# TIME DECAY AND THETA

Time decay is an option’s loss of value as time passes. We know that every option has two components in its value. One is intrinsic value which shows the value based on the moneyness of an option. And the second is the time value component which shows the value of the option that is based on the time left to maturity (Calculated as days left to maturity in practice).

Time value is probably the most crucial part to understand in an option because it’s very common that investors deal with options without necessary knowledge in this component and the results may seem surprising to them. Also, it doesn’t help that it is slightly more complex than common intuition. But let me explain some of the most important details of time decay and share some examples that will help you have a better understanding of options’ relation with time.

### MINI glossary

 Time value: Time value is one of the two components of an option’s value (the other one is intrinsic value) Theta: Theta is the greek parameter of an option that is related to time left to maturity. It shows the time value loss of an option per one day.(Also it is a dynamic value and it changes based on other parameters as well) Time decay: Time decay is the decline of an options price by time and it is not a linear concept. Time decay is expected to increase as the option approaches its maturity.

### what is time value?

In simplest words, time value component of an option is the value that an option provides you based on probabilities in a duration of time. We have touched upon the normal distribution formula it is calculated with named Black & Scholes. This formula looks like complex mathematics at first glance. But what it really does is, it takes into account certain variables that might affect the future outcomes regarding the price changes of underlying security. When you think about those variables, it is all very intuitive. Time left to maturity defines the base of the probability of a price change. If you give any security long enough time, it is expected to see many different price fluctuations. If there is only a couple of days left this option normally doesn’t really provide a huge time value. Unless something very impactful happens in the markets in exactly those days. That’s where the second important variable comes into play: volatility.

The formula, that won the Nobel Prize to its authors is complex enough to take into account many variables other than time value. The more volatile a stock, the more time value is calculated for it since this volatility promises more potential price changes in any direction and that potential means more potential value for your option.

During the course of an option it is usually not so simple to observe what part of the losses might be coming from the loss in time value as the intrinsic value might be independently moving up or down in big or small amounts. If the option’s intrinsic value goes up a lot, option price would increase despite the time value. And if the intrinsic value goes down, option price might go down as well but it might not be so easy to see how much of that is due to time value. Some of the upcoming illustrations will make this concept clearer.

But before that, it is also important to note that time value itself gets affected by other parameters. Normally as the “time to maturity” decreases, an option’s “loss rate of time value” increases. This phenomenon is called time decay. Typically, an option only loses one third of its time value during first half of its life and other two thirds in the second half of its life. Especially the last 30 days, rate of time value loss becomes quite steep.

In the upcoming pages you will find some of the basic examples and simulations that will assist understanding the options’ time value concept further.

### Time value vs intrinsic value

Previously we discussed that, time value calculation is related to all the factors of in the moneyness, time to expiry, risk free interest rate and volatility.

Let’s look at the time value effects of “In the moneyness”. Given that, all the other parameters are stationary, if an option (call or put regardless) is deep in the money, its time value decreases while the intrinsic value increases. If an option is deep out of the money time value again decreases and intrinsic value also decreases. However, when an option is at the money, its time value is at maximum while intrinsic value is at zero. The logical explanation for this is that, when an option is at the money, the probabilities of it becoming in the money is the highest. This probability is what makes the options time value maximum.

Here is an example of an option on Amazon shares. Suppose AMZN is trading at \$1850. We have a call options with different maturity dates and the same strike price of \$1900 below.

Let’s isolate volatility and interest rate so we can observe the time value better. (Volatility approx.: %35, interest rate r:2% – price calculations are approximated and might have small inconsistencies)

Example below demonstrates a progressive increase in the rate of time value loss. Theta here is showing the daily value loss and as the time left to maturity decreases theta becomes a much larger value.

Here is an example:

 AMZN: \$1850, Call, Strike: \$1900, Vol: 35%, r: 2% Days to maturity Price* Theta 210 117.65 -0.25 165 109.50 -0.37 135 76.10 -0.34 105 63.30 -0.36 75 49.09 -0.43 35 40.80 -0.77 20 20.20 -0.51 15 14.75 -0.73 8 8.80 -0.87 1 1.75 -1.74

Source : Nasdaq Option Chain (AMZN)

This shows the exponentially increasing nature of theta which means options lose more and more time value as they approach the expiry date.

You can also see that price of the option in the first interval drops from \$103 to \$86 in almost 50 days. But in the last interval 4 day causes a \$10+ drop in the price. This non-linear acceleration is a very important characteristic of options.

Note how quickly time premium decays around 30 days prior to expiration. We can see that Theta is not a linear progression as the option advances toward expiration. Rather, options with the least remaining time to maturity will tend to decay the most.

### example

Let’s look at different Amazon Call options with an “at the money strike price”. If we look at different examples with different maturity dates in the future, our options will show their time value with regards to time left to expiration.

Let’s suppose Amazon shares are trading at \$1850.

All the call options above are approximately at the money with the same strike price of \$1850.

 Calls Last Chg Bid Ask Vol Open Root Strike 3-Jan-2020 17.2 -15.85 16.85 17.15 3609 716 AMZN 1850 7-Feb-2020 66.3 -16.5 66.85 67.55 73 44 AMZN 1850 20-Mar-2020 89.75 -14.75 90.4 90.95 137 353 AMZN 1850 17-Apr-2020 101.5 -30.85 103.65 105.55 53 124 AMZN 1850 19-Jun-2020 134.9 -14.25 135.05 136.65 127 340 AMZN 1850 17-Jul-2020 147 -15.24 145.25 147.65 8 35 AMZN 1850

Option chain table shows how rapidly the option’s value decreases from February to January maturity dates compared to July to June. Price difference in the former spread is almost \$50 while in the latter spread it’s nearly \$12 although both spreads are approximately

1 month apart. This example clearly shows the time decay phenomenon in options’ time value.