# Option Theta

This is part of option Greeks tutorials. See also delta, gamma, vega, rho.

## What Is Theta

Options generally lose value with passing time. For example, an option which is worth \$4.83 today may only be worth \$4.79 tomorrow and \$4.55 next week, without the market moving. This process is known as time decay. Theta measures the speed of time decay – how much option premium will decrease in one day.

## Example

Consider a \$100 strike call option with 8 weeks (56 days) left to expiration. Its underlying stock is trading at \$101. The option’s premium, currently at \$4.83, consists of intrinsic value (101 – 100 = \$1) and time value (4.83 – 1 = \$3.83).

The option’s theta is -0.04. It means the option premium will decrease by 0.04 to \$4.79 until the next day (as number of days to expiration decreases by 1), if everything else remains the same. We could also say the option’s time value will decrease by 0.04 to \$3.79, because passing time only affects time value; intrinsic value can change only if underlying stock moves.

Let’s assume nothing happens in the market until the next day. The stock stays at \$101 and the option is indeed trading at \$4.79. Now it has 55 days left to expiration and theta of -0.04, indicating that the premium will decrease to \$4.75 until the next day.

## Theta Values

Options generally have negative theta – lose value with passing time. There are some rare exceptions, but for now it is safe to take it as a universal rule.

Short option positions have positive theta and benefit from time decay (unless other factors like underlying price or volatility move against them).

There is no theoretical limit on how large an option’s theta can be. In general, at the money options have greatest (most negative) theta, as they have more time value to decay than out of the money or in the money options.

Moneyness (the relationship between underlying price and strike price) is only one of several factors affecting theta. Like all the other Greeks, theta can also change with changing volatility, interest rates, or passing time. But before we discuss these, let’s make a quick note about theta sign.

## Positive or Negative Sign?

There is some inconsistency in expressing theta among market practitioners and resources. Some write it with positive sign (theta of 0.04 means the option will lose \$0.04 in one day), while others write it as negative number. We stick to the latter approach in this tutorial and website.

There are good logical reasons for either approach. The positive sign format fits better with option model mathematics, where theta is the derivative of option premium with respect to time to expiration. When time to expiration increases (going back in time), option premium increases, and therefore the derivative must be positive. In practical use, traders are rarely interested in what will happen to their position yesterday – they are more concerned about the future. Negative sign is more consistent with the way traders normally read and work with theta, as well as the other Greeks – a negative number means the position will lose money.

Regardless of the sign you prefer, when we say “theta increases”, we usually mean it in absolute terms – the size of theta increases. For example, theta increases from -0.04 to -0.08, although mathematically it is incorrect. Similarly, we say that theta of -0.08 is “greater” than theta of -0.04.

While all this attention to signs and wording may seem overly meticulous when talking about a single option, it becomes more important with more complex portfolios composed of both long and short options, when the direction of the aggregate exposure may not be immediately clear.

## Theta Units

Let’s make another quick note about units. While theta is most commonly expressed as dollars per calendar day, some traders prefer trading days. Those with very long time horizon may measure theta in weeks, while short term traders or those working with options very close to expiration may measure theta in hours or shorter intervals.

Markets are closed on weekends and holidays. As a result, the number of trading days per year is much smaller (252 on average in the US) than the number of calendar days (365). Most option pricing models work with time in years (time to expiration enters the model as % of year). After they calculate theta, they convert it to days by dividing by the number of days per year. Dividing by 365 or 252 obviously leads to very different results.

While numerous academic papers have been written on which approach is more accurate (most tend to side with calendar days), neither is incorrect and neither is perfect – we must keep in mind that any model is just a simplified picture of reality. It is OK to use either calendar days or trading days. The only requirement is consistency and knowing what exactly your theta figures mean.

When using calendar days, theta means how much the option or portfolio value will change in one calendar day. When using trading days, it is per one trading day.

With the formats and units out of the way, let’s now explore how theta changes with passing time and changing volatility.

## How Theta Changes with Passing Time

The rate of time decay is not constant. Even when all the other factors (underlying price, volatility, interest rates) stay the same, theta changes as an option gets closer to expiration. It can increase or decrease, depending on the option’s moneyness.

Theta of options which are near or at the money tends to increase (in absolute terms) with passing time. When there are still several months left before expiration, the rate of time decay is relatively slow, and it accelerates as expiration approaches. At the money options have greatest theta in the final days before expiration. If we measured theta for infinitely short intervals instead of days, an at the money option’s theta would be infinitely large at the moment of expiration.

Options which are further out of the money or deeper in the money tend to lose most of their time value earlier. With long time left to expiration, their theta may also increase with passing time. However, as their remaining time gets closer to zero, theta starts to decrease, until both time value and theta get to almost zero. This can happen several weeks before expiration (the further from at the money the option is, the earlier this happens). In the final days, far out of the money and deep in the money options have virtually no time value left and no theta.

Note that although out of the money and in the money options lose a bigger part of their (relatively small) time value earlier, at the money options always have higher theta – even with more time left to expiration, because their time value is much bigger to start with.

## How Theta Changes with Volatility

The effect of volatility on theta is simple: Higher volatility means more time value and higher theta, other things being equal.

The idea of making money “automatically” from passing time can tempt some inexperienced traders to treating positive theta strategies as sure ways to profits. Some resources describe such strategies, like short straddles or iron condors, as “income strategies”, which may further contribute to this illusion.

It is important to understand that mere passage of time can never, on its own, generate returns higher than the risk-free interest rate.

This doesn’t mean that positive theta strategies are bad. Indeed, some of them have shown very good risk-adjusted performance in a wide range of underlying markets in the long run. However, the source of their profits is not passage of time itself; it is passage of time combined with mispriced volatility.

Options on many underlyings tend to trade at inflated prices, higher than what actual realized volatility would justify. This is called volatility risk premium.

An iron condor is not a good trade just because it has positive theta. It is a good trade if (and only if) you can make it at a good price, or in other words, if the positive theta is large enough to compensate you for the risk of negative gamma. Being able to make such decision requires experience and good understanding of volatility, option prices and Greeks.

## Theta and Gamma Relationship

You can think of theta (time decay) as the other side of gamma (optionality). These two Greeks represent the cost and benefit of options.

Long option positions generally have negative theta and positive gamma (you pay for buying optionality).

Short option positions have positive theta and negative gamma (you get paid for providing optionality).

Positive theta is good: you make money with passing time. Negative theta is bad.

Positive gamma is good: if the underlying price moves in your favor, your profits accelerate; if it moves against you, your losses slow down. Negative gamma is bad: accelerating losses and decelerating profits.

An ideal position would have positive gamma and positive theta. Unfortunately, there is no such option strategy.

Every strategy has strengths and weaknesses. It can position you favorably to either market moves or passage of time – but not both, as one pays for the other. Option theta and gamma provide quick information on where you are in this tradeoff.

## How to Calculate Theta

Mathematically, theta is the derivative of option premium with respect to time to expiration (multiplied by -1 when using the negative sign as we do here). This tutorial focuses mainly on the logic and practical use of theta. If you are interested in the mathematics, you can find exact theta formulas in Black-Scholes Greeks Formulas and Option Greeks Excel Formulas.

## Summary

• Theta measures how much an option’s price will change in one day.
• All options (with some rare exceptions) have negative theta – lose value with passing time.
• Theta is greatest at the money. At the money theta is greatest just before expiration.
• An increase in volatility increases time value and thereby theta.
• Short option positions have positive theta and profit from passing time.
• Positive theta goes hand in hand with negative gamma. There is no free lunch.