Gamma risk near expiration

Every seller of options eventually discovers the same thing: the last week before expiration pays the most theta per day and loses the most money per dollar of stock movement. Both facts come from the same source. Gamma and theta of an at-the-money option both explode as time runs out, and they are two faces of one coin.

How gamma and theta grow

XYZ at $50, IV 30%, per-share values of the at-the-money $50 call:

Both columns grow roughly with 1/√time. Halving the days remaining raises gamma by about 40%; going from 45 days to 1 day raises it seven-fold. The theta column looks attractive: $0.16 a day on a $0.32 option is half the premium per day. The gamma column is the bill for it.

A useful way to see it: at 45 days, an option with gamma 0.07 changes its delta by 0.07 for a $1 move, so a $1 move against a short option costs about ½ × 0.07 × 1² = $0.035 per share from gamma, roughly one day of theta. At 1 day, gamma 0.50 means the same $1 move costs about $0.25 per share, more than a day and a half of theta on an option that only has one day left. Away from the money the picture is even sharper: a $48 call with one day left has almost no gamma and almost no theta, until the stock reaches $48, at which point it has both in enormous quantity.

A short straddle at 1 day versus 30 days

Sell the at-the-money $50 straddle (call and put) on XYZ. A rule of thumb prices an ATM straddle at about 0.8 × stock × IV × √(time in years):

• 30 days: 0.8 × 50 × 0.30 × √(30/365) ≈ $3.45 credit.
• 1 day: 0.8 × 50 × 0.30 × √(1/365) ≈ $0.63 credit.

Now compare what a $2 move does. For the 30-day straddle, the combined gamma is about 2 × 0.09 = 0.18 per share, so the move costs about ½ × 0.18 × 2² = $0.36 per share, minus a day of theta on both legs (about $0.06): −$30 per straddle, on a $345 credit. Annoying, recoverable.

For the 1-day straddle, gamma arithmetic breaks down because the option simply becomes intrinsic. The P&L at expiration is the credit minus the size of the move:

A $2 move loses more than twice the entire credit. The stock's typical daily move at 30% IV is about $0.80–0.95, so the breakevens sit at roughly one standard deviation; you win a bit more often than you lose, and the losses are much bigger than the wins. That is the exact profile of the credit spread expectancy in the fundamentals course, compressed into one afternoon.

Short ATM straddle, P&L at expiry (per share)

  +0.63 |        /\
        |       /  \
    0   |------/----\------   breakevens 49.37 / 50.63
        |     /      \
  -1.37 |    /        \
        |   /          \
  -4.37 |  /            \
        +--+--+--+--+--+--+
        45 47 49 50 51 53 55   XYZ

Why "close before the last week" is a rule

The exposure a seller carries in the last few days is not the same trade they entered. At 45 days, the 45/40 bull put spread from the fundamentals course had a net gamma of about −2.2 per contract. With XYZ at $45.50 and two days left, the short $45 put alone has gamma near 0.40 per share and the long $40 put has none: net gamma about −40 per contract, eighteen times what you signed up for. The spread can go from worth $0.10 to worth $3.00 in one session, and the theta you were collecting on it in those last days was a few cents.

That is why three widely used rules exist, and all of them are gamma rules in disguise:

• Close short premium at 21 days or when 50% of max profit is reached, whichever is first.
• Never hold a short strike that is at the money into the last 48 hours unless you intend to take the shares.
• If you must be short near expiration, be far from the money, where gamma is still small, and accept the tiny credit.

The mirror image is the reason long-option buyers who want a big move like short-dated options: cheap, huge gamma, and a payoff that becomes stock-like within hours if the move arrives. The buyer's cost is that the theta column above runs against them, and it usually does.

Key idea: Near expiration, at-the-money gamma and theta both grow as 1/√time. The rich daily theta in the last week is payment for a position whose delta can swing by 40 or 50 per contract on a $1 move. Close, roll, or move far from the money before that week; do not collect the last few cents.

Try it: In the options profit calculator, set up a short $50 straddle on a $50 stock with IV 30% at 30 days, and again at 1 day. Move the stock $1 and $2 in each and compare the P&L. Then do the same with a 45/40 put spread at 20 days and at 2 days with the stock at $45.50. Note how many times larger the 2-day loss is.

Recap

• ATM gamma and theta both grow roughly with 1/√(time to expiry); from 45 days to 1 day, gamma rises about seven-fold.
• A 1-day short ATM straddle collects about $0.63 and loses $1.37 on a $2 move; the 30-day straddle loses about $0.30 on the same move.
• Breakevens on short-dated straddles sit near a one-standard-deviation daily move: win slightly more often, lose much bigger.
• "Close at 21 days" and "take profit at 50%" are gamma-management rules.
• Short strikes near the money in the last 48 hours are a different trade from the one you entered.