This is a research note, not a tutorial. It assumes familiarity with Black-Scholes Greeks through third order.

Research Note RN-001 (May 23, 2026)

What We Observed

On May 7, 2026, we pulled the Ratio Scatter panel on Cayø Largo's Greeks Tectonics page for all six coins tracked on Deribit, i.e. BTC, ETH, SOL, XRP, AVAX, and TRX. Each panel plots Speed/Gamma (x) against Veta/Vega (y) across the ORIA cohort grid. Every bubble is one cohort, i.e. one moneyness bucket crossed with one expiry bucket, and its coordinates are the median ratio values for all options in that cohort. Colour encodes the expiry bucket. Blue is options expiring within 7 days, yellow 8-30 days, pink 31-60 days, green 61 days and beyond. Bubble size scales with the number of options in each cohort.

Figure 1: Speed/Gamma vs Veta/Vega across all six Deribit coins, 7 May 2026 19:10 UTC. Same axes, same colour coding. The parabola appears in every coin. What changes is the x-axis scale, the arm depth, and the noise level.

The same downward arc appears in all six coins. Every scatter shows the same layout, i.e. long-dated cohorts clustered near the origin, short-dated ones pulling away toward deeply negative Veta/Vega. The parabolic shape is actually easiest to see in XRP and TRX, where the wider Speed/Gamma range gives the arc enough horizontal room to show its curvature. BTC and ETH have the deepest arms (below -700) but their x-axes are a bit irregular at the first glance. SOL and AVAX sit in between.

May 7 was simply when we first noticed the pattern. The parabola is not a one-off snapshot. It reappears every 10 minutes, across all six coins, in every update of the Ratio Scatter panel. The question is whether this is a coincidence of the vol surface, or something structural.

The Two Ratios

Cayø Largo computes 37 Black-Scholes Greeks per option every 10 minutes, then derives 22 cross-Greek ratios across five analytical tiers (see Crypto-Greek Atlas). Each ratio divides a higher-order Greek by a lower-order one, converting raw sensitivity values into proportional rates of change. Two of those ratios are the subject of this note:

Speed / Gamma. Speed is the third derivative of option price with respect to spot, $\partial\Gamma / \partial S$. It measures how fast gamma itself is changing as the underlying moves. Dividing by gamma normalises the result into a proportional rate. By the chain rule, $d(\ln f)/dx = (1/f) \cdot df/dx$, so Speed/Gamma $= \partial(\ln\Gamma)/\partial S$. A Speed/Gamma of -0.003 means gamma is declining at 0.3% per dollar of spot movement.

Veta / Vega. Veta is $\partial\text{Vega} / \partial t$, the rate at which volatility exposure erodes as calendar time passes. Dividing by vega, the option's sensitivity to implied volatility, gives the same log-derivative form, Veta/Vega $= \partial(\ln\text{Vega})/\partial t$. It answers the question, what fraction of my current vega am I losing per unit of time?

Both ratios strip out the absolute size of the Greek and expose the rate of change. That is what makes them comparable across strikes, expiries, and coins with very different notional scales.

The Derivation

We want to show algebraically that the parabola in Figure 1 is not a coincidence but an exact relationship under Black-Scholes. Set $r = 0$ for clarity (the risk-free rate adds a small correction but does not change the structure). All Greeks below are for a European option on an underlying with spot price $S$, strike $K$, annualised volatility $\sigma$, and time to expiry $T$ (in years).

Starting point: $d_1$ and its sensitivities

In Black-Scholes, every Greek is a function of $d_1$, the normalised log-moneyness adjusted for drift. We start here because both Speed/Gamma and Veta/Vega reduce to expressions in $d_1$, and eliminating it between them is what produces the parabola.

$$d_1 = \frac{\ln(\frac{S}{K}) + \left( r + \frac{\sigma^2}{2} \right) T} {\sigma \sqrt{T}}$$ $$= \frac{\ln(S/K)}{\sigma\sqrt{T}} + \frac{\sigma\sqrt{T}}{2}$$

Two partial derivatives of $d_1$ appear repeatedly. We need both before touching any Greek.

Sensitivity to spot. Differentiating $d_1$ with respect to $S$:

$$\boxed{\frac{\partial d_1}{\partial S} = \frac{1}{S \sigma \sqrt{T}}}$$

Sensitivity to time. Differentiate each piece of $d_1$ with respect to $T$ separately.

