Speed (Greek)
The third-order Greek that tells you how fast your gamma is changing as the market moves — critical for anyone running a large gamma-hedged book near expiry.
Definition
Speed is a third-order option Greek defined as the rate of change of gamma with respect to a move in the underlying price. Mathematically, it is the third partial derivative of the option's theoretical price with respect to the underlying — equivalently, the second derivative of delta and the first derivative of gamma. Speed answers a specific question that the first two derivatives cannot: if the spot price moves by one unit, by how much does my gamma change? A non-zero speed means your gamma estimate will be stale almost immediately after a market move, and any delta hedge built on that gamma will accumulate error faster than a first-order approximation suggests.
Why it matters
For retail traders hedging one or two lots, speed is an academic curiosity. For institutional desks, prop firms, and market makers running large option books on NSE indices — particularly in the final days of a weekly expiry cycle — speed becomes operationally significant. Near expiry, the gamma profile of an at-the-money option spikes sharply and then collapses, and that spike is not symmetric or smooth: it changes very rapidly with small underlying moves. A market maker who quotes a gamma of 0.05 when Nifty is at 23,500 may find that gamma has jumped to 0.12 when Nifty ticks 50 points higher, and the delta hedge calculated using the old gamma is now substantially wrong. Speed quantifies precisely how wrong it will be. When running gamma exposure reports across a book of thousands of lots across multiple strikes and expiries, including speed in the risk grid allows for tighter pre-emptive hedging, reducing the realised slippage cost of reactive rebalancing after large market moves. Speed is also relevant for anyone pricing exotic or barrier options where the path of gamma through spot levels matters.
Formula
Speed is defined as:
Speed = ∂Γ / ∂S = ∂3V / ∂S3
Under the Black-Scholes framework:
Speed = − Γ / S × (d1 / (σ√T) + 1)
where Γ is the option's gamma, S is the current underlying price, d1 is the standard Black-Scholes d1 term, σ is implied volatility, and T is time to expiry in years. The negative sign means that for standard vanilla options, speed is typically negative for calls (gamma decreases as the underlying rallies past the strike) and follows a similar sign structure for puts. Speed is quoted as the change in gamma per one unit move in the underlying — e.g., a speed of −0.0002 means gamma falls by 0.0002 for each point the underlying rises.
Example
Suppose a hypothetical Nifty 23,500 CE expiring in two days has a delta of 0.50, a gamma of 0.08, and a speed of −0.0003. At this moment, the delta-neutral hedger shorts 0.50 Nifty futures per lot. Now suppose Nifty moves up 100 points to 23,600. Using gamma alone, the new estimated delta would be 0.50 + (0.08 × 100) = 0.58. But including speed, the gamma at 23,600 has drifted to approximately 0.08 + (−0.0003 × 100) = 0.05, which changes the second-order delta correction. The true delta will be closer to 0.54 than 0.58. A hedger ignoring speed would over-hedge, creating an unintended short delta bias. On a book of 10,000 lots, this 0.04 delta error translates to 400 lots of unintended exposure — material in size. Figures are hypothetical and illustrative only.
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