Delta And Gamma For Equity Options

Why Delta and Gamma Matter

An option’s value does not generally move dollar for dollar with the stock price. Its sensitivity changes as the stock price moves, so a single linear estimate may miss a material part of the change. Delta measures the local first-order sensitivity; gamma measures how quickly delta itself changes. Together, they provide a practical way to estimate the value change of an equity-option position for a specified stock-price movement.

For an insurer or other financial institution, these sensitivities help describe the market risk of an option position and support scenario analysis. They are local estimates, not guarantees: accuracy depends on the size of the stock-price move and on whether other inputs, such as volatility and time to expiry, are held fixed. This tutorial is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: For an option value $V(S)$, where $S$ is the underlying stock price, delta is $\Delta=\frac{\partial V}{\partial S}$, and gamma is $\Gamma=\frac{\partial^2 V}{\partial S^2}=\frac{\partial \Delta}{\partial S}$. Delta is the approximate change in option value per one-dollar stock-price change; gamma is the approximate change in delta per one-dollar stock-price change. For a position, the units are dollars of position value per dollar of stock-price change for delta, and dollars of position value per squared dollar of stock-price change for gamma.

From Option Greeks to Portfolio Greeks

Greeks reported for an individual option are commonly quoted per share. To obtain the sensitivity of a position, multiply each option’s Greek by the number of contracts, the number of shares per contract, and the position sign. Use a positive sign for a long position and a negative sign for a short position. Then sum across the positions.

\Delta_{\text{portfolio}}=\sum_j n_j m_j\Delta_j, \qquad \Gamma_{\text{portfolio}}=\sum_j n_j m_j\Gamma_j,

Here $n_j$ is the signed number of contracts, $m_j$ is the number of shares per contract, and $\Delta_j$ and $\Gamma_j$ are the per-share Greeks of option $j$. The contract multiplier is applied once to each per-share Greek. A short option has the opposite position delta and gamma from the corresponding long option.

For a stock-price change $\Delta S$, the first-order estimate uses delta alone. The second-order estimate adds the gamma correction:

\Delta V_{\text{first}}\approx \Delta_{\text{portfolio}}\Delta S, \qquad \Delta V_{\text{second}}\approx \Delta_{\text{portfolio}}\Delta S+ \frac{1}{2}\Gamma_{\text{portfolio}}(\Delta S)^2.

The estimated ending value is the starting value plus the estimated change. The formula assumes that the Greeks are evaluated at the starting market conditions and that the stock-price change is the only modeled change. It does not include changes in volatility, time to expiry, interest rates, dividends, or other risk factors.

Why the Gamma Term Has a One-Half

The second-order approximation follows from the Taylor expansion of $V(S+\Delta S)$ around the current stock price $S$. The first derivative gives the local linear change, while the second derivative accounts for the curvature of the option value:

V(S+\Delta S) \approx V(S)+V'(S)\Delta S+\frac{1}{2}V''(S)(\Delta S)^2.

Subtract $V(S)$ from both sides. By definition, $V'(S)=\Delta$ and $V''(S)=\Gamma$, so the estimated change is $\Delta\Delta S+\frac{1}{2}\Gamma(\Delta S)^2$. The factor one-half is part of the Taylor expansion; omitting it doubles the gamma contribution. Higher-order terms are omitted, so the approximation can become less accurate for large moves or highly curved option values.

Black–Scholes Greeks Under Stated Assumptions

A standard way to calculate option Greeks is the Black–Scholes model. For the formulas below, assume a European option, no dividends, constant continuously compounded risk-free rate $r$, constant volatility $\sigma$, and time to expiry $T$ in years. Under the risk-neutral model, the stock follows a geometric Brownian motion with drift $r$ and volatility $\sigma$. These formulas do not directly apply to American options or to a model with dividends unless the appropriate model adjustments are made.

Let $K$ be the strike price, and let $N(z)$ and $\phi(z)$ denote the standard normal cumulative distribution function and density. Define:

d_1=\frac{\ln(S/K)+(r+\sigma^2/2)T}{\sigma\sqrt{T}}, \qquad d_2=d_1-\sigma\sqrt{T}.

