Scaling Laws for Greeks¶
This section provides a unified framework for understanding how Greeks scale with time-to-maturity \(\tau\) and moneyness.
Near-the-money ATM scaling table¶
For at-the-money options (\(S \approx K\)) in Black–Scholes:
| Greek | ATM Formula | Scaling in \(\tau\) | Behavior as \(\tau \to 0\) |
|---|---|---|---|
| \(\Delta\) | \(N(d_1) \approx \frac{1}{2}\) | \(\mathcal{O}(1)\) | \(\to \frac{1}{2}\) |
| \(\Gamma\) | \(\frac{1}{S\sigma\sqrt{2\pi\tau}}\) | \(\tau^{-1/2}\) | \(\to \infty\) |
| \(\Theta\) | \(-\frac{S\sigma}{2\sqrt{2\pi\tau}}\) | \(\tau^{-1/2}\) | \(\to -\infty\) |
| \(\nu\) (Vega) | \(\frac{S\sqrt{\tau}}{\sqrt{2\pi}}\) | \(\tau^{1/2}\) | \(\to 0\) |
| \(\rho\) | \(\frac{K\tau}{2}e^{-r\tau}\) | \(\tau\) | \(\to 0\) |
Theta-Gamma relationship¶
A fundamental identity links theta and gamma for delta-hedged positions:
For ATM options where \(\Delta \approx \frac{1}{2}\) and \(V \approx \frac{S\sigma\sqrt{\tau}}{\sqrt{2\pi}}\):
This shows that time decay and gamma are two sides of the same coin: options with high gamma (convexity benefit) must pay for it through time decay.
Dimensional analysis¶
Greeks can be understood through dimensional analysis. Let \([X]\) denote the dimension of \(X\):
| Quantity | Dimension |
|---|---|
| \(V\) (price) | \(\$\) |
| \(S\) (spot) | \(\$\) |
| \(\tau\) (time) | \(T\) |
| \(\sigma\) (volatility) | \(T^{-1/2}\) |
Then: - \(\Delta = \partial V/\partial S\) is dimensionless - \(\Gamma = \partial^2 V/\partial S^2\) has dimension \(\$^{-1}\) - \(\Theta = \partial V/\partial t\) has dimension \(\$/T\) - \(\nu = \partial V/\partial \sigma\) has dimension \(\$ \cdot T^{1/2}\)
The ATM scalings follow from \(V_{\text{ATM}} \sim S\sigma\sqrt{\tau}\):
Far from the money: exponential decay¶
For OTM options with log-moneyness \(x = \ln(K/S)\):
| Greek | OTM decay rate |
|---|---|
| \(\Delta\) | \(\exp(-x^2/(2\sigma^2\tau))\) |
| \(\Gamma\) | \(\frac{1}{S}\exp(-x^2/(2\sigma^2\tau))\) |
| \(\nu\) | \(S\sqrt{\tau}\exp(-x^2/(2\sigma^2\tau))\) |
The characteristic scale separating ATM from OTM is \(|x| \sim \sigma\sqrt{\tau}\).
Unified asymptotic expansion¶
For options near ATM as \(\tau \to 0\), all Greeks can be expanded in powers of \(\sqrt{\tau}\):
Price: [ V = V_0 + V_1 \sqrt{\tau} + V_2 \tau + \mathcal{O}(\tau^{3/2}) ]
where \(V_0 = (S-K)^+\) (intrinsic), \(V_1 = \frac{S\sigma}{\sqrt{2\pi}}\mathbf{1}_{\text{ATM}}\) (ATM time value).
Greeks: [ \Gamma = \frac{1}{S\sigma\sqrt{2\pi\tau}} + \mathcal{O}(1) ] [ \Theta = -\frac{S\sigma}{2\sqrt{2\pi\tau}} + \mathcal{O}(1) ] [ \nu = \frac{S\sqrt{\tau}}{\sqrt{2\pi}} + \mathcal{O}(\tau) ]
Medium maturity: vega dominance¶
For moderate maturities (\(\tau \sim 1\) year), vega often dominates portfolio risk:
- Vega is maximized around \(\tau \sim 1/(2r)\) for typical interest rates
- Gamma remains bounded: \(\Gamma \sim \frac{1}{S\sigma}\)
- Theta-vega ratio: \(\Theta/\nu \sim -\sigma/(2\tau)\)
Long maturity asymptotics¶
For \(\tau \to \infty\):
Call price: \(C \to S\) (assuming no dividends)
Greeks: - \(\Delta \to 1\) - \(\Gamma \to 0\) as \(\tau^{-1/2}\) - \(\nu \to S\sqrt{\tau}N'(d_1) \sim S\sqrt{\tau}/\sqrt{2\pi}\) grows without bound - \(\rho \to K\tau e^{-r\tau}N(d_2)\) grows then decays
Practical implications¶
- Short-dated options: Large gamma, unstable delta hedging, high theta decay
- Medium-dated options: Vega dominance, sensitivity to implied vol changes
- Long-dated options: Low gamma, stable delta, model risk concerns
Greeks scaling summary¶
What to remember¶
- Short-dated ATM options: large gamma (\(\tau^{-1/2}\)), unstable higher Greeks
- Vega scales as \(\sqrt{\tau}\), peaking at moderate maturities
- Theta and gamma are linked: \(\Theta \approx -\frac{1}{2}\sigma^2 S^2 \Gamma\) for ATM
- Extreme strikes: tail-dominated sensitivities with exponential decay
- Dimensional analysis provides quick scaling estimates
Exercises¶
Exercise 1. Using the dimensional analysis framework, verify that \(\Gamma_{\text{ATM}} \sim 1/(S\sigma\sqrt{\tau})\) has dimension \(\$^{-1}\) and that \(\Theta_{\text{ATM}} \sim -S\sigma/(2\sqrt{\tau})\) has dimension \(\$/T\). Check that the theta-gamma identity \(\Theta + \frac{1}{2}\sigma^2 S^2 \Gamma = r(V - S\Delta)\) is dimensionally consistent.
