Option Theta: What conditions are needed for Theta > P/N, where P = option price, and N = days to expiration?

Question

The wikipedia definition for Theta is:

And people frequently refer to the picture below, to show what typically happens with an option's value as a function of time; we can note this is non-linear, with the highest time loss occurring as one gets close to expiration. As such, generally speaking, one (I) would expect that at any given point in time, the theta of an option would generally be less than its current price, divided by the days remaining; the reason being, it's going to continue to decay faster, in a non-linear manner.

So, if an out-of-the-money option (all time value) has a price P (say $3.00), and there are N days to expiration (say 45), can you help me understand scenarios where theta would be 1) greater than P/N ( > $3.00/45) and 2) less than P/N?

Final Note: From the recent far out-of-the-money PUT options I looked at on SPY, in each case they fell into case 1 to my surprise. With 45 days to to expiration, their theta was significantly higher than their price divided by 45. So, I'm wondering what conditions result in case 1 -- what causes this to happen?

Answer 1

"So, if an out-of-the-money option (all time value) has a price P (say $3.00), and there are N days..."

The extrinsic value isn't solely determined by time value as your quote suggests. It's also based on volatility and demand. Here is a quote from http://www.tradingmarkets.com/options/trading-lessons/the-mystery-of-option-extrinsic-value-767484.html distinguishing between extrinsic time value and extrinsic non-time value:

The time value of an option is entirely predictable. Time value premium declines at an accelerating rate, with most time decay occurring in the last one to two months before expiration. This occurs on a predictable curve.

Intrinsic value is also predictable and easily followed. It is worth one point for every point the option is in the money. For example, a call with a strike of 30 has three points of intrinsic value when the current value of the underlying stock is $33 per share; and a 40 put has two points of intrinsic value when the underlying stock is worth $38.

The third type of premium, extrinsic value, increases or decreases when the underlying stock changes and when the distance between current value of stock and strike of the option get closer together. As a symptom of volatility, extrinsic value may be greater for highly volatile underlying stock, and lower for less volatile stocks. Extrinsic value is the only classification of option premium that is unpredictable.

The SPYs you point out probably had a volatility component affecting value. This portion is a factor of expectations or uncertainty. So an event expected to conclude prior to expiration, but of unknown outcome can cause theta to be higher than p/n. For example, a drug company is being sued and the outcome of a trial will determine whether that company pays out millions or not. The extrinsic will be higher than p/n prior to the outcome of the trial then drops after.

Of course, the most common situation where this happens is earnings. After the announcement, it's not unusual to see a dramatic drop in the extrinsic portion of options. This is why sometimes a new option trader gets angry when buying calls prior to earnings. When 'surprise' good earnings are announced as hoped, the rise is stock price is largely offset by a fall in extrinsic value giving call holders little or no gain!

As for the reverse situation where theta is lower than p/n would expect? Well you can actually have negative theta meaning the extrinsic portion rises over time. (this statement is a little confusing because theta is usually described as negative, but since you describe it as a positive number, negative here means the opposite of what you'd expect).

This is a quote from "Option Volatility & Pricing". Keep in mind that they use 'positive' theta to mean the time value increases up over time:

Is it ever possible for an option to have a positive theta such that if nothing changes the option will be worth more tomorrow than it is today? When futures options are subject to stock-type settlement, as they currently are in the United States, the carrying cost on a deeply in-the-money option, either a call or a put, can, under some circumstances, be greater than the volatility component. If this happens, and the option is European (no early exercise permitted), it will have a theoretical value less than parity (less than intrinsic value). As expiration approaches, the value of the option will slowly rise to parity. Hence, the option will have a positive theta.

Sheldon Natenberg. Option Volatility & Pricing: Advanced Trading Strategies and Techniques (Kindle Locations 1521-1525). Kindle Edition.