*This article was originally published in The Option Strategist Newsletter Volume 8, No. 22 on November 24, 1999. *

At a recent seminar or conference (don’t ask which one – there have been too many to distinguish one from another!), the subject was raised regarding the effect of time decay on an option. As the discussion progressed, it dawned on me that many (perhaps novice) option traders seem to think of time as the main antagonist to an option buyer. However, when one really thinks about it, he should realize that the portion of an option that is not intrinsic value is really much more related to stock price movement and/or volatility than anything else – at least in the short term.

For this reason, it might be beneficial to more closely analyze just what the “excess value” portion of an option represents and why a buyer should not primarily think of it as time value premium.

An option’s price is composed of two parts: 1) intrinsic value, which is the “real” part of the option’s value – the distance by which the option is in-themoney, and 2) “excess value” – often called time value premium. There are actually five factors that affect the “excess value” portion of an option. Eventually, time will dominate them all, but the longer the life of the option, the more that the other factors influence the “excess value”.

**The five factors influencing “excess value” are:**

- Stock price movements
- Changes in implied volatility
- The passage of time
- Changes in the dividend (if any exists)
- Changes in interest rates

You can see that each is stated in terms of a movement or change – that is, these are not static things. In fact, to measure them we use the “greeks”: delta, vega, theta, (there is no “greek” for dividend change), and rho. Typically, the effect of a change in dividend or a change in interest rate is small (although a large dividend change or an interest rate change on a very long-term option can produce visible changes in the prices of options).

If everything remains static, then time decay will eventually wipe out all of the “excess value” of an option. That’s why it’s called time value premium. But things don’t ever remain static, and on a daily basis, time decay is small, so it is the other factors that are most important.

Example: XYZ is trading at 82 in late November. The Jan 80 call is trading at 8. Thus, the intrinsic value is 2 (82 minus 80) and the “excess value” is 6 (8 minus 2). If the stock is still at 82 at January expiration, the option will of course only be worth 2 and we will say that the 6 points of “excess value” that was lost was due to time decay. But on that day in late November, the other factors were much more dominant.

On this particular day, the implied volatility of this option is just over 50%. We can determine that the call’s “greeks” are:

- Delta: 0.60
- Vega: 0.13
- Theta: -0.06

This means, for example, that time decay is only 6 cents per day. It would increase as time went by, but even with a day or so to go, theta would not increase above about 20 cents unless volatility increased or the stock moved closer to the strike price.

From the above figures, you can see – and this should appeal to your intuition – that the biggest factor influencing the price of the option is stock price movement (delta). It’s a little unfair to say that because it’s conceivable, although unlikely, that volatility could jump by a large enough margin to become a greater factor than delta for a one day’s move in the option. Furthermore, since this option is composed mostly of “excess value”, these more dominant forces influence the “excess value” more than time decay does.

There is a direct relationship between vega and “excess value”. That is, if implied volatility increases, the “excess value” portion of the option will increase and, if implied volatility decreases, so will “excess value”.

The relationship between delta and “excess value” is not so straightforward. The farther the stock moves away from the strike, the more this will have the effect of shrinking the “excess value”. If the call is in-the-money (as in the above example), then an increase in stock price will result in a decrease of “excess value”. That is, a deeply inthe- money option is composed primarily of time value premium, while “excess value” is quite small. However, when the call is out-of-the-money, the effect is just the opposite: then, an increase in call price will result in an increase in “excess value” because the stock price increase is bringing the stock closer to the option’s striking price. I don’t know if this helps or hurts you to conceptualize it, but the part of the delta that addresses “excess value” is this:

Out-of-the-money call: 100% of the delta affects the “excess value”

In-the-money call: “1.00 minus delta” affects the “excess value” (so, if a call is very deeply in the money, and has a delta of 0.95, then the delta only has 1.00 – 0.95, or 0.05, “room” to increase. Hence it has little effect on what small amount of “excess value” remains in this deeply in-the-money call).

These relationships are not static, of course. Suppose, for example, that we have the same situation of the stock trading at 82, the Jan 80 call trading at 8, but now there is only week remaining until expiration! Then the implied volatility would be 155% (high, but not unheard of in these volatile times). The “greeks” would bear a significantly different relationship to each other in this case, though:

- Delta: 0.59
- Vega: 0.044
- Theta: –0.51

This very short-term option has about the same delta as its counterpart in the previous example (the delta of an at-themoney option is generally slightly above 0.50). Meanwhile, vega has shrunk. The effect of a change in volatility on such a short-term option is actually about a third of what it was in the previous example. However, time decay in this example is huge, amounting to a half point per day in this option.