First piece: $\frac{\ln(S/K)}{\sigma\sqrt{T}} = \frac{\ln(S/K)}{\sigma} \cdot T^{-1/2}$

$$\frac{\partial}{\partial T}\left[\frac{\ln(S/K)}{\sigma} \cdot T^{-1/2}\right] = -\frac{\ln(S/K)}{2\sigma T^{3/2}}$$

Second piece: $\frac{\sigma\sqrt{T}}{2} = \frac{\sigma}{2} \cdot T^{1/2}$

$$\frac{\partial}{\partial T}\left[\frac{\sigma}{2} \cdot T^{1/2}\right] = \frac{\sigma}{4\sqrt{T}}$$

Sum the two pieces.

$$\frac{\partial d_1}{\partial T} = -\frac{\ln(S/K)}{2\sigma T^{3/2}} + \frac{\sigma}{4\sqrt{T}}$$

Rewrite the second term with denominator $4\sigma T^{3/2}$ by multiplying top and bottom by $\sigma T$, giving $\sigma^2 T / (4\sigma T^{3/2})$. Rewrite the first term by multiplying top and bottom by $2$, giving $-2\ln(S/K) / (4\sigma T^{3/2})$. Combine.

$$\frac{\partial d_1}{\partial T} = \frac{\sigma^2 T - 2\ln(S/K)}{4\sigma T^{3/2}}$$

This is the raw form (verifiable in Wolfram Alpha). For the steps below it is more convenient to express it in terms of $d_1$. Start from the unsimplified sum.

$$\frac{\partial d_1}{\partial T} = -\frac{\ln(S/K)}{2\sigma T^{3/2}} + \frac{\sigma}{4\sqrt{T}}$$

From the definition of $d_1$, we have $\ln(S/K) = \sigma\sqrt{T} \cdot d_1 - \sigma^2 T / 2$. Substitute into the first term:

$$-\frac{\ln(S/K)}{2\sigma T^{3/2}} = -\frac{\sigma\sqrt{T} \cdot d_1 - \sigma^2 T/2}{2\sigma T^{3/2}}$$

split the fraction,

$$= -\frac{\sigma\sqrt{T} \cdot d_1}{2\sigma T^{3/2}} + \frac{\sigma^2 T / 2}{2\sigma T^{3/2}} = -\frac{d_1}{2T} + \frac{\sigma}{4\sqrt{T}}$$

add back the second piece $+\sigma/(4\sqrt{T})$ from the original sum:

$$\frac{\partial d_1}{\partial T} = -\frac{d_1}{2T} + \frac{\sigma}{4\sqrt{T}} + \frac{\sigma}{4\sqrt{T}} = -\frac{d_1}{2T} + \frac{\sigma}{2\sqrt{T}}$$

Therefore,

$$\boxed{\frac{\partial d_1}{\partial T} = -\frac{d_1}{2T} + \frac{\sigma}{2\sqrt{T}}}$$

We will also need the derivative of the standard normal PDF with respect to its argument. The PDF is

$$\phi(x) = \frac{1}{\sqrt{2\pi}} \, e^{-x^2/2}$$

Differentiating with respect to $x$,

$$\phi'(x) = \frac{1}{\sqrt{2\pi}} \, e^{-x^2/2} \cdot (-x) = -x \cdot \phi(x)$$

so

$$\boxed{\phi'(d_1) = -d_1 \cdot \phi(d_1)}.$$

This appears in Step 2 when differentiating Gamma, because $\Gamma$ contains $\phi(d_1)$ and the chain rule pulls out $\phi'(d_1)$.

Step 1: Gamma

Gamma is the second derivative of option price (the same for calls and puts) with respect to spot at $r = 0$:

$$\Gamma = \frac{\phi(d_1)}{S \sigma \sqrt{T}}$$

Step 2: Speed

Speed is $\partial\Gamma / \partial S$. Write $\Gamma$ as a product of two functions of $S$:

$$\Gamma = \phi(d_1) \cdot \frac{1}{S\sigma\sqrt{T}}$$

Differentiating with the product rule, the first factor depends on $S$ through $d_1$ and the second depends on $S$ directly.

$$\frac{\partial\Gamma}{\partial S} = \phi'(d_1) \cdot \frac{\partial d_1}{\partial S} \cdot \frac{1}{S\sigma\sqrt{T}} \;+\; \phi(d_1) \cdot \frac{\partial}{\partial S}\!\left[\frac{1}{S\sigma\sqrt{T}}\right]$$

The second derivative is $-1/(S^2\sigma\sqrt{T})$, so

$$\text{Speed} = \frac{1}{S\sigma\sqrt{T}} \cdot \boxed{\phi'(d_1)} \cdot \boxed{\frac{\partial d_1}{\partial S}} \;-\; \frac{\phi(d_1)}{S^2 \sigma\sqrt{T}}$$

Substituting $\phi'(d_1) = -d_1 \cdot \phi(d_1)$ and $\partial d_1 / \partial S = 1/(S\sigma\sqrt{T})$.

$$\text{Speed} = \frac{-d_1 \cdot \phi(d_1)}{S^2 \sigma^2 T} \;-\; \frac{\phi(d_1)}{S^2 \sigma\sqrt{T}} = -\frac{\Gamma}{S}\left(\frac{d_1}{\sigma\sqrt{T}} + 1\right)$$

the same for calls and puts.

