Theta

14.1 – Time is money

Remember the adage “Time is money”, it seems like this adage about time is highly relevant when it comes to options trading. Forget all the Greek talk for now, we shall go back to understand one basic concept concerning time. Assume you have enrolled for a competitive exam, you are inherently a bright candidate and have the capability to clear the exam, however, if you do not give it sufficient time and brush up the concepts, you are likely to flunk the exam – so given this what is the likelihood that you will pass this exam? Well, it depends on how much time you spend to prepare for the exam right? Let’s keep this in perspective and figure out the likelihood of passing the exam against the time spent preparing for the exam.

Quite obviously higher the number of days for preparation, the higher is the likelihood of passing the exam. Keeping the same logic in mind, think about the following situation – Nifty Spot is 8500, you buy a Nifty 8700 Call option – what is the likelihood of this call option to expire In the Money (ITM)? Let me rephrase this question in the following way –

• Given Nifty is at 8500 today, what is the likelihood of Nifty moving 200 points over the next 30 days and therefore 8700 CE expiring ITM?
  • The chance for Nifty to move 200 points over the next 30 days is quite high, hence the likelihood of an option expiring ITM upon expiry is very high

• What if there are only 15 days to expiry?
  • An expectation that Nifty will move 200 points over the next 15 days is reasonable, hence the likelihood of an option expiring ITM upon expiry is high (notice it is not very high, but just high).

• What if there are only 5 days to expiry?
  • Well, 5 days, 200 points, not really sure hence the likelihood of 8700 CE expiring in the money is low

• What if there was only 1 day to expiry?
  • The probability of Nifty to move 200 points in 1 day is quite low, hence I would be reasonably certain that the option will not expire in the money, therefore the chance is ultra-low.

Is there anything that we can infer from the above? Clearly, the more time for expiry the likelihood for the option to expire In the Money (ITM) is higher. Now keep this point in the back of your mind as we now shift our focus on the ‘Option Seller’. We know an option seller sells/writes an option and receives the premium for it. When he sells an option he is very well aware that he carries an unlimited risk and limited reward potential. The reward is limited to the extent of the premium he receives. He gets to keep his reward (premium) fully only if the option expires worthless. Now, think about this – if he is selling an option early in the month he very clearly knows the following –

1. He knows he carries unlimited risk and limited reward potential
2. He also knows that by virtue of time, there is a chance for the option he is selling to transition into ITM option, which means he will not get to retain his reward (premium received)

In fact at any given point, thanks to ‘time’, there is always a chance for the option to expire in the money (although this chance gets lower and lower as time progresses towards the expiry date). Given this, an option seller would not want to sell options at all right? After all, why would you want to sell options when you very well know that simply because of time there is scope for the option you are selling to expire in the money. Clearly, time in the option sellers context acts as a risk. Now, what if the option buyer in order to entice the option seller to sell options offers to compensate for the ‘time risk’ that he (option seller) assumes? In such a case it probably makes sense to evaluate the time risk versus the compensation and take a call right? In fact, this is what happens in real-world options trading. Whenever you pay a premium for options, you are indeed paying towards –

1. Time Risk
2. The intrinsic value of options.

In other words – Premium = Time value + Intrinsic Value Recall earlier in this module we defined ‘Intrinsic Value’ as the money you are to receive if you were to exercise your option today. Just to refresh your memory, let us calculate the intrinsic value for the following options assuming Nifty is at 8423 –

1. 8350 CE
2. 8450 CE
3. 8400 PE
4. 8450 PE

We know the intrinsic value is always a positive value or zero and can never be below zero. If the value turns out to be negative, then the intrinsic value is considered zero. We know for Call options the intrinsic value is “Spot Price – Strike Price” and for Put options, it is “Strike Price – Spot Price”. Hence the intrinsic values for the above options are as follows –

1. 8350 CE = 8423 – 8350 = +73
2. 8450 CE = 8423 – 8450 = -ve value hence 0
3. 8400 PE = 8400 – 8423 = -ve value hence 0
4. 8450 PE = 8450 – 8423 = + 27

So given that we know how to calculate the intrinsic value of an option, let us attempt to decompose the premium and extract the time value and intrinsic value.

Details to note are as follows –

• Spot Value = 8531
• Strike = 8600 CE
• Status = OTM
• Premium = 99.4
• Today’s date = 6th July 2015
• Expiry = 30th July 2015

Intrinsic value of a call option – Spot Price – Strike Price i.e 8531 – 8600 = 0 (since it’s a negative value) We know – Premium = Time value + Intrinsic value 99.4 = Time Value + 0 This implies Time value = 99.4! Do you see that? The market is willing to pay a premium of Rs.99.4/- for an option that has zero intrinsic value but ample time value! Recall time is money ☺ Notice the underlying value has gone up slightly (8538) but the option premium has decreased quite a bit! Let’s decompose the premium into its intrinsic value and time value – Spot Price – Strike Price  i.e  8538 – 8600 = 0 (since it’s a negative value) We know – Premium = Time value + Intrinsic value 87.9 = Time Value + 0 This implies Time value = 87.9! Notice the overnight drop in premium value? We will soon understand why this happened. Note – In this example, the drop in premium value is 99.4 minus 87.9 = 11.5. This drop is attributable to a drop in volatility and time. We will talk about volatility in the next chapter. For the sake of argument, if both volatility and spot were constant, the drop in premium would be completely attributable to the passage of time. I would suspect this drop would be around Rs.5 or so and not really Rs.11.5/-. 

