Free lesson · Options
Delta and gamma are local sensitivities
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Start with the idea
Delta measures the first response to a small spot move; gamma describes how that response itself changes.
Symbols, units & horizon
- ΔV: option price change per underlying unit over a short scenario
- ΔS: spot change in currency per underlying unit
- Δ: first spot derivative, option-price units per spot unit
- Γ: second spot derivative, option-price units per squared spot unit
- approximation holds other inputs fixed
When and why to use this
Explain local option risk and compare exact repricing with a sensitivity approximation.
Delta measures the first response to a small spot move; gamma describes how that response itself changes.
Hold time, implied volatility, rates and other inputs fixed when defining spot partial derivatives. A real market move can change those inputs simultaneously. A gamma approximation is local and can fail for large jumps or near-expiry discontinuities.
Write the unit of each Greek before multiplying. Quoted contract Greeks may already include the multiplier; applying it again is a frequent accounting error.
Delta and gamma are local sensitivities
- Expand the price function in a Taylor series around current spot.
- Keep the linear and quadratic spot terms and omit higher powers.
- Compute delta and gamma contributions separately before adding; this is a numerical approximation rather than a cash-flow identity.
Delta .5, gamma .02 per dollar and spot rise $2 give .5×2+.5×.02×4=1.04 price units. A delta-only approximation gives 1.00.
Apply it in a strategy
- Freeze inputs at the stated decision time and record their units.
- Explain local option risk and compare exact repricing with a sensitivity approximation.
- Recompute the example, then change the material assumption and explain the difference.
Research deliverable
Delta and gamma are local sensitivities: produce the worked calculation, a timestamped input record and a written decision addressing this limitation: Large moves, changing implied volatility and elapsed time can overwhelm omitted terms.
These are synthetic mechanics examples, not historical performance or paper replications. Module evidence and research boundaries record the 12 September 2026 review.
Python implementation
Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.
# Python 3.10+; standard library and NumPy only.
# Synthetic teaching inputs; conventions and units are defined in the notation above.
def delta_gamma_change(delta,gamma,spot_change):
linear=delta*spot_change
curvature=.5*gamma*spot_change**2
return linear,curvature,linear+curvature
assert abs(delta_gamma_change(.5,.02,2)[2]-1.04)<1e-12
print(delta_gamma_change(.5,.02,2))Continue learning
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- Replicate a one-step option with stock and cash
- Black–Scholes as a conditional benchmark
- Delta and gamma are local sensitivities
- Cash accounting for a discretely hedged option
- Early exercise compares immediate and continuation value
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