9.8 Expiry-day gamma · M09 Volatility trading

Motivation

On a Nifty Tuesday expiry, ATM gamma in the last hour is many times its value a week earlier. Small index moves flip option deltas from 0 to 1. This drives hedging flows, makes short-dated option selling a high-variance business, and creates the incentive that SEBI's expiry-day rules and the Jane Street case (9.12.2) are about.

9.8.1ATM gamma near expiry#

Proposition 9.10ATM gamma and theta as $\tau\to0$

For a European option with $S=K$ , volatility $\sigma$ , rate $r$ and time to expiry $\tau$ ,

\Gamma_{ATM}=\frac{\varphi(d_1)}{S\sigma\sqrt\tau}=\frac{1}{\sigma S\sqrt{2\pi\tau}}\left(1+O(\tau)\right), \tag{9.26}

\Theta_{ATM}=-\frac{\sigma S}{2\sqrt{2\pi\tau}}\left(1+O(\sqrt\tau)\right),\qquad\tfrac12\Gamma S^2=\frac{S}{2\sigma\sqrt{2\pi\tau}}\left(1+O(\tau)\right). \tag{9.27}

Proof

At $S=K$ , $d_1=(r/\sigma+\sigma/2)\sqrt\tau$ , so $d_1^2=O(\tau)$ and $\varphi(d_1)=(2\pi)^{-1/2}e^{-d_1^2/2}=(2\pi)^{-1/2}(1+O(\tau))$ . This gives (9.26), and multiplying by $\tfrac12S^2$ gives the dollar gamma. For theta, the theta–gamma relation (M08 (8.27), with $q=0$ ) gives $\Theta=-\tfrac12\sigma^2S^2\Gamma+r(V-S\Delta)$ ; at the money $V=O(\sqrt\tau)$ and $S\Delta=O(S)$ , so $r(V-S\Delta)=O(1)$ , while $\tfrac12\sigma^2S^2\Gamma=O(\tau^{-1/2})$ .∎

So ATM gamma scales as $\tau^{-1/2}$ : quartering the time to expiry doubles it (tested in test_atm_gamma_scaling). The same identity shows why the theta collected by a seller and the gamma risk she runs are two sides of one number: $\Theta\approx-\tfrac12\sigma^2S^2\Gamma$ .

9.8.2Width of the gamma zone#

How wide is the "gamma zone"? Delta $N(d_1)$ moves from $N(-1)=0.16$ to $N(1)=0.84$ as $\ln(S/K)$ moves across $\pm\sigma\sqrt\tau$ . With $\sigma=12\%$ and 30 minutes left, $\sigma\sqrt\tau\,S=0.12\sqrt{0.5/8760}\times25{,}000=22.7$ Nifty points. A move from 24,950 to 25,050 (50 points either side of the strike) takes a 25,000 call's delta from 0.014 to 0.986.

Table (S = 25,000, σ = 12%, r = 0, per lot of 65; theta is the instantaneous rate per calendar day).

Time to expiry	Γ per unit	ATM straddle (points)	Dollar gamma per lot per 1% move² (₹)	Theta per lot per day (₹)
7 days	0.000960	331.5	1,950	−769
1 day	0.00254	125.3	5,161	−2,036
3 hours	0.00719	44.3	14,596	−5,759
30 minutes	0.0176	18.1	35,754	−14,106

(The fourth column is $\tfrac12\Gamma S^2(0.01)^2\times65$ : the gamma P&L per lot from a 1% move. The gamma, dollar-gamma and theta columns are for one ATM option (call or put); the straddle has twice these. Calendar time is used throughout; on expiry day the market is open 6.25 hours, so a "trading-time" convention would give different, larger numbers.)

9.8.3Hedging flows, incentives and margin#

Consequences on a Tuesday Nifty expiry

Hedging flows. A dealer short 1,000 lots of the ATM straddle with 30 minutes left needs to buy about 622 lots of futures for a 20-point rise (exact Black–Scholes delta change; the linear gamma approximation says 704). These trades move the index, which changes deltas again. If dealers are net short gamma, hedging amplifies moves; net long gamma dampens them and can pin spot to large open-interest strikes.
Lottery tickets. An OTM call 100 points away with 3 hours left costs a few points. Retail buys these because the payoff is large relative to the premium. Under $\mathbb Q$ they are fairly priced; under $\mathbb P$ with a variance premium they are expensive; after costs they are worse. SEBI's FY26 study found that 59% of index-option turnover was in contracts on their expiry day (9.12.3).
Manipulation incentive. An option book's value near expiry is very sensitive to the index level at settlement. A participant whose option delta exposure is many times the cash/futures position they can trade has an incentive to move the index. This is the alleged mechanism in the SEBI interim order of 3 July 2025 (facts in M08 · 8.12 Case B; the gamma angle in 9.12.2), and the reason settlement is computed as an average over a window rather than one print.
Margin. Exchanges add margin when gamma is highest: an extra 2% extreme loss margin (ELM) on short index options on expiry day, in force since 20 Nov 2024 (9.10).