Step 3: Speed / Gamma

The ratio takes form:

$$\frac{\text{Speed}}{\Gamma} = \underbrace{-\frac{\Gamma}{S}\left(\frac{d_1}{\sigma\sqrt{T}} + 1\right)}_{\text{Speed}} \;\cdot\; \underbrace{\frac{S\sigma\sqrt{T}}{\phi(d_1)}}_{1/\Gamma}$$

Expanding and cancelling $\phi(d_1)$,

$$\frac{\text{Speed}}{\Gamma} = \underbrace{\frac{-d_1 \cdot \phi(d_1)}{S^2 \sigma^2 T}}_{\text{Speed, } 1^{st} \text{ term}} \cdot \underbrace{\frac{S\sigma\sqrt{T}}{\phi(d_1)}}_{1/\Gamma} \;-\; \underbrace{\frac{\phi(d_1)}{S^2 \sigma\sqrt{T}}}_{\text{Speed, } 2^{nd} \text{ term}} \cdot \underbrace{\frac{S\sigma\sqrt{T}}{\phi(d_1)}}_{1/\Gamma}$$ $$\boxed{\frac{\text{Speed}}{\Gamma} = -\frac{d_1}{S\sigma\sqrt{T}} - \frac{1}{S}}$$

Call this $u$. We will use it as the x-axis variable.

Step 4: Vega

Vega is the sensitivity of option price to volatility:

$$\text{Vega} = S \cdot \phi(d_1) \cdot \sqrt{T}$$

Step 5: Veta

Veta is $-\partial\text{Vega}/\partial T$ (negative sign so that the passage of time gives negative Veta for a decaying position). Let's differentiate $\text{Vega}$ with respect to $T$:

$$\frac{\partial\text{Vega}}{\partial T} = S \cdot \boxed{\phi'(d_1)} \cdot \boxed{\frac{\partial d_1}{\partial T}} \cdot \sqrt{T} \;+\; S \cdot \phi(d_1) \cdot \frac{1}{2\sqrt{T}}$$

Substituting highlighted terms, we get:

$$\frac{\partial\text{Vega}}{\partial T} = S \cdot \bigl[-d_1 \cdot \phi(d_1)\bigr] \cdot \left[-\frac{d_1}{2T} + \frac{\sigma}{2\sqrt{T}}\right] \cdot \sqrt{T} \;+\; S \cdot \phi(d_1) \cdot \frac{1}{2\sqrt{T}}$$

Factor out $S \cdot \phi(d_1)$,

$$\frac{\partial\text{Vega}}{\partial T} = S \cdot \phi(d_1) \left[ -d_1 \left( -\frac{d_1}{2T} + \frac{\sigma}{2\sqrt{T}} \right) \sqrt{T} \;+\; \frac{1}{2\sqrt{T}} \right]$$

Expanding the bracket:

$$\frac{\partial\text{Vega}}{\partial T} = S \cdot \phi(d_1) \left[ \frac{d_1^2}{2\sqrt{T}} - \frac{d_1 \sigma}{2} + \frac{1}{2\sqrt{T}} \right]$$ $$\frac{\partial\text{Vega}}{\partial T} = \frac{S \cdot \phi(d_1)}{2\sqrt{T}} \left[ d_1^2 + 1 - d_1 \sigma \sqrt{T} \right]$$

Therefore,

$$\boxed{\text{Veta} = -\frac{S \cdot \phi(d_1)}{2\sqrt{T}} \left[ d_1^2 + 1 - d_1 \sigma \sqrt{T} \right]}$$

Step 6: Veta / Vega

The ratio takes form:

$$\frac{\text{Veta}}{\text{Vega}} = \underbrace{-\frac{S \cdot \phi(d_1)}{2\sqrt{T}} \left[ d_1^2 + 1 - d_1 \sigma \sqrt{T} \right]}_{\text{Veta}} \;\cdot\; \underbrace{\frac{1}{S \cdot \phi(d_1) \cdot \sqrt{T}}}_{1/\text{Vega}}$$

The $S \cdot \phi(d_1)$ cancels,

$$\frac{\text{Veta}}{\text{Vega}} = -\frac{1}{2\sqrt{T}} \cdot \frac{d_1^2 + 1 - d_1 \sigma\sqrt{T}}{\sqrt{T}}$$ $$\boxed{\frac{\text{Veta}}{\text{Vega}} = -\frac{d_1^2 + 1}{2T} + \frac{d_1 \sigma}{2\sqrt{T}}}$$

This is the y-axis variable, expressed in terms of $d_1$, $\sigma$, and $T$.

