How Does Delta Change Over Time?

How accurate is our delta model in the real world of options? The following table illustrates current- and near-month deltas on Nvidia Corp. (NVDA) when the stock was $40, with about three weeks remaining until July expiration:

Figure 5.6

Current- and Near-Month Option Deltas

Nvidia (NVDA) Stock $40

Status; Predicted July Call Delta August Call Delta Strike August Put Delta July Put Delta Status; Predicted
ITM – .90 87.22% 77.59% 35.0 -23.03% -13.02% OTM – .20
ITM – .80 72.06% 66.88% 37.5 -33.94% -29.11% OTM – .30
ATM – .55 50.14% 54.63% 40.0 -46.45% -50.46% ATM – .50
OTM – .30 29.81% 42.06% 42.5 -59.34% -70.95% ITM – .80

Figure 5.6 notes the predicted current-month deltas for both Nvidia’s calls and puts. Though not exactly on the money, the real deltas closely match up to the textbook deltas. The ITM deltas in particular seem a bit low. Another week or ten days closer to July expiration, however, and the ITM deltas would have been much higher.

But something else is going on in Figure 5.6. Notice how for both puts and calls, the option deltas for August are either higher or lower than for July. Here is what the table tells us about August option deltas compared to those for July:

ITM Strikes     ATM Strikes    OTM Strikes

August Options  LOWER            CLOSE              HIGHER


If this seems odd, it should not. As we will see, it devolves to the notion already discussed that delta expresses probabilities and ITM options are considered to be “ahead” and OTM options “behind.” The following segment will explore this further.

Effect of Time on Delta (Delta Horse Race)

But our discussion of delta has so far concentrated on the near-dated options. Does the amount of time to expiration affect delta? Are the deltas of all strikes affected the same?

The effect of strike and time on delta can be better understood, perhaps, by analogy to a horse race. Imagine that odds were not frozen at race time but could change dynamically as the race progressed and that bets could continue to be placed almost up to the finish line; the finish line in this case being option expiration.

As the horses burst from the gates, they all have – at least theoretically – a chance to win. In a real horse race, odds will vary by horse due to perceptions of its likelihood of winning. Deltas work the same way. Though the future is unknown and any option strike might win, ITM strikes are considered to have a higher probability of winning and thus a higher delta; and OTM strikes have a lesser probability of winning and therefore lower delta.

Essentially, think of ITM options as the most-favored horses; and OTM options as the least favored horses.

As the race progresses, some horses are out front and some have fallen behind. The ones out front are the “ITM” horses, which are more likely to win than the “OTM” horses. The further the horses run – the more time elapses – the ITM horses have a progressively higher probability of winning. And their delta increases correspondingly.

The trailing (OTM) horse has a very low chance of winning, thus a low delta. With this back drop, let’s dig into Figure 5.7’s “horse race” diagram to understand the effect of time and strike price on option deltas:

Figure 5.7

Figure 5.7 displays the changing deltas over time of various call strikes for the January 2009 LEAPS call options, beginning in December (the starting line). January is of course the finish line (expiration). We only care how delta for the January 2009 calls changes over time. Figure 5.7 does not take into account any changes in stock price and assumes the stock is $50 from start to finish. Sure, this is not realistic, but the purpose of this analogy is to understand the effect of time and strike on delta.

The first thing to notice is that in December, the deltas of the different call strikes are somewhat bunched together. Yes, the ITM calls have the highest deltas due to a higher probability of winning, and the OTM calls have the lowest ones, as we would expect. But their respective deltas are bunched up much like horses at the starting gates. However, as time passes the ITM calls are rewarded with increasing deltas, and the OTM calls are rewarded with falling deltas. The ATM calls start out slightly higher than the average 55% for a current-month call and then fall to about the 55% level as time elapses.

In other words, deltas are bunched up far from expiration and spread out as expiration approaches.

Now, look at the January 2009 deltas as expiration grows very near. The deepest ITM call’s delta has shot from 78% to 98% over time. This is because the ITM $40 Call was the likeliest to win, and that probability is only increasing close to January 2009 expiration; delta increased accordingly.

But the deeply OTM $60 Call’s delta has absolutely collapsed from 53% to 8%. It started with the lowest probability of winning, and close to expiration it seems a virtual certainty that the $60 Call will expire worthless. Accordingly, its delta has fallen to only 8%.

Why do we care about delta? People often buy or write an option that is months out and then are staggered when it hardly moves as the stock price changes. Figure 5.7 makes the reason for this abundantly clear.

What the market giveth, it also taketh away. Recall that the amount of time value premium per month declines the further out in time the expiration month, due to premium compression. This compression is great for the option buyer, because it cheapens the cost of buying long-term options, all other factors being equal. Yet to get decent delta in a long-dated option, it is really necessary to buy an ITM option.

Yet the lower delta of ITM long-term options undercuts this objective. Put differently, the premium compression “discount” realized upon buying the long-term option is to some extent recaptured by the market in the form of lower delta.

The delta of long-term options also hurts the call writer, because the calls are so slow to lose value if a fall in the stock price requires repurchasing the short calls. Yet the call writer was hurt, not benefited, by premium compression upon writing them. This is why it seldom makes sense to write long-term calls.

Note that implied volatility (IV) also can affect delta, because it affects time value; higher IV adds time value and lower IV takes it away. An increase in IV will increase the deltas of OTM options and decrease the deltas of ITM options. A spike or collapse in IV can therefore dramatically affect option delta. Actual volatility has the same effect; higher volatility increases OTM delta and decreases ITM delta, because the OTM calls have a greater chance of winning and the ITM calls a greater chance of losing on volatile stocks.

How to Make Use of Delta

Here are some rough guidelines that may help orient you to using delta in your trades:

Figure 5.8

Using Delta in Options Positions
Strategy Outlook Action
Short-term speculation on the stock You expect a move up in the next two months Buy front- or near-month ITM options with low time value. Delta justifies the higher cost.
Long-term speculation on the stock You expect a move up but it could take many months Buy longer-dated ITM options to get better delta, which will increase with time.
Buying puts as a long-term hedge for portfolio stock (married put) You fear a decline in a stock being held for the long term. Buy long-term OTM or ATM puts in order to reduce cost, and sell calls each month to pay for them.
Writing covered calls Goal is buy-writing to maximize income. Sell ATM calls for the current or next month to maximize rate of return and time decay. Sell ITM calls if bearish or conservative, OTM calls if short-term bullish.
Buying LEAPS calls to write calls against them Goal is to write calls against the LEAPS calls instead of buying the underlying stock Buy ITM LEAPS calls with low time value; though delta will be low, it still will be the highest of all.

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