Theta (Options)

Theta, in options trading, is the rate at which the price of an option changes as time passes, assuming the price of the underlying asset, its volatility and interest rates stay the same. It is one of the option Greeks. Theta is normally quoted as the change in the option's value over one day, and for an option that has been bought it is usually negative: the option loses a little value each day. The effect is called time decay.

Time decay exists because part of an option's price reflects the chance that the underlying will move favourably before expiry. As the remaining time shrinks, that chance shrinks with it. Theta therefore works against buyers of options and in favour of sellers, and it is the cost that buyers pay for the benefits measured by gamma.

In the Black-Scholes-Merton model, theta for a European call on an asset paying no dividends is:

• $S$ is the price of the underlying asset and $K$ the strike price.
• $\sigma$ is the annual volatility and $T$ the time to expiry in years.
• $r$ is the continuously compounded risk-free interest rate.
• $N'(d_1)$ is the standard normal density at $d_1$, and $N(d_2)$ is the cumulative standard normal probability at $d_2$.

The formula gives theta per year. Dividing by 365 gives the figure per calendar day.

Take a one-year call with a share price of $100, a strike of $100, a rate of 5 percent and volatility of 20 percent. Then $d_1 = 0.35$ and $d_2 = 0.15$, with $N'(d_1) = 0.3752$ and $N(d_2) = 0.5596$. The first term is $-100 \times 0.3752 \times 0.20 / 2 = -3.752$. The second is $-0.05 \times 100 \times 0.9512 \times 0.5596 = -2.662$. Theta is $-6.414$ per year, or about $-0.0176$ per day. The option, worth $10.45, loses a little under two cents a day at first.

Decay is not even over an option's life. The time value of an at-the-money option is roughly proportional to the square root of the time remaining. An option with 60 days left therefore loses about 29 percent of its time value over the first 30 days and the remaining 71 percent over the last 30. Decay accelerates sharply in the final weeks before option expiry.

• Moneyness. Theta is largest for options with strikes near the current price, which hold the most time value. Deep in-the-money and far out-of-the-money options have little time value left to lose.
• Volatility. Higher implied volatility means a larger premium and therefore more value to decay each day.
• Interest and dividends. The second term of the formula reflects financing. In unusual cases, such as a deep in-the-money European put, theta can be positive.

Models count calendar days, although markets do not trade at weekends. Market makers typically adjust their volatility inputs so that the weekend's decay is spread across the preceding trading days.

Theta and gamma are two sides of one bargain. For a position that is hedged against small price moves, and ignoring interest, the model implies:

In the example, gamma is 0.01876, and $0.5 \times 0.04 \times 10{,}000 \times 0.01876 = 3.752$, which matches the first term of theta. A trader who owns options has positive gamma and profits from large moves in the underlying but pays theta every day. A trader who has sold options collects theta but loses when the underlying moves sharply. The owner comes out ahead only if the realized volatility of the underlying exceeds the implied volatility built into the price, which is the principle behind gamma scalping.

Strategies that sell options are often described as collecting theta. They include the covered call, the short straddle and the iron condor. Each earns income from the passage of time provided the underlying stays within a range. A calendar spread sells a short-dated option and buys a longer-dated one at the same strike, seeking to benefit from the faster decay of the near contract.

Positive theta is not free income. It is compensation for accepting the risk of a large adverse move, and a steady stream of daily gains can be reversed by a single sharp change in the underlying or in volatility. For that reason theta is always read together with gamma and vega.

Theta describes what happens if nothing else changes, which is never exactly the case. Over a day the underlying price moves and implied volatility shifts, and those effects usually outweigh time decay. Theta itself changes as expiry approaches and as the price moves relative to the strike, so a figure calculated today is only a local estimate. Because it is derived from a pricing model, it also inherits that model's assumptions about volatility and interest rates, the latter being the subject of rho.