For a European call, delta is $N(d_1)$; for a European put, delta is $N(d_1)-1$. Both options have gamma $\phi(d_1)/(S\sigma\sqrt{T})$. Thus a call and a put with the same underlying stock, strike, expiry, and model inputs have the same gamma, although their deltas differ by one.

Worked Example: Black–Scholes Greeks for a Short Put Position

A hypothetical account is short 5 European put contracts, each covering 100 shares. The stock price and strike are both $100. The options have six months to expiry, volatility is 20% per year, and the continuously compounded risk-free rate is 4% per year. Assume no dividends, constant volatility and interest rate, and the Black–Scholes risk-neutral lognormal stock model. Calculate the position Greeks and estimate the value change if the stock rises by $4.

Method. First calculate the per-share put Greeks, then apply the short-position sign and 500-share multiplier. Convert six months to $T=0.5$ years.

Since $S=K=100$, $\ln(S/K)=0$. Also, $\sigma^2/2=0.20^2/2=0.02$, so:

d_1=\frac{(0.04+0.02)(0.5)}{0.20\sqrt{0.5}} =\frac{0.03}{0.1414214} \approx 0.212132, \qquad d_2=0.212132-0.20\sqrt{0.5}\approx 0.070711.

The standard normal values are $N(d_1)\approx0.5840$ and $\phi(d_1)\approx0.39007$. Therefore, the put delta per share is $0.5840-1=-0.4160$. Its gamma per share is:

\Gamma=\frac{0.39007}{100(0.20)\sqrt{0.5}} =\frac{0.39007}{14.14214} \approx 0.0275823.

The short position is equivalent to a signed share multiplier of $-5(100)=-500$. Hence:

\Delta_{\text{portfolio}}=(-500)(-0.4160018)\approx 208.0009, \qquad \Gamma_{\text{portfolio}}=(-500)(0.0275823)\approx -13.7912.

For $\Delta S=+$4$, the delta contribution is $208.0009(4)=+$832.0036$. The gamma contribution is $\frac12(-13.7912)(4^2)=-$110.3296$. The second-order estimated change is therefore $832.0036-110.3296=$721.6740$, or +$721.67 to the nearest cent. The positive estimated change is consistent with the short puts’ positive portfolio delta at these inputs, while the negative portfolio gamma reduces the linear estimate. The calculation checks by applying the short sign to both Greeks before using the approximation.

Common mistake avoided: the numerator in $d_1$ is the full product $(r+\sigma^2/2)T=0.03$, not $0.04T=0.02$. Also, a short put has positive position delta here but negative position gamma.

Worked Portfolio Estimates

Worked Example: Combining Calls and Puts Before a Stock Decline

A hypothetical position is long 3 European calls and short 2 European puts on the same stock. Each contract covers 100 shares. At current market conditions, each call has per-share delta $0.62$ and gamma $0.018$; each put has per-share delta $-0.38$ and gamma $0.018$. Assume these supplied Greeks are valid at the starting stock price and that volatility, time, interest rates, and dividends do not change. Estimate the position value change for a $5 decline in the stock.

Method. Aggregate signed Greeks first. The three long calls represent $+300$ shares of option exposure, and the two short puts represent $-200$ shares of option exposure.

\Delta_{\text{portfolio}}=300(0.62)-200(-0.38)=186+76=262, $$ $$ \Gamma_{\text{portfolio}}=300(0.018)-200(0.018)=5.4-3.6=1.8.

The stock change is $\Delta S=-$5$. The delta contribution is $262(-5)=-$1{,}310$. The gamma contribution is $\frac12(1.8)(-5)^2=0.9(25)=+$22.50$. Thus the second-order estimate is $-1{,}310+22.50=\mathbf{-\$1{,}287.50}$. The squared stock move makes the gamma term positive here, even though the stock falls; the net position remains a loss because the delta term is much larger. The arithmetic checks because the short-put delta changes sign with the short position, while the short-put gamma is negative.

Common mistake avoided: do not multiply the put delta by a negative position sign and then treat its gamma as positive. The same short-position sign applies separately to both Greeks.

Worked Example: A Portfolio with Negative Net Gamma

A hypothetical portfolio is long 1 call contract and short 2 put contracts on one stock. Each contract covers 100 shares. The call has delta $0.55$ and gamma $0.020$ per share; each put has delta $-0.45$ and gamma $0.015$ per share. Assume the supplied Greeks apply at the starting price, with no change in volatility, time, interest rates, or dividends. Estimate the portfolio value change if the stock falls by $6.

