Gamma

Delta tells you how fast an option's price moves when the underlying moves. Gamma tells you how fast the delta itself moves.

The workbook's image is the right one: if delta is speed, gamma is acceleration. An option with a large gamma changes character quickly — it can travel from behaving like a lottery ticket to behaving almost like the underlying itself over a modest move in the market.

Mathematically it is the second derivative of the option price with respect to the price of the underlying. Practically it is the answer to one question: how wrong will my delta be after the next move?

Delta is only accurate for an infinitesimal move. Over a real move, the option price traces a curve and delta traces the tangent to it, so a delta-based estimate is always a straight line drawn against something bent. Gamma measures the bend.

Three consequences:

• Long options are long gamma; short options are short gamma. Gamma is positive for both a long call and a long put. That means a bought option's delta always moves in the owner's favour: it grows as the position goes right and shrinks as it goes wrong.
• Gamma is highest at the money and near expiry. A deep in-the-money or deep out-of-the-money option has a delta pinned near 1 or near 0 and very little left to change.
• Gamma is what breaks a delta hedge. A trader who is short options and has neutralised delta is neutral only at that instant. The moment the underlying moves, gamma drags the delta away from zero and the hedge has to be rebalanced — which is the whole cost of running a short-option book.

Gamma = Change in option delta ÷ Unit change in price of the underlying

New delta = Old delta + (Gamma × Change in underlying)

Gamma is positive for long calls and long puts, and negative for the writer of either.

The workbook's own case, on a stock. A call has a delta of 0.50 and a gamma of 0.08. The stock rises by Re 1:

Change in delta = 0.08 × 1 = 0.08
New delta       = 0.50 + 0.08 = 0.58

A put, where the sign catches people. An at-the-money put has a delta of −0.50 and a gamma of 0.004 — negative delta, positive gamma. If the underlying rises 10 points:

Change in delta = 0.004 × 10 = +0.04
New delta       = −0.50 + 0.04 = −0.46   ← less negative; the put is losing its bite

If the underlying falls 10 points instead:

Change in delta = 0.004 × (−10) = −0.04
New delta       = −0.50 − 0.04 = −0.54   ← more negative; the put bites harder

Now the money. Nifty is at 22,250 and you hold one at-the-money call, lot size 25, with a delta of 0.50 and a gamma of 0.004. The index rises 50 points.

Delta-only estimate:   0.50 × 50           = Rs 25.00 per unit
Delta after the move:  0.50 + 0.004 × 50   = 0.70
Average delta:         (0.50 + 0.70) ÷ 2   = 0.60
Gamma-adjusted change: 0.60 × 50           = Rs 30.00 per unit

Delta-only, per contract:  Rs 25 × 25 = Rs   625
Gamma-adjusted:            Rs 30 × 25 = Rs   750
Curvature you would have missed:       Rs   125

Twenty per cent of the gain was gamma, on a move of less than a quarter of a per cent. For the writer of that call, the same Rs 125 is an unbudgeted loss — and it grows with the square of the move.

Chapter 16.7 (Option Greeks) defines gamma with the delta 0.50 / gamma 0.08 illustration, and Chapter 21.6 repeats it for interest rate options with the −0.50 put delta and 0.004 gamma used above. Chapter 17 uses the same idea in delta hedging, where a delta-neutral position has to be rebalanced as the underlying moves. Expect a straight "new delta = old delta + gamma × move" computation, and a conceptual question on which Greek measures the rate of change of delta.

• Gamma is not delta. Delta measures the option price against the underlying; gamma measures the delta against the underlying. A question asking for the sensitivity of the premium wants delta.
• Gamma is positive for a long put too, even though the put's delta is negative. Adding a positive gamma to a negative delta makes it less negative on a rise.
• Gamma is largest at the money, not deep in the money. Deep ITM options have a delta near 1 and almost no gamma left.
• Delta hedging is not a one-off. A delta-neutral book is neutral only at a point; gamma is what forces continuous rebalancing.
• Do not confuse it with theta (time decay) or vega (volatility sensitivity) — the second-derivative Greek with respect to price is gamma, and nothing else.
• The writer of an option is short gamma. His losses accelerate and his gains decelerate, which is the structural reason option selling carries unlimited risk.