Options Greeks

The Greeks measure the sensitivity of an option’s modeled value to changes in its pricing inputs. Each isolates one source of change while holding the others fixed, such as a movement in the underlying, elapsed time, or a change in volatility. They are local rates of change rather than fixed characteristics: as the inputs change, the sensitivities generally change as well.

The units matter when applying a Greek to a position. A sensitivity quoted per share must be multiplied by contract size and quantity to obtain a dollar position effect. A short position reverses the sign of the corresponding long-option exposure.

Delta

Delta measures the option-value response to a small change in the underlying price. For a call on shares trading at $750 with delta of 0.46, a rise to $751 adds approximately $0.46 per share to the call’s value when other inputs are unchanged. One purchased 100-share contract therefore gains about $46.

Purchased calls normally have delta between zero and one, while purchased puts normally have delta between negative one and zero. A put with delta of -0.54 loses approximately $0.54 per share after a $1 underlying rise and gains approximately that amount after a $1 fall. Deeply in-the-money options generally respond more like the underlying, whereas far-out-of-the-money options are usually less responsive to a small price change.

Delta is a local price sensitivity. Using it as an exact forecast for a large underlying move ignores the fact that the option’s response changes along the price path.

Gamma

Gamma measures the change in delta for a small change in the underlying. If the example call has gamma of 0.0075, a $1 rise increases its delta from approximately 0.46 to 0.4675. Its sensitivity to the next price increment is consequently greater than at the starting point.

Purchased calls and puts normally have positive gamma. As the underlying rises, a call’s delta increases and a put’s negative delta moves toward zero. As the underlying falls, the put’s delta becomes more negative, increasing its gain from a further decline. Gamma thus describes the curvature of the option-value relationship.

A short option reverses this exposure. A short call’s position delta becomes more negative as the underlying rises, and a short put’s positive position delta increases as the underlying falls. In both cases, continued movement against the position can produce a progressively larger response.

Near expiration, the transition between an option with little payoff and one that closely follows the underlying becomes concentrated around the strike. Gamma can be high in that region. Farther in or out of the money, delta is generally more stable and gamma lower.

Theta

Theta measures the effect of elapsed time, commonly reported per calendar day. A theta of -$0.26 per share estimates a one-day decline of about $26 for one purchased 100-share contract if the underlying, volatility, and other inputs remain unchanged.

Purchased options usually have negative theta because a shorter remaining term reduces the opportunity for their exercise rights to become more valuable. Near the money, the rate of time decay often accelerates toward expiration. Deeply in-the-money or out-of-the-money options may have little time value remaining and therefore follow a different pattern.

Financing can qualify this usual relationship. A European put cannot receive its strike proceeds through exercise until expiration. At positive interest rates, the present value of those proceeds increases as the receipt date approaches. For a deeply in-the-money put with little remaining uncertainty, this effect can outweigh the loss of optionality and produce positive theta. It follows from the discounting term in the European put formula, rather than from a general rule that purchased options always lose value with time.

Vega

Vega measures the option-value response to a one-percentage-point change in implied volatility, such as an increase from 20% to 21%. A vega of $1.04 per share estimates a $1.04 increase in the premium for that change, or $104 for one purchased 100-share contract, with other inputs unchanged.

Purchased standard calls and puts normally have positive vega. Higher volatility increases the value of potential favorable outcomes without requiring exercise after an unfavorable outcome. Vega is often greater near the money and with more time remaining. It approaches zero at expiration as the payoff becomes determined by the final underlying price. Its value also changes as volatility and moneyness change.

Rho

Rho measures sensitivity to a one-percentage-point change in the interest rate. A call rho of $0.40 per share estimates a $0.40 premium increase when the rate rises from 4% to 5%, holding other inputs fixed.

Higher rates reduce the present value of the strike payment. This generally benefits call holders, who may pay the strike later, and reduces the value of the future proceeds available to put holders. Rho is therefore usually positive for purchased calls and negative for purchased puts. The effect tends to be more significant for longer-dated contracts because the potential strike payment is further in the future.

Position Greeks

Position sensitivities combine the option’s quoted Greek with the multiplier, quantity, and long or short sign. Two purchased 100-share calls, each with delta of 0.46, have position delta of 92. A small $1 underlying rise therefore produces an estimated $92 gain. Selling those calls to open instead gives position delta of -92 because a rise increases the short liability.

A multi-leg position’s Greek is the sum of its legs’ signed, scaled sensitivities. Nearly offsetting deltas at one underlying price can become less balanced after a move if the legs have different gamma. Similarly, offsetting vegas describe a common change in IV; they do not eliminate exposure when the IVs at different strikes change by different amounts.

Mathematical definitions and units

For per-share option value $V$ and underlying price $S$ , delta is the first derivative of value with respect to price, and gamma is the second derivative:

\Delta=\frac{\partial V}{\partial S},\qquad \Gamma=\frac{\partial^2V}{\partial S^2}

These expressions describe a smooth value curve. At expiration, call and put payoffs have a corner at the strike, where a unique slope and a finite gamma are not defined. Pre-expiration sensitivities should not be treated as ordinary derivatives at that corner.

If remaining time $T$ is measured in years, daily theta under a 365-day reporting convention is:

\Theta=-\frac{1}{365}\frac{\partial V}{\partial T}

The minus sign reflects the reduction in remaining time as calendar time passes. The reporting convention must be distinguished from the year fraction used in a particular pricing example, which may use a different basis.

When annual volatility $\sigma$ and interest rate $r$ are expressed as decimals, dividing their derivatives by 100 reports vega and rho per percentage point:

\text{Vega}=\frac{1}{100}\frac{\partial V}{\partial\sigma},\qquad \rho=\frac{1}{100}\frac{\partial V}{\partial r}

Combining small changes

For an underlying-price change $x$ , elapsed calendar days $h$ , an IV change of $u$ percentage points, and a rate change of $w$ percentage points, a local approximation is:

\text{Change in }V\approx \Delta x+\tfrac12\Gamma x^2+\Theta h+\text{Vega}\,u+\rho w

The gamma term adjusts for curvature over the underlying move. The remaining terms apply the initial sensitivities to the specified changes. This approximation omits higher-order effects and interactions between inputs, so full repricing is more reliable for substantial changes.

Using the rounded illustrative call sensitivities above, consider a $1 underlying rise over one day while IV falls from 20% to 19%, with rates and dividend assumptions unchanged:

0.46+\tfrac12(0.0075)(1)^2-0.26-1.04\approx-\$0.84

The estimated decline is per share; quantity and the contract multiplier scale it to the position. In this example, time passing and lower IV more than offset the benefit of the underlying rise. The calculation demonstrates why one favorable input change does not determine an option’s total price movement.