• Spot Value = 8514.5
• Strike = 8450 CE
• Status = ITM
• Premium = 160
• Today’s date = 7th July 2015
• Expiry = 30th July 2015

Intrinsic value of call option – Spot Price – Strike Price  i.e  8514.5 – 8450 = 64.5 We know – Premium = Time value + Intrinsic value 160 = Time Value + 64.5 This implies the Time value = 160 – 64.5 = 95.5 Hence out of the total premium of Rs.160, traders are paying 64.5 towards intrinsic value and 95.5 towards the time value. You can repeat the calculation for all options (both calls and puts) and decompose the premium into the Time value and intrinsic value.

14.2 – Movement of time

Time as we know moves in one direction. Keep the expiry date as the target time and think about the movement of time. Quite obviously as time progresses, the number of days for expiry gets lesser and lesser. Given this let me ask you this question – With roughly 18 trading days to expiry, traders are willing to pay as much as Rs.100/- towards time value, will they do the same if the time to expiry was just 5 days? Obviously, they would not right? With lesser time to expiry, traders will pay a much lesser value towards time.

• Date = 29th April
• Expiry Date = 30th April
• Time to expiry = 1 day
• Strike = 190
• Spot = 179.6
• Premium = 30 Paisa
• Intrinsic Value = 179.6 – 190 = 0 since it’s a negative value
• Hence time value should be 30 paisa which equals the premium

With 1 day to expiry, traders are willing to pay a time value of just 30 paise. However, if the time to expiry was 20 days or more the time value would probably be Rs.5 or Rs.8/-. The point that I’m trying to make here is this – with every passing day, as we get closer to the expiry day, the time to expiry becomes lesser and lesser. This means the option buyers will pay lesser and lesser towards time value. So if the option buyer pays Rs.10 as the time value today, tomorrow he would probably pay Rs.9.5/- as the time value. This leads us to a very important conclusion – “All other things being equal, an option is a depreciating asset. The option’s premium erodes daily and this is attributable to the passage of time”. Now the next logical question is – by how much would the premium decrease on a daily basis owing to the passage of time? Well, Theta the 3rd Option Greek helps us answer this question.

14.3 – Theta

All options – both Calls and Puts lose value as the expiration approaches. The Theta or time decay factor is the rate at which an option loses value as time passes. Theta is expressed in points lost per day when all other conditions remain the same. Time runs in one direction, hence theta is always a positive number, however, to remind traders it’s a loss in options value it is sometimes written as a negative number. A Theta of -0.5 indicates that the option premium will lose -0.5 points for every day that passes by. For example, if an option is trading at Rs.2.75/- with a theta of -0.05 then it will trade at Rs.2.70/- the following day (provided other things are kept constant). A long option (option buyer) will always have a negative theta meaning all else equal, the option buyer will lose money on a day by day basis. A short option (option seller) will have positive theta. Theta is a friendly Greek to the option seller. Remember the objective of the option seller is to retain the premium. Given that options lose value on a daily basis, the option seller can benefit by retaining the premium to the extent it loses value owing to time. For example, if an option writer has sold options at Rs.54, with a theta of 0.75, all else equal, the same option is likely to trade at – =0.75 * 3 = 2.25 = 54 – 2.25 = 51.75 Hence the seller can choose to close the option position on T+ 3 day by buying it back at Rs.51.75/- and profiting Rs.2.25 …and this is attributable to theta!

This is the graph of how premium erodes as a time to expiry approaches. This is also called the ‘Time Decay’ graph. We can observe the following from the graph –

1. At the start of the series – when there are many days for expiry, the option does not lose much value. For example, when there were 120 days to expiry the option was trading at 350, however, when there were 100 days to expiry, the option was trading at 300. Hence the effect of theta is low
2. As we approach the expiry of the series – the effect of theta is high. Notice when there were 20 days to expiry the option was trading around 150, but when we approach towards expiry the drop in premium seems to accelerate (option value drops below 50).

So if you are selling options at the start of the series – you have the advantage of pocketing a large premium value (as the time value is very high) but do remember the fall in premium happens at a low rate. You can sell options closer to the expiry – you will get a lower premium but the drop in premium is high, which is advantageous to the options seller. Theta is a relatively straightforward and easy Greek to understand. We will revisit theta again when we will discuss cross dependencies of Greeks. But for now, if you have understood all that’s being discussed here you are good to go. We shall now move forward to understand the last and the most interesting Greek – Vega!

Key takeaways from this chapter

1. Option sellers are always compensated for the time risk
2. Premium = Intrinsic Value + Time Value
3. All else equal, options lose money on a daily basis owing to Theta
4. Time moves in a single direction hence Theta is a positive number
5. Theta is a friendly Greek to option sellers
6. When you short naked options at the start of the series you can pocket a large time value but the fall in premium owing to time is low
7. When you short option close to expiry the premium is low (thanks to time value) but the fall in premium is rapid