Options Theta Explained
Options Theta Math
It's not necessary to understand the math behind theta (please feel free to go to the next section if you want), but for those interested theta is defined more formally as the partial derivative of options price with respect to time.
The formula for a call option is below (some knowledge of the normal distribution is required to understand it).
Whether you’re an options holder or writer, you need to understand theta.
This Greek metric will help you make the right decisions and see a successful investment.
As theta has different meanings in other fields (including in economics, where it refers to the reserve ratio of banks), it is important that you learn what theta means in regard to options trading.
How Is Theta Different from the Other Greeks?
All the other Option Greek metrics measure how the price of an option is sensitive to a particular variable. For instance, vega measures how price is sensitive to a change in implied volatility by one percentage point.
Option Delta indicates how the price of the option is sensitive to every $1 change in the underlying asset and Option Gamma shows how a change of $1 to the underlying security affects the delta.
Finally, rho measures sensitivity to a change in interest rates.
Theta, unlike all the above, is not about price sensitivity. Instead, it measures time decay.
What Is Theta?
Theta measures how the value of an option deteriorates over the passage of time. Put simply, it’s the time decay of an option as represented as a dollar or premium amount. Whereas you can calculate the theta on a weekly basis, it is more common for theta to represent a day-to-day time decay.
When all other factors are constant, the option will lose value as it approaches its expiry date. For this reason, the theta is usually a negative value. However, you always need to bear in mind that a significant increase or drop in the price of the underlying asset or a change in implied volatility will also impact option price.
To calculate how theta impacts option price, let’s imagine that a call option is currently $3 and the theta is -0.06. This means that the option will drop in price by $0.06 per day. After one day, the price of the option will have fallen to $2.94. After one week, the price will be $2.58.
How the Passage of Time Impacts Theta
Longer-term options have a theta close to 0 since, there’s no loss of value on a daily basis. Options with a shorter term have a higher theta, since the time value is at its highest and there is more premium to lose on a day-to-day basis.
The theta is at its highest when options are at the money and lowest when they are out of the money or in the money. The theta value rises for options at or near the money as the option nears expiration.
However, in options that are deep in or out of the money, the theta value falls as the option approaches expiration.
Furthermore, when an option is out of the money, the time decay is particularly noticeable. Bear in mind that when an option is out of the money, the underlying asset is lower than the strike price in the case of a call and higher than the strike in the case of a put.
Therefore, when an option that is out of the money moves closer to expiration, the likelihood that it will ever be in the money diminishes.
Theta Curve
An important point to make is that, even if all the other factors do remain equal, time decay is not a linear descent. The theoretical time decay becomes greater (meaning the theta increases dramatically) as options near their expiration date because there is less time for the option to move when it is close to expiration. This leads to what is called the theta curve — where there is a gradual decay early on and an accelerated decay as the option approaches expiration.
Pricing models take weekends and trading holidays into account, either by adjusting volatility or time expiration. This means that you’ll see a decay over seven days, no matter how many trading days are actually in the week. It also means that you cannot cheat the system, such as by opening a new short position late on Friday and closing it early on Monday to collect two free days of time decay.
For the same reason, it can be a good idea to close a position on Friday if it’s showing a reasonable profit — you’re unlikely to see a greater payoff if you wait until Monday. Plus, it’s often possible on the Monday to reenter the position for almost the same price as you exited, should you change your mind.
Nonetheless, the lack of a standardized method of representing the time decay of options means that you may see a different time decay according to which model you use.
Why Does Theta Matter?
Theta gives a numerical value to the risk that options buyers and writers will face due to the passage of time. This risk exists because you only have the right to buy or sell the underlying asset of an option at strike price before the expiry date in options trading.
Therefore, in the case that two options have similar characteristics but one has an expiry date further in the future, the longer option will be more valuable. This is because there is a greater chance that the option will exceed the strike price due to the longer amount of time it has.
This is all down to the fact that the value of an option has intrinsic and extrinsic value. Intrinsic value refers to the profit from an option based on the difference between strike price and market price.
Extrinsic value refers to all the rest of the premium: the value of holding the option and the chance for the option to grow in value as the underlying stock price moves. When all else is equal, the extrinsic value of options will drop over time, leaving only the intrinsic value at expiration.
Volatility and Theta
Typically, an option with a higher volatility of its underlying asset will have a higher theta than a similar option with a low-volatility stock. The reason for this is the higher time value premium of high-volatility options, which means the potential loss each day is greater.
To put this into context, let’s use another example. This time, imagine that our call option is currently $5 and that the underlying stock is trading at $1,030 with a strike price of $1,045. Let’s also say that the option will expire in 10 days and has a theta of -0.5, meaning that the value of the option will decrease by $0.50 each day.
If everything remains the same, the option will already have lost $2.50 by the end of five days. However, if volatility leads the underlying stock to increase in price, this could offset the loss for the option holder that the theta calculated. In the above example, the price of the underlying asset would need to increase to at least $1,050 to give the option $5 in intrinsic value.
Positive and Negative Theta
We previously mentioned that theta is generally negative — it follows, then, that theta can also be positive. This is because both option buyers and option writers can use theta.
Theta is negative when you are in net long in a position. To see a profit as a buyer, therefore, one of two things is necessary: you can either respond quickly and be directionally right or you need implied volatility to be on your side. For the latter, you want to see implied volatility expand more than the theta is able to decay the value of your option.
Negative theta is a reason why it’s important to hedge your long options with short options. For instance, it is better to opt for calendar spreads, vertical spreads, and diagonal spreads than long naked options, as this will allow you to eliminate some (or perhaps all) of the time decay.
Theta is positive when you are net short in a position. Since option writers want their position to lose value, positive theta is favorable. In addition, it’s cheaper to buy back an option to close out a short position.
How to Use Theta
As we already mentioned, theta drops every day when all other factors remain equal. This means you lose money every day after you buy an option. When you choose to buy an option, then, you are expecting that factors will not remain equal — that the price of the underlying asset will move significantly.
Alternatively, if you believe that you’ll see little change in the underlying asset price, theta gives you a good opportunity to short the option. Time decay will bring you a profit, as the option’s value will drop.
Of all the Greeks, theta is the most indefinite. Since the calculation has to assume that implied volatility and price movement is steady (when, of course, it can be anything but), theta is often inaccurate.
For this reason, it’s necessary to consider theta as part of the bigger picture and never in isolation.
List of positive theta options strategies
- Short Call
- Short Put
- Short Straddle
- Short Strangle
- Covered Call Write
- Covered Put Write
- Long Calendar Spread
- Vertical Credit Spread
- Iron Condor
- Butterfly Spread
List of negative theta options strategies
About the Author: Chris Young has a mathematics degree and 18 years finance experience. Chris is British by background but has worked in the US and lately in Australia. His interest in options was first aroused by the ‘Trading Options’ section of the Financial Times (of London). He decided to bring this knowledge to a wider audience and founded Epsilon Options in 2012.
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