Option Greeks: Theta

In this next part of our series on the option Greeks, after delta and gamma, we will study the next of the four Greek letters: theta.

Option Greeks (Part 3) – Theta

Time is money

Suppose NIFTY is trading at 8500 on the spot market and you buy a call option

 with a strike of 8700, expiring in one month. The question is: by expiration, will your option be ITM, ATM or OTM? Naturally, you always want your option to end up ITM. We need to judge how likely it is that the spot price rises from 8500 to above 8700 (a 200-point gain) within the next 30 days, based on a few assumptions:

• The chance of the spot price rising 200 points in the next 30 days is fairly high, because there is still plenty of time.
• The chance of it rising 200 points in the next 15 days is also relatively high.
• But with only five days left to expiration, a 200-point rise would be quite difficult.

Clearly, the more time there is until expiration, the more likely the call buyer’s option is to end up ITM. The call seller collects a fixed premium and faces unlimited risk if the spot price rises. The seller’s risk therefore grows the longer the option’s term. To compensate for that risk, sellers charge a higher premium for longer-dated options than for shorter-dated ones.

Time alone, however, does not capture everything that affects the option premium. We also need to look at the option’s intrinsic value to understand what drives the premium.

Option premium = Time value + Intrinsic value

Intrinsic value here means the amount you would receive if you exercised the option today.

Back to our example:

Suppose NIFTY is trading at 8423 on the spot market.

1. Option with a strike of 8350 (call option)
2. Option with a strike of 8450 (call option)
3. Option with a strike of 8400 (put option)
4. Option with a strike of 8450 (put option)

We know that intrinsic value is always greater than or equal to zero. For a call, Intrinsic value = Spot price – Strike price; for a put, Intrinsic value = Strike price – Spot price.

So the intrinsic value of option (1) = 8423 – 8350 = +73

The intrinsic value of (2) = 8423 – 8450 = (-), so it equals 0 (meaning the call would not be exercised)

The intrinsic value of (3) = 8400 – 8423 = (-), so it equals 0 (meaning the put would not be exercised)

The intrinsic value of (4) = 8450 – 8423 = +27

Now consider another example:

Today, intrinsic value = 8531 – 8600 = (-), so it equals 0; since the premium (99.4) = intrinsic value (0) + time value, the time value here is 99.4. In other words, people are willing to pay a premium of 99.4 for an option with zero intrinsic value, because it has time value.

• Spot price = 8531.
• Strike price = 8600 (call option).
• Option premium = 99.4.
• Today is 1 March 2022.
• The option expires on 31 March 2022.

Today, intrinsic value = 8531 – 8600 = (-), so it equals 0; since the premium (99.4) = intrinsic value (0) + time value, the time value here is 99.4. In other words, people are willing to pay a premium of 99.4 for an option with zero intrinsic value, because it has time value. Remember: “Time is money”.

On 2 March 2022,

On 2 March 2022, the premium equals  the time value, 87.9 (because intrinsic value is still zero).

You can see that the spot price rose slightly from 8531 (1 March 2022) to 8538 (2 March 2022), yet the premium fell from 99.4 to 87.9. On 1 March 2022, intrinsic value = 8531 – 8600 = (-), so it equals 0, meaning there is no intrinsic value and the premium equals the time value of 99.4. Likewise, on 2 March 2022 the premium equals  the time value, 87.9 (because intrinsic value is still zero). So the premium fell the next day because of the passage of time and changes in volatility. We will study volatility in the next chapter.

What theta means

Both calls and puts lose value as time passes. Theta is the measure of the value lost with the passage of time. Option buyers have negative theta, meaning they lose money with every day that passes. For option sellers, theta is positive, meaning they benefit with every day that passes. For example, suppose you sell a put at 54 with a theta of 0.75; three days later, all else being equal, the put will be worth 54 – 0.75*3 = 51.75. If you buy the put back at that point to close your original position, you pay 51.75, so you gain 54 – 51.75 = 2.25.

Next, let us look at a chart of how fast option premiums decay over time.

Option Premium vs Time to Expiry

• Early on, when there is plenty of time left, an option does not lose much value. For example, with 120 days to expiration the premium is 350; with 100 days to expiration it is 300, a drop of 50. The effect of time, theta, is relatively small.
• With 20 days to expiration the premium is 150, and as expiration approaches it falls to about 50 points.
• You can see that if you sell longer-dated options, you collect a higher premium than with shorter-dated ones.

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