Gamma and theta

Chapter 8 ended with a delta-hedged short call losing money on a 400-point move and making money on a quiet day. This chapter explains that asymmetry, and the explanation turns out to be the most elegant result in the subject.

The two quantities

Gamma is the rate of change of delta.

$\Gamma = \frac{\partial \Delta}{\partial S} = \frac{\partial^2 C}{\partial S^2} = \frac{n(d_1)}{S\sigma\sqrt{T}}$

where $n(\cdot)$ is the standard normal density. Gamma is curvature — how much the price-versus-underlying line bends. It is positive for both bought calls and bought puts, and identical for the two at the same strike and expiry, which follows from differentiating parity twice.

Theta is the rate of change with the passage of time.

$\Theta = \frac{\partial C}{\partial t} = \underbrace{-\frac{S\,n(d_1)\,\sigma}{2\sqrt{T}}}_{\text{curvature part}} \;\underbrace{-\; rKe^{-rT}N(d_2)}_{\text{financing part}}$

Theta is negative for bought options: waiting costs the owner. Conventionally it is quoted per day, which is the annual figure divided by 365.

The split in theta is worth keeping. The second term is interest on the deferred strike — chapter 1's financing effect, which has the opposite sign for a put. The first term is the one that matters here, and it is not about time preference at all.

The identity that joins them

Compare the curvature part of theta with the gamma formula:

$-\tfrac{1}{2}\Gamma S^2 \sigma^2 = -\tfrac{1}{2} \cdot \frac{n(d_1)}{S\sigma\sqrt{T}} \cdot S^2\sigma^2 = -\frac{S\,n(d_1)\,\sigma}{2\sqrt{T}}$

They are the same expression. Not approximately — identically. So

$\Theta_{\text{curvature}} = -\tfrac{1}{2}\Gamma S^2\sigma^2$

For the running contract — index 24,000, strike 24,000, thirty days, 14% volatility — both sides come to −₹2,310.52 a year, or −₹6.33 a day.

This is not a coincidence and it is not a curiosity. It is the pricing argument of chapters 3 to 6, visible from the inside. The model sets the decay to exactly offset the expected benefit of the curvature, because if it did not, a delta-hedged position would have a predictable profit or loss and the replication would not be riskless. Derman and Kani's "instantaneously riskless portfolio" is this identity.

What a delta-hedged owner actually earns

Hold a call, hedge the delta, and wait one day. Two things happen.

Curvature pays you. Because the price curve bends upwards, the option gains more on an up move than the hedge loses, and loses less on a down move than the hedge gains. The gain over a move of $\Delta S$ is approximately

$\tfrac{1}{2}\Gamma(\Delta S)^2$

Note the square. The direction is irrelevant; only the size matters. A delta-hedged option owner does not care which way the index goes.

Decay charges you. One day of theta, which is a fixed amount, whatever happens.

So the day's result is a race: $\tfrac{1}{2}\Gamma(\Delta S)^2$ against one day of $|\Theta|$.

The breakeven move, and why it is the volatility

Set the two equal and solve for the move that breaks even against the curvature part of decay:

$\tfrac{1}{2}\Gamma(\Delta S)^2 = \tfrac{1}{2}\Gamma S^2\sigma^2 \Delta t \quad\Longrightarrow\quad \Delta S = S\sigma\sqrt{\Delta t}$

The gamma terms cancel. The breakeven move is $S\sigma\sqrt{\Delta t}$ — which is exactly one standard deviation of the underlying over that interval, at the volatility the option was priced with.

For the running contract, with $\Delta t = 1/365$:

$24{,}000 \times 0.14 \times \frac{1}{\sqrt{365}} = 175.9 \text{ points}$

And solving the breakeven from the actual gamma and theta figures gives 175.9 points. The same number.

Check that it does not depend on the maturity:

Time left	Gamma	Theta per day	Price	Theta as % of price	Gamma breakeven
90 days	0.00023084	−₹6.00	₹868.45	0.69%	175.9 pts
30 days	0.00040932	−₹8.65	₹450.57	1.92%	175.9 pts
7 days	0.00085503	−₹15.45	₹200.85	7.69%	175.9 pts
2 days	0.00160274	−₹26.97	₹103.54	26.05%	175.9 pts
1 day	0.00226750	−₹37.24	₹72.31	51.50%	175.9 pts

Gamma rises almost tenfold and theta more than sixfold, and the breakeven does not move at all. Longer-dated options have less curvature and cheaper decay; shorter-dated ones have more of both, in exactly offsetting proportion.

Which gives the cleanest statement of what buying an option is. Not a bet on direction, and not a bet on the index. A delta-hedged option owner is long realised volatility and short implied volatility — they profit if the underlying moves more than its own price said it would, and lose if it moves less. That is the trade, stated exactly, and chapter 10 is about how it is priced.

(The full breakeven, including the financing part of theta, is a little wider — about 206 points for the thirty-day contract. The identity is exact for the curvature term only.)

Where gamma and theta live

Both peak in the same place, and that is not an accident either.