Solution to Exercise 1
Checking \(\Gamma_{\text{ATM}}\): The dimension of \(\Gamma = \partial^2 V / \partial S^2\) is \([\$] / [\$]^2 = \$^{-1}\).
For \(\Gamma_{\text{ATM}} \sim 1/(S\sigma\sqrt{\tau})\): the dimension of the denominator is \([\$] \cdot [T^{-1/2}] \cdot [T^{1/2}] = [\$]\), so \(\Gamma_{\text{ATM}}\) has dimension \(\$^{-1}\). Confirmed.
Checking \(\Theta_{\text{ATM}}\): The dimension of \(\Theta = \partial V / \partial t\) is \([\$]/[T]\).
For \(\Theta_{\text{ATM}} \sim -S\sigma/(2\sqrt{\tau})\): the dimension is \([\$] \cdot [T^{-1/2}] / [T^{1/2}] = [\$] / [T]\). Confirmed.
Checking the theta-gamma identity \(\Theta + \frac{1}{2}\sigma^2 S^2 \Gamma = r(V - S\Delta)\):
- Left side, first term: \([\Theta] = \$/T\)
- Left side, second term: \([\sigma^2 S^2 \Gamma] = [T^{-1}][\$^2][\$^{-1}] = \$/T\). Consistent.
- Right side: \([r(V - S\Delta)] = [T^{-1}][\$] = \$/T\). Consistent.
All three terms have dimension \(\$/T\), so the identity is dimensionally consistent.
Exercise 2. For an OTM call with log-moneyness \(x = \ln(K/S) = 0.10\) (roughly 10% OTM), \(\sigma = 0.20\), and \(\tau = 0.25\) years, compute the exponential decay factor \(\exp(-x^2/(2\sigma^2\tau))\). How does this compare to the ATM vega at the same maturity?
Solution to Exercise 2
With \(x = 0.10\), \(\sigma = 0.20\), and \(\tau = 0.25\):
The ATM vega at the same maturity is \(\nu_{\text{ATM}} \approx S\sqrt{\tau}/\sqrt{2\pi} = 100 \times 0.5 / 2.507 \approx 19.95\).
The OTM vega is approximately \(\nu_{\text{OTM}} \approx 19.95 \times 0.6065 \approx 12.10\).
So the 10% OTM option retains about \(61\%\) of the ATM vega. This is because \(|x|/(\sigma\sqrt{\tau}) = 0.10/0.10 = 1\), meaning the option is only one characteristic width away from ATM. At this moderate distance, vega is reduced but far from negligible.
Exercise 3. The theta-gamma relationship states that for ATM options, \(\Theta \approx -\frac{1}{2}\sigma^2 S^2 \Gamma\). Verify this numerically for \(S = K = 100\), \(\sigma = 0.20\), \(r = 0.05\), \(\tau = 0.5\) by computing both sides using the closed-form Black--Scholes formulas.
Solution to Exercise 3
Using the Black-Scholes formulas with \(S = K = 100\), \(\sigma = 0.20\), \(r = 0.05\), \(\tau = 0.5\):
Computing \(\Gamma\):
Computing \(\Theta\): The exact Black-Scholes theta for a call is
Left side of identity: \(-\frac{1}{2}\sigma^2 S^2 \Gamma = -\frac{1}{2}(0.04)(10000)(0.02736) = -5.472\).
Right side: \(r(V - S\Delta)\). We need \(V\) and \(\Delta\):
Checking: \(\Theta + \frac{1}{2}\sigma^2 S^2\Gamma = -8.116 + 5.472 = -2.644 \approx -2.645 = r(V - S\Delta)\).
The identity is verified numerically, confirming \(\Theta \approx -\frac{1}{2}\sigma^2 S^2\Gamma\) up to the correction term \(r(V - S\Delta)\).
Exercise 4. A portfolio manager holds ATM options at three maturities: \(\tau = 0.08\) (1 month), \(\tau = 0.5\) (6 months), and \(\tau = 2\) (2 years). Using the scaling laws, rank these positions by: (a) gamma exposure, (b) vega exposure, (c) daily theta bleed. Which maturity dominates each risk?
