Theta and time decay

An option is a right with an end date, and part of its price pays for the time until that date. As the days pass that part shrinks. Theta is the measure of how fast.

What the theta of an option measures

Theta is a theoretical estimate of the change of the premium, in dollars per share, when one day passes and every other pricing factor stays the same.

A call with a premium of $1.80 and a theta of -0.03 is expected to cost about $1.77 tomorrow if nothing else changes. One contract covers 100 shares, so the contract loses about $3 a day.

A bought option has a negative theta and a written option a positive one. The writer of an option gains what its holder loses to time.

Why only time value decays

A premium is intrinsic value plus time value. Intrinsic value depends on the stock price and the strike price, and the calendar does not change it. Time value is what the market pays for the time left, and at expiration none is left.

Time decay is this loss of time value. An option at the money, whose premium is all time value, has the most to lose. An option deep in the money has a large premium and little time value, and it decays slowly.

How time decay speeds up before expiration

The table follows a put with a strike price of $50 on a stock that stays at $50, in the Black-Scholes model with a volatility of 30%, an interest rate of 4% and no dividend.

The premium falls by $1.09 in the 60 days from day 90 to day 30 and by $1.63 in the last 30 days. Time decay is not linear. It is gradual at first and speeds up as expiration comes closer.

This holds for options at the money. An option far out of the money has lost most of its time value well before the last week. In the same model a 45 put on the $50 stock is worth $0.20 with 30 days left and $0.04 with 14.

Example of time decay for a holder and for a writer

What else moves the premium while time passes

Theta assumes that nothing else changes, and something always does. A move of $1 in the stock changes the premium of an option at the money by about $0.50, which is more than two weeks of decay in the table. A change in implied volatility moves the premium as well.

Time decay is for this reason the one part of the result that is certain in direction, and often the smallest part. A writer who collects theta has been paid to carry the risk of those other moves.

What can go wrong with time decay

A holder can be right about the direction of the stock and lose money, because the move took longer than the option had. Each day without the move costs the holder the theta of that day, and the cost per day rises as expiration comes closer.

A writer can collect decay for weeks and lose more than all of it in a day. In the model of the table, the put written at $1.63 with the stock at $50 is worth $3.32 a week later if the stock has fallen to $47, twice what was received. The decay received is small and steady, and the loss arrives at once.

Theta is an estimate of a model. It is computed for one day and changes every day, with the price of the stock and with implied volatility. The industry has no single method for it, so two pricing models can show two values of theta for the same option.