Options Greeks Visualised

Once you've memorised the payoff diagrams (long call, short put, bull spread, straddle), the next layer is the Greeks — the partial derivatives of option price with respect to its inputs. Delta, Gamma, Vega, Theta, and Rho. They show up on CFA Level 1, CISI Diploma derivatives papers, and the harder ICWIM Chapter 3 questions. This guide visualises each one.

Quick reference table

Delta (Δ): sensitivity to the underlying

Delta is the change in option price for a £1 change in the underlying. It's the slope of the option-value curve at the current spot.

• Long call delta: 0 to +1. Deeply OTM calls have ~0; deeply ITM calls have ~+1.
• Long put delta: 0 to −1. Deeply OTM puts have ~0; deeply ITM puts have ~−1.
• At-the-money options: delta ≈ ±0.5.

Delta and hedging

Delta hedging means offsetting the directional exposure of an option position by trading the underlying. A trader long 100 calls with delta 0.6 has effective long exposure to 60 shares. To delta-hedge, they short 60 shares — leaving them neutral to small spot moves (until delta changes, which is Gamma's job).

Gamma (Γ): the curvature

Gamma is the rate of change of Delta — how fast Delta moves when spot moves. It's the second derivative of option price with respect to spot.

• Always positive for long options (calls AND puts). Long options have positive gamma.
• Always negative for short options.
• Peaks at-the-money. Deep ITM or OTM options have low gamma (delta is stable).

Gamma scalping

Long-gamma traders benefit from spot volatility. As spot rises, delta rises (positive gamma) so they sell some underlying to re-hedge. As spot falls, delta falls, so they buy back. Each round-trip is a small profit. Long gamma is implicitly a bet on realised volatility being higher than implied.

Vega (ν): sensitivity to volatility

Vega is the change in option price for a 1 percentage point change in implied volatility. Higher implied volatility = higher option prices (more chance of ending ITM by a lot, more chance of expiring OTM but the limited downside caps that loss).

• Always positive for long options. Higher vol = higher option value.
• Peaks at-the-money like Gamma.
• Increases with time to expiry — longer-dated options have more vega.

Theta (Θ): time decay

Theta measures the change in option price per day (or year, depending on convention) as time passes. Almost always negative for long options — the option loses value as expiry approaches.

• Negative for long options. Time decay erodes the option value.
• Positive for short options. Sellers benefit from time passing.
• Accelerates as expiry approaches — particularly steep in the last few weeks.
• Greatest at ATM in absolute terms.

Rho (ρ): sensitivity to interest rates

Rho measures the change in option price for a 1pp change in the risk-free rate. Usually the smallest Greek by impact for short-dated options, but material for long-dated options.

• Calls: positive rho (higher rates → higher call value via PV of strike being lower)
• Puts: negative rho (mirror image)
• Larger for longer-dated options

Rho is rarely an active consideration for most option traders — interest rates move slowly relative to other variables. It matters most for long-dated rate-sensitive instruments like LEAPS and warrant-style derivatives.

The Greeks summary chart

The risk-management hierarchy

Professional option desks manage exposures in roughly this priority order:

1. Delta — first-order directional risk. Hedged continuously via the underlying.
2. Gamma — second-order directional risk. Managed via positions in other options or by adjusting delta hedge frequency.
3. Vega — volatility risk. Managed via cross-strike or cross-tenor option positions.
4. Theta — time decay. Often acts as the "income" from a portfolio's long-gamma or short-vega bias.
5. Rho — generally low priority unless long-dated portfolio.

Common exam traps

Practical use cases

Related guides

• Derivatives Payoffs in 8 Diagrams — the payoff prerequisites for the Greeks
• Bond Pricing Primer: YTM, Duration, Convexity — the fixed-income equivalent of options sensitivities
• ICWIM Calculation Formulas Cheat Sheet — companion formulas
• Modern Portfolio Theory in 6 Charts — the broader framework

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