Strike	Gamma	Theta per day	Price	Theta as % of price
23,000	0.00019832	−₹6.65	₹1,176.02	0.57%
23,500	0.00032917	−₹8.16	₹771.04	1.06%
24,000	0.00040932	−₹8.65	₹450.57	1.92%
24,500	0.00038809	−₹7.50	₹230.21	3.26%
25,000	0.00028516	−₹5.22	₹101.39	5.15%

Gamma is largest at the money, because that is where delta is changing fastest — chapter 8's polarising deltas. Deep in or out of the money, delta is pinned near 1 or 0 and barely moves, so there is little curvature.

And gamma grows as expiry approaches, from 0.00023 at ninety days to 0.00227 at one day — a factor of ten. The $\sqrt{T}$ in the denominator is doing it.

So the maximum of gamma, and of theta, is the at-the-money option on expiry day. The Derivatives chapter on what an option price is made of described expiry-day behaviour from the price side; this is the mechanism. At one day to expiry the at-the-money option loses 51.5% of its own value per day to decay, and the whole premium is a wager on one day's move.

SEBI's regulatory response names the same thing. On the day of options expiry, "given the heightened speculative activity around options positions and the attendant risks," an additional 2% extreme loss margin applies to short options, effective 20 November 2024 — to open shorts at the start of the day and to shorts opened during it that expire that day. That is a margin surcharge on short gamma at its maximum.

Short gamma, which is the position that hurts people

Reverse the signs. A seller is short gamma and long theta: they collect decay every day and pay for movement.

Chapter 8's arithmetic, restated. Short one lot of the thirty-day at-the-money call, delta-hedged with futures, lot size 75:

What the index does overnight	Result
Falls 400 points	−₹1,908
Rises 400 points	−₹1,716
Does nothing	+₹653

Both tails lose and the middle pays, which is the shape of an insurance business. The seller is being paid ₹653 a quiet day to carry a liability that costs about ₹1,900 on a 400-point day. Over a long run of quiet days the position looks like a yield; it is not a yield, it is a premium received for a risk not yet realised.

And three features make it worse than that table suggests.

The loss grows with the square of the move. A 400-point gap costs about ₹1,900; an 800-point gap costs roughly four times as much, not twice. The Derivatives chapter on selling options and tail risk makes this argument from the payoff; gamma is why it holds before expiry too.

The hedge cannot be adjusted through a gap. Chapter 8's overnight move happened with the market closed. Short gamma is precisely the exposure that discrete trading cannot hedge away, which is chapter 7's broken assumption with a number attached.

Gamma is largest exactly when premiums are smallest. Weekly expiries sell for a few rupees and carry the highest curvature in the market. Selling the cheapest-looking options is selling the most gamma, and the two facts are the same fact.

Working the problem

Gamma 0.00040932, curvature decay ₹6.33 a day, index 24,000, 14% volatility.

Step 1 — the breakeven from the Greeks.

$\tfrac{1}{2} \times 0.00040932 \times (\Delta S)^2 = 6.33 \quad\Longrightarrow\quad (\Delta S)^2 = \frac{2 \times 6.33}{0.00040932} = 30{,}929$

$\Delta S = 175.9 \text{ points}$

Step 2 — the daily volatility implied by 14% a year. Volatility scales with the square root of time, so

$\sigma_{\text{daily}} = \frac{0.14}{\sqrt{365}} = 0.7328\% , \qquad 0.7328\% \times 24{,}000 = 175.9 \text{ points}$

Step 3 — they match, and here is why. Substitute the identity $\Theta_{\text{curvature}} = -\tfrac{1}{2}\Gamma S^2\sigma^2$ into the breakeven condition. Gamma appears on both sides and cancels, leaving $\Delta S = S\sigma\sqrt{\Delta t}$ with no reference to the option at all.

What that means in plain terms. The decay you pay each day is the fair price of the curvature you own, assessed at the volatility the option is quoted at. The market is selling you movement at the rate it has told you movement costs. If the index obliges by moving one standard deviation a day, you break even; if it moves more, you win; less, you lose.

And the three honest qualifications.

Realised volatility is not constant either. A month with one 800-point day and twenty-nine flat days can deliver the same annualised figure while producing a quite different hedging result, because the gains go as the square of each day's move. Buying options is long the variance of the path, not merely its average.

The hedging is not free. Every rebalance pays a spread, brokerage, exchange charges and tax. The breakeven move is wider than 175.9 points by however much your costs come to — chapter 12's figures — which is a structural tilt in the seller's favour and part of why implied volatility usually exceeds realised.

And nobody reading this is delta hedging. The 175.9-point result describes a hedged position. An unhedged buyer owns direction as well, which dominates everything in this chapter. The Behavioural finance subject's measured cost of activity is what happens when the hedging is skipped and the trading is not.

The point

Gamma is the curvature of the price in the underlying and theta its decay in time, and they are tied by an exact identity: the curvature part of theta equals $-\tfrac{1}{2}\Gamma S^2\sigma^2$. So a delta-hedged owner gains about $\tfrac{1}{2}\Gamma(\Delta S)^2$ from any move, pays a fixed daily decay, and breaks even at a move of $S\sigma\sqrt{\Delta t}$ — exactly one standard deviation at the quoted volatility, the same 175.9 points whether ninety days or one day remain. Both Greeks peak for the at-the-money option nearest expiry, where decay reaches half the premium a day and SEBI adds 2% extreme loss margin to short positions. Being short gamma loses on moves in either direction, by amounts that grow with the square, through gaps that cannot be hedged — and the cheapest options carry the most of it.