Step 7: eliminating $d_1$

From Step 3, $u = -d_1/(S\sigma\sqrt{T}) - 1/S$, so $d_1 = -(u + 1/S) \cdot S\sigma\sqrt{T}$.

Substitute $d_1$ into the Veta/Vega expression. First the quadratic term.

$$d_1^2 = (u + 1/S)^2 \cdot S^2\sigma^2 T = u^2 S^2 \sigma^2 T + 2u S\sigma^2 T + \sigma^2 T$$

Then the linear term.

$$d_1 \sigma = -(u + 1/S) \cdot S\sigma^2\sqrt{T} = -u S\sigma^2\sqrt{T} - \sigma^2\sqrt{T}$$

Plug both into Step 6.

$$\frac{\text{Veta}}{\text{Vega}} = -\frac{u^2 S^2 \sigma^2 T + 2uS\sigma^2 T + \sigma^2 T + 1}{2T} + \frac{-uS\sigma^2\sqrt{T} - \sigma^2\sqrt{T}}{2\sqrt{T}}$$

Simplify each piece:

$$-\frac{u^2 S^2 \sigma^2 T}{2T} = -\frac{S^2\sigma^2}{2} \cdot u^2 \ ,$$ $$-\frac{2uS\sigma^2 T}{2T} = -S\sigma^2 \cdot u \ ,$$ $$-\frac{\sigma^2 T}{2T} = -\frac{\sigma^2}{2} \ ,$$ $$-\frac{1}{2T} = -\frac{1}{2T} \ ,$$ $$\frac{-uS\sigma^2\sqrt{T}}{2\sqrt{T}} = -\frac{S\sigma^2}{2} \cdot u \ ,$$ $$\frac{-\sigma^2\sqrt{T}}{2\sqrt{T}} = -\frac{\sigma^2}{2} \ ,$$

Grouping by powers of $u$,

$$\frac{\text{Veta}}{\text{Vega}} = \left(-\frac{S^2\sigma^2}{2}\right) \cdot u^2 \;+\; \left(-S\sigma^2 - \frac{S\sigma^2}{2}\right) \cdot u \;+\; \left(-\frac{\sigma^2}{2} - \frac{\sigma^2}{2} - \frac{1}{2T}\right)$$

reduces to:

$$\boxed{\frac{\text{Veta}}{\text{Vega}} = -\frac{S^2\sigma^2}{2} \cdot u^2 \;-\; \frac{3S\sigma^2}{2} \cdot u \;-\; \sigma^2 \;-\; \frac{1}{2T}}$$

where $u = \text{Speed}/\Gamma$.

This is

$$y = -au^2 - bu - c$$

an inverted quadratic function form, a downward-opening parabola in Speed/Gamma, exact under BSM with no approximations.

So What Does This Actually Give Us

The identity is exact under BSM. Every option on every coin, at every strike and expiry, must land on this parabola if the model holds. That is a strong constraint, and it has a few consequences worth thinking through.

The first is diagnostic. If we compute Speed/Gamma and Veta/Vega from live data and a cohort falls off the curve, something is inconsistent in the pricing. It could be a data issue, a stale quote, or a Greek computation bug. The parabola becomes a free sanity check that runs across all 24 cohorts simultaneously, without needing a reference value for any individual Greek.

The second is structural. The relationship tells us that gamma curvature and vega decay are not independent risk dimensions. They are algebraically coupled through spot, vol, and time. A desk monitoring both ratios is, in a BSM world, monitoring one thing expressed two ways. That does not mean one is redundant in practice (skew and jumps break the coupling), but it changes how you think about hedging both at once.

The third, and we think most interesting, is what happens when the parabola breaks. BSM assumes flat vol, no jumps, continuous hedging. Deribit options violate all three. Skew should bend the arms asymmetrically. Jump risk should scatter short-dated cohorts off the curve. A clean parabola on a given day probably means the volatility surface is well-behaved and internally consistent. A fragmented one might be an early sign that the market is pricing something BSM cannot capture.

We have not built a signal from this yet. The next step would be to measure the residual, i.e. the distance of each cohort from the theoretical curve, and track it over time. If the residual spikes before large moves, there is something tradeable here. If it stays noisy, the identity remains a useful diagnostic but not a signal. We plan to test this.

---

This observation was made using the Ratio Scatter panel in Cayø Largo's Greeks Tectonics page. The panel plots any two Greek ratios against each other across the 24