Options Greeks

A stock can rise while a call you bought loses value. The rise helps the call, but elapsed time and falling implied volatility (IV) can take away more value than the stock move adds. The Greeks help separate these effects by estimating the response to each input with the others held fixed. They are local sensitivities, not fixed rules for the option’s next price. 1

Suppose the stock rises from $750 to $751 over one calendar day while IV falls from 20% to 19%, with interest rates and dividend assumptions unchanged. Use the rounded illustrative sensitivities below, quoted per share, for one purchased contract with a 100-share multiplier.

Start with the $1 stock rise. At a delta of 0.46, it adds approximately $0.46 per share, or $46 to the purchased contract, if the other inputs stay fixed. Delta measures this local response to the underlying price.

That response is not quite constant along the move. Gamma measures how delta changes with the stock price. With gamma of 0.0075, the $1 rise increases the call’s delta from about 0.46 to 0.4675, making it a little more sensitive to the next dollar of movement. A gamma adjustment will account for this curvature in the estimate below.

Meanwhile, the call has one fewer day remaining. Its theta estimates the effect of time passing, reported here per calendar day. Theta of -$0.26 per share contributes an estimated one-day decline of $26 for the purchased 100-share contract, holding the stock price, volatility and other inputs fixed.

The fall in IV contributes another decrease. Vega measures sensitivity to a one-percentage-point change in IV. With vega of $1.04 per share, a rise from 20% to 21% would add approximately $1.04 per share, or $104 to the purchased contract, all else equal. Our example goes the other way, from 20% to 19%, so the estimated contribution is negative. These are percentage-point changes, not proportional percentage changes in IV.

Combining small changes

Adding the stock-price, time and volatility effects, including gamma’s curvature adjustment, gives:

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

The estimated decline is about $0.84 per share, or $84 for one purchased 100-share contract. Time passing and lower IV more than offset the benefit of the stock’s rise. That explains how the call could lose value without the stock moving against it; delta alone describes only part of the change.

More generally, let $x$ be the underlying-price change in dollars, $h$ the elapsed calendar days, $u$ the IV change in percentage points and $w$ the interest-rate change in percentage points. A local estimate of the per-share value change is:

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

The gamma term adjusts delta’s estimate for curvature; rho accounts for interest rates, which are unchanged in this example. The expression uses the starting sensitivities and omits higher-order effects and interactions between inputs. It is an approximation, not a full repricing or a guaranteed trading result. For substantial changes, especially when several inputs move together, reprice the option at the new inputs rather than extending the starting Greeks across the entire move.

Why the stock-price sensitivity changes

The call’s positive delta explains why the stock rise helped it. Purchased calls normally have delta between zero and one; purchased puts, between negative one and zero. A put with delta of -0.54 loses about $0.54 per share for a $1 stock rise and gains about that amount for a $1 fall, with other inputs fixed. Deeply in-the-money options generally respond more like the underlying, while far-out-of-the-money options respond less to small moves.

Purchased calls and puts normally have positive gamma: as the stock rises, call delta increases and put delta becomes less negative. When the stock falls, put delta becomes more negative. This changing slope is the curvature of the option-value relationship. 2

A short option reverses that exposure. A short call’s delta becomes more negative as the stock rises; a short put’s delta becomes more positive as the stock falls. A further move against either position can therefore cause a larger loss.

Gamma can be especially high near the strike close to expiration, where a small stock move sharply changes the option’s sensitivity. Farther in or out of the money, delta is generally steadier and gamma lower. Delta is neither an exact guide to a large price change nor an exact real-world probability of exercise.

How time and volatility change the exposure

The example’s negative theta is common for purchased options: less time remains for a favorable payoff to develop. Decay often accelerates near expiration for at-the-money options, while deeply in-the-money or out-of-the-money options can follow a different pattern.

Negative theta is not universal. A European put must wait until expiration to receive its strike proceeds through exercise. At positive interest rates, those proceeds become worth more in present-value terms as the payment date approaches. For a deeply in-the-money put with little remaining uncertainty, that financing effect can outweigh the loss of optionality and produce positive theta. 2

The example’s positive vega is also typical of purchased standard calls and puts. Higher volatility increases the potential for a larger payoff without obliging the holder to exercise after an unfavorable move. Vega is often larger near the money and with more time remaining. It changes with volatility and moneyness, and approaches zero at expiration, when the stock price determines the payoff.

This calculator uses whole days. Expiration today means zero remaining model time and intrinsic value, not a forecast of intraday decay.

Where interest rates enter

The worked example holds rates fixed, but a rate change would add another contribution. Rho measures sensitivity to a one-percentage-point interest-rate change. A call rho of $0.40 per share implies an approximately $0.40 premium increase when the rate rises from 4% to 5%, with other inputs fixed.

Higher rates reduce the present value of the strike payment. That generally helps call holders, who may pay the strike later, and hurts put holders, who may receive it. Purchased calls therefore usually have positive rho and purchased puts negative rho. The effect tends to be larger for longer-dated contracts. 3

Position Greeks

The same sensitivities apply to a position after allowing for its size and direction. Multiply each leg’s per-share Greek by its contract multiplier and signed quantity: positive for a long leg, negative for a short leg. Then add the legs.

Two purchased 100-share calls with delta of 0.46 have position delta of $2\times100\times0.46=92$ . A small $1 stock rise gives an estimated $92 gain with other inputs fixed. Selling those two calls to open instead produces delta of -92. A short leg likewise reverses the corresponding long option’s other sensitivities.

Offsetting Greeks do not necessarily stay balanced. Legs with different gamma can lose their delta balance after a stock move. Similarly, offsetting vegas describe a common IV change, not protection against different volatility changes at different strikes.

The calculator’s position Greeks already include side, quantity and multiplier; do not scale them again. They use today’s date and Entry price, even when Analysis date changes the stock-price graph.

Mathematical definitions and units

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

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

These definitions assume a smooth value curve. At expiration, the payoff has a corner at the strike, where a unique slope and finite gamma are not defined.

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

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

The minus sign accounts for remaining time decreasing as a day passes. This reporting divisor is distinct from the 365.25-day year fraction used in this calculator’s pricing examples.

When annual volatility $\sigma$ and interest rate $r$ are decimals, divide their derivatives by 100 to report 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}