So now we have the idea of how the “excess value” is affected by the big three of stock price movement, change in implied volatility, and passage of time. How can one use this to his advantage? First of all, one can see that an option’s “excess value” may be due much more to the potential volatility of the underlying stock and therefore to the option’s implied volatility than to time. Selling Options Is Not Necessarily A High Probability Strategy:

As a result of the above information regarding “excess value”, one shouldn’t think that he can easily go around selling what appear to be options with a lot of “excess value” and then expect time to bring in the profits for him. In fact, there may be a lot of volatility – both actual and implied – keeping that “excess value” nearly intact for a fairly long period of time.

In fact, this gives me a chance to insert another “commercial” against selling options without proper calculation of the probabilities of success. I received a letter from a customer this week suggesting that the sale of stock options would be a better strategy than straddle buying. Randomly, that might be true, but with the amount of probability analysis we do with straddle buying, the latter strategy should be superior. Furthermore, stocks can gap on earnings news, takeovers, etc., and – as has been pointed out in several of our articles lately – can move far more than the “normal” distance of 3 standard deviations in a day (or in any time period, for that matter). These are not the situations in which one wants to sell naked options.

Another point that always gripes me is the myth that a great majority of options expire worthless. Numbers as high as 90% have been bandied about – even on television. These numbers are so preposterous that it’s not even funny. So, to verify that such statements are lies, we took the closing prices from last Friday’s November expiration and analyzed them. There were just over 100,000 options in our data that day. Not all of them traded, but they all had a bid and offer.

We totaled the open interest in them, and those expiring in-the-money (where the last bid was at least a half point) had open interest of 38.2 million, while those expiring out-of-the-money had open interest of 12.5 million. Wow! What numbers – who would have thought 26 years ago when options were first listed – that a single monthly expiration might have over 50 million contracts of open interest?

Back to the subject: that means that 75% of the open interest expired in-the-money. Of course, we are coming off of one of the strongest one-month moves in market history – and this breakdown doesn’t distinguish between puts and calls, nor between stock and index options (but neither does the critic’s statement about 90% of options expiring worthless). Still I would think that at nearly every expiration, we would find something fairly similar to what was seen in November.

How do statements like this get started and why are they perpetuated? The cynical view might be that shrewd option buyers are behind the whole thing – trying to convince hapless (stock) option sellers to wade into the marketplace to sell options to these shrewd buyers. Perhaps a less cynical and more plausible theory might be the fact that novice traders tend to buy options that are too far out-of-the-money initially. Such options have a larger chance of expiring worthless (although I doubt if it’s as high as 90% unless they’re really far out-of-the-money), and hence the novice might be willing to believe and spread a false statement about the probabilities of options expiring worthless. Certain members of the news media and others, uneducated about options, believe the statements and spread them. Nevertheless, you – our readers and subscribers – should certainly know the difference and not be swayed by such gibberish.

**Option Buyers Can Benefit** Does this mean that option buyers are really the big winners? Well, if they’re careful they can be. What many option sellers fail to do is consider what can happen during the life of the option. That’s why we use our Monte Carlo probability calculator for such situations – the probability of a stock hitting a price at any time during the option’s life may be a lot greater than one imagines. The delta of the option is a reasonably good estimate of the probability of the stock being above the strike at the end of the option’s life, but is a low estimate of the probability of it hitting the strike during the option’s life.

Just as we use the probability calculator – which is based strictly on statistical volatility – for straddle buying, one can use it for straight option buying. In essence, this incorporates the “delta” portion of “excess value” in one’s analysis via stock price movement. In addition, if one is careful to buy options that are cheap with respect to their implied volatility – as we do with our straddles – then the “vega” portion of “excess value” is incorporated as well. Only “theta” remains a constant enemy. In order to see how this might work, let’s look at an example of using the Monte Carlo probability calculator to evaluate an outright call purchase:

Example: assume once again that at the current time (November), XYZ is trading at 82 and the Jan 80 call is selling at 8, as above. The implied volatility in this case was just over 50%. For the purposes of this example, let’s also assume that statistical volatility is 50%. That would not necessarily be the case in the “real” world, but it will suffice for this example. Using the Monte Carlo probability calculator, we find that:

Inputs: Current Stock Price: 82 Target Price: 88 Trading days to expiration: 40 Volatility: 50% Outputs: Probability of ever hitting 88: 70% Probability of being over 88 at expiration: 38%

These numbers should raise your eyebrows! Yes, it’s actually true that there’s a 70% chance of the stock trading at 88 at sometime during the option’s life, assuming that our volatility estimate of 50% is correct. In other words, there are good odds of making money by buying this call – far better than you’d surmise if you only knew the “ending” probability of 38%. Add onto this the possibility of implied volatility increasing – certainly a possibility if “cheap” options are purchased initially – and you can see that call option buying is probably a lot better strategy than you’d been lead to believe, according to the false and misleading statements routinely circulated.

*This article was originally published in The Option Strategist Newsletter Volume 8, No. 22 on November 24, 1999. *

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