Solution to Exercise 4
Using the ATM scaling formulas with \(S = K = 100\), \(\sigma = 0.20\):
(a) Gamma exposure (\(\Gamma \sim 1/(S\sigma\sqrt{2\pi\tau})\)):
- \(\tau = 0.08\): \(\Gamma \approx 1/(100 \times 0.20 \times \sqrt{2\pi \times 0.08}) = 1/(20 \times 0.7090) = 0.0706\)
- \(\tau = 0.5\): \(\Gamma \approx 1/(20 \times 1.7725) = 0.0283\)
- \(\tau = 2\): \(\Gamma \approx 1/(20 \times 3.5449) = 0.0141\)
Ranking (highest to lowest gamma): 1-month \(>\) 6-month \(>\) 2-year.
(b) Vega exposure (\(\nu \sim S\sqrt{\tau}/\sqrt{2\pi}\)):
- \(\tau = 0.08\): \(\nu \approx 100 \times 0.2828 / 2.5066 = 11.28\)
- \(\tau = 0.5\): \(\nu \approx 100 \times 0.7071 / 2.5066 = 28.21\)
- \(\tau = 2\): \(\nu \approx 100 \times 1.4142 / 2.5066 = 56.42\)
Ranking (highest to lowest vega): 2-year \(>\) 6-month \(>\) 1-month.
(c) Daily theta bleed (\(\Theta \sim -S\sigma/(2\sqrt{2\pi\tau})\), converted to daily):
- \(\tau = 0.08\): \(\Theta/365 \approx -(100 \times 0.20)/(2 \times 0.7090 \times 365) = -20/517.6 = -0.0387\) per day
- \(\tau = 0.5\): \(\Theta/365 \approx -20/(2 \times 1.7725 \times 365) = -20/1293.9 = -0.0155\) per day
- \(\tau = 2\): \(\Theta/365 \approx -20/(2 \times 3.5449 \times 365) = -20/2587.8 = -0.00773\) per day
Ranking (highest daily theta bleed): 1-month \(>\) 6-month \(>\) 2-year.
Summary: Short-dated options dominate gamma and theta risk, while long-dated options dominate vega risk. This reflects the fundamental scaling: gamma and theta scale as \(\tau^{-1/2}\) (growing as maturity shrinks) while vega scales as \(\tau^{1/2}\) (growing with maturity).
Exercise 5. The characteristic scale separating ATM from OTM is \(|x| \sim \sigma\sqrt{\tau}\). For \(\sigma = 0.30\) and \(\tau = 1\) month, compute this scale in both log-moneyness units and as a percentage of spot price. An option with \(K = 105\) when \(S = 100\) --- is it effectively ATM or OTM at this maturity?
Solution to Exercise 5
With \(\sigma = 0.30\) and \(\tau = 1/12\) (1 month):
This is the scale in log-moneyness units (dimensionless).
As a percentage of spot price: since \(|x| = |\ln(S/K)| \approx |S - K|/S\) for small deviations, the characteristic scale is about \(8.66\%\) of spot.
For \(K = 105\) and \(S = 100\):
The ratio of this log-moneyness to the characteristic scale is:
Since this ratio is less than 1, the \(K = 105\) option is well within one characteristic width of ATM. At this maturity, the option is effectively ATM --- its Greeks (delta, gamma, vega) will be only modestly reduced from their peak ATM values. The exponential decay factor is \(\exp(-0.564^2/2) = e^{-0.159} \approx 0.853\), so Greeks are at about \(85\%\) of their ATM levels.
Exercise 6. In the long-maturity regime (\(\tau \to \infty\)), vega grows as \(S\sqrt{\tau}/\sqrt{2\pi}\) while gamma decays as \(1/(S\sigma\sqrt{2\pi\tau})\). Show that the ratio \(\nu/\Gamma = \sigma S^2 \tau\) grows linearly with \(\tau\). What does this imply about the relative importance of vega versus gamma hedging for long-dated options?
Solution to Exercise 6
Using the ATM scaling formulas:
Taking the ratio:
This grows linearly with \(\tau\), confirming the claim.
Implications for hedging long-dated options: As \(\tau\) increases, vega grows much faster relative to gamma. For a portfolio of long-dated options:
- A \(1\%\) change in implied volatility (\(\Delta\sigma = 0.01\)) causes P&L of \(\nu \times 0.01 \sim S\sqrt{\tau}/\sqrt{2\pi} \times 0.01\), which grows with \(\sqrt{\tau}\).
- A \(1\%\) move in spot (\(\Delta S = 0.01 S\)) causes gamma P&L of \(\frac{1}{2}\Gamma(\Delta S)^2 \sim S/(2\sigma\sqrt{2\pi\tau}) \times (0.01)^2\), which decays with \(\sqrt{\tau}\).
For long-dated options (\(\tau \gg 1\)), vega dominates portfolio risk by a factor proportional to \(\tau\). Therefore, vega hedging is far more important than gamma hedging for long-dated positions. This is why traders managing long-dated books focus primarily on volatility surface risk rather than delta-gamma risk.