month, on the next line down, the option has lost another 0.94$. As another 29 days pass the option loses 1.87$. With 1 day left to delivery it’s only worth 0.42$. Thus another 1.46$ lost in just 29 days.
Let’s compare that with a different course of events. Say that when the first month has past, the price of the underlying has fallen to 94$. In the chart this puts us at the point where the second curve from the top meets the y-axis.
If the price of the underlying remains 94$ when the next month passes, we move from the second curve from the top to the second curve from the bottom. Doing so, the value of the call goes from 1.02$ to 0.40$, a loss in time value of 0.62$. In the case when the price was 100$ the loss over the same time period was 0.94$. Why does the option lose more time value if its closer to being at-the-money? As you can see in the chart this is true whether the option is below or above the money. Recall the discussion above of options as lottery tickets vs. Options as futures. The time value comes from the probability of moving from the lottery-ticket state to the future contract state. Or to put it differently: At the money there is a lot of hope. Is the option then going to move in the direction of hopelessness, or the direction where no hope is needed cause the option will have intrinsic value instead? Hope is most meaningful when an option is at the money. Deeply out of the money hope is futile. Deeply in the money you already have what you hoped for, intrinsic value. It is thus at the money the option has the most hope to lose. So therefore the time value has to be higher the closer the strike price and the price of the underlying are.