Theta: time decay
What it costs to hold an option for one more day, and why the cost speeds up in the final weeks.
5 minute read
Theta is how much value an option loses in one day, if the stock price and everything else stay the same.
Picture an ice cube on a warm table. It melts a little every day, and a little faster each day as it gets smaller. Time value melts the same way as expiry gets closer.
Our $100 call is worth $3.02 with 30 days left. Its theta is −0.053. So if the stock is still at $100 tomorrow, the call should be worth about $2.97. On one contract, that is about $5.30 gone overnight.
Why the melting speeds up
Time value is the price of what could still happen. With more days left, the stock has more room to move, so the chance is worth more. Every day that passes takes some of that room away.
The loss is not even, though. Going from 60 days to 30 costs this call $1.34. The last 30 days cost all of the remaining $3.02. Watch the dot below as the clock runs down. That call is priced a little differently from ours, so its dollar amounts differ, but the shape is the same.
60 days left
$5.49 call value
The next diagram turns time into the horizontal axis. The left edge is 90 days before expiry, and the right edge is expiry itself. The steeper the curve, the faster the option is losing value. The tangent line shows the slope on the day you pick, and that slope is theta.
Value of the $100 call (at the money) as expiry approaches, stock held at $100
Drag across the chart, or focus it and use the arrow keys, to change the days left.
With 30 days left, the $100 call is worth $3.02 and loses about $0.05 a day, or $5.30 a contract. With 60 days left it lost $0.04 a day. With 7 days left it loses $0.10.
It depends on the strike
An at-the-money option loses the most dollars near the end. Our $100 call is worth $1.42 a week before expiry and loses $0.10 a day, roughly double its rate with a month left.
An out-of-the-money option melts earlier. The $110 call is worth $0.35 with 30 days left. With 7 days left it is worth less than a cent. By the final week there is almost nothing left to lose.
An in-the-money option is mostly intrinsic value, and time can’t touch that. The $90 call decays toward its $10 floor, not toward zero.
Who pays and who collects
If you buy an option, theta is a cost you pay every day, and weekends count. It is the rent on the gamma from chapter 4. If you sell an option, you collect that rent. In exchange, you carry the risk of a big move.
Time is the one input everyone agrees on. The next chapter covers the input people argue about most: how much the stock is likely to swing.
Try it on real data
Check theta on a SPY contract a week from expiry and on the same strike three months out. The short one bleeds faster every day.
See theta on SPY