Options that expire the same day you trade them, called 0DTE options (zero days to expiration), are now a large share of all index option trading. They are the same contracts you have studied all course with one input pushed to its limit: time. When time runs out, gamma and theta stop being gentle curves and become the whole trade.
An expiration every weekday
SPY and QQQ list an expiration for every weekday, as SPX does, so every trading morning has a chain that settles at the 4pm close. SPY options are American and settle into shares: an in-the-money SPY option at the close is exercised, and a small option position becomes 100 shares a contract held overnight. SPX options settle in cash. Module 13 went through settlement and pin risk; on a same-day expiry both arrive within hours.
Gamma at the extreme
Take an ETF at $600 with 15% implied volatility. Its at-the-money gamma is about 0.016 with 30 days left, 0.085 with one day left and 0.17 with six hours left. Gamma grows as one over the square root of time, so a day holds five and a half times the gamma of a month, and the last six hours double it again.
In deltas: a $3 rise, half a percent, takes the 30-day at-the-money call from a delta of about 0.50 to 0.55. The same move takes a one-day call to about 0.74 and a six-hour call to about 0.90. On the last afternoon an ordinary move turns a coin flip into nearly stock, or nearly nothing. That is why gamma dominates a same-day position.
An at-the-money option on a $100 stock at 15% volatility. With a week left gamma is a low hill; with six hours left it is a spike more than five times as tall, and it vanishes a dollar or two either side. Drag volatility up and the spike widens and falls.
For a buyer that is the appeal: a small premium whose delta can swing from 0.5 to nearly 1 by lunch. For a seller it is the danger: a short same-day straddle has no time for the market to calm down before a loss is locked in.
Theta at the extreme
Every dollar of time value must be gone by 4pm. With one day left, the $600 ETF's at-the-money call is worth about $1.88, all time value, and its theta is about minus $0.94 a day. Theta is a rate, and on the last day the rate keeps climbing as the hours run out: if the ETF sits still, the call loses the whole $1.88 by the close, twice the theta printed that morning. With six hours left the printed rate is already about minus $1.88.
Read from right to left as expiration nears. The at-the-money theta keeps steepening all the way to the close, while options 1% and 2% out of the money peak and then fade, because their value is already close to zero.
Gamma and theta are two sides of one price, as Module 6 showed with the hedged straddle. The one-day straddle, two options with a combined gamma of 0.17 and theta of minus $1.88 a day, breaks even on a move of about $4.71, 0.79% of $600: exactly the one-day, one-standard-deviation move at 15% vol.
The same-day expected move
Add the at-the-money call and put mids and you have the straddle, the market's price for movement either way until the close. For the $600 ETF with one day left it is about $3.76, 0.63% of the price. As Module 4 showed, a straddle is worth about 0.8 of a one-standard-deviation move, so 0.63% implies a one-sigma day near 0.79%. That is the expected move for the session: sell the straddle and you win if the day moves less; buy it and you need more.
The range for a $100 stock by the next close, priced at 15% volatility: one standard deviation is about 79 cents either way, and the straddle costs about 80% of that. Drag volatility to 30 for a stressed day and the range doubles.
Read the straddle in dollars rather than the IV. On a same-day expiration the implied volatility swings through the session and depends on how the hours left are counted, while the straddle is simply what the day's movement costs.
Event days
Module 11 showed how a scheduled event adds a bump to every expiration that contains it. A same-day expiration holds at most one session, so the event can be most of what it prices. A Fed decision lands at 2pm Eastern, inside that day's expiration: the morning straddle is dearer than usual, and once the decision is out most of the extra drains away, because only two hours of ordinary movement are left. An inflation report comes out at 8:30am, before the open, so it is carried by that day's expiration bought the afternoon before; by the opening bell the number is known. Selling a same-day straddle on a Fed morning is selling the decision itself, with no time to recover from a surprise.
The maths: the Greeks with hours to go
At the money with little time left, where S is the price, σ the volatility and T the time in years: call value ≈ 0.4 × S × σ × √T, gamma ≈ 0.4 / (S × σ × √T), theta per day ≈ -(0.4 × S × σ) / (2 × √T × 365). The 0.4 is the height of the bell curve at its center.
For S = 600, σ = 0.15 and one day, √T = √(1/365) = 0.0523: the call is 0.4 × 600 × 0.15 × 0.0523 = $1.88, gamma is 0.4 / (600 × 0.15 × 0.0523) = 0.085, and theta is -(0.4 × 90) / (2 × 0.0523 × 365) = -$0.94 a day. Cut T to a quarter and √T halves: gamma and theta double to 0.17 and -$1.88, and the call halves to $0.94. Because the rate grows like one over √T, the last day burns $1.88 though it starts with a theta of $0.94. The hedged straddle breaks even at √(2 × 1.88 / 0.169) ≈ $4.71, equal to S × σ × √T = 600 × 0.15 × 0.0523.
Your turn
Open the SPY chain on the expiration that settles next. Add the at-the-money call and put mids, divide by the price for the day's expected move, then read the gamma and theta on the same row. Module 16 will test all of this course in one final exam.