Method. Apply the signed contract-share amounts, $+100$ for the call and $-200$ for the puts:

\Delta_{\text{portfolio}}=100(0.55)-200(-0.45)=55+90=145, $$ $$ \Gamma_{\text{portfolio}}=100(0.020)-200(0.015)=2.0-3.0=-1.0.

For $\Delta S=-$6$, the delta contribution is $145(-6)=-$870$. The gamma contribution is $\frac12(-1.0)(-6)^2=-$18$. The second-order estimate is $-870-18=\mathbf{-\$888.00}$. Here negative net gamma makes the second-order estimate more negative than the delta-only estimate. The result checks against the position composition: short puts add positive delta at the supplied inputs, but their negative position gamma outweighs the long call’s positive gamma.

Common mistake avoided: the gamma term is not negative merely because the stock falls. Its sign is the sign of portfolio gamma, because $(\Delta S)^2$ is nonnegative.

Interpreting the Estimate and Avoiding Traps

Delta and gamma answer different questions. Delta estimates the local direction and scale of the value change. Gamma adjusts that estimate for curvature. A long vanilla call or put has positive gamma under the standard Black–Scholes assumptions; shorting that option reverses the position gamma. A portfolio can have positive, negative, or near-zero net gamma after positions are combined.

The second-order estimate is most useful for a moderate move when the starting Greeks remain informative. For a large move, the option’s delta and gamma can change materially, and omitted higher-order effects may matter. Repricing the options under the new stock price is preferable when a precise value is needed. The approximation also isolates stock-price risk: it is not a complete valuation change if implied volatility, time to expiry, rates, or dividends move at the same time.

Wrong: for a short option, use the quoted per-share Greeks as if the position were long. Correction: multiply both Greeks by the signed contract count and shares per contract.
Wrong: calculate the gamma term as $\Gamma(\Delta S)^2$. Correction: use $\frac12\Gamma(\Delta S)^2$.
Wrong: use a negative squared stock move when the stock falls. Correction: square the change; direction affects the delta term, not the sign of the square.
Wrong: use a per-share Greek as though it were already the portfolio Greek. Correction: include the number of contracts and the contract multiplier before estimating dollars.

Key takeaway: Calculate signed portfolio delta and gamma first, then estimate the value change as $\Delta_{\text{portfolio}}\Delta S+\frac12\Gamma_{\text{portfolio}}(\Delta S)^2$. Keep the units, position signs, and assumptions explicit.

Check Your Understanding

For each hypothetical position, use the stated Greeks and assume that all inputs other than the stock price remain unchanged. Round dollar estimates to the nearest cent.

A long position holds 2 call contracts, each for 100 shares. Per-share delta is $0.48$ and gamma is $0.016$. Estimate the value change for a $2 stock-price rise. (Answer: $198.40)
An account is short 3 put contracts, each for 50 shares. Per-share delta is $-0.35$ and gamma is $0.020$. Estimate the change for a $1.50 stock-price rise. (Answer: $75.38)
A position is long 4 calls and short 2 puts on the same stock, with 25 shares per contract. Each call has delta $0.70$, each put has delta $-0.25$, and both have gamma $0.030$ per share. Calculate portfolio delta and gamma, then estimate the change for a $3 stock-price decline. (Answer: Delta $82.5$, gamma $1.5$, estimated change $-\$240.75$)
A European put has $S=K=$50$, $T=1$ year, volatility $20\%$, continuously compounded risk-free rate $0\%$, and no dividends. Under the Black–Scholes model, $d_1=0.10$, $N(d_1)=0.5398$, and $\phi(d_1)=0.39695$. Find its per-share delta and gamma. Then estimate the change for a long 100-share contract if the stock rises by $2. (Answer: Delta $-0.4602$, gamma $0.039695$, estimated change $-\$84.10$)
A portfolio has delta $120$ and gamma $-8$, with values in dollars per dollar and dollars per squared dollar, respectively. Estimate the value change for a $3.50 stock-price decline and compare it with the delta-only estimate. (Answer: Second-order estimate $-\$469.00$; delta-only estimate $-\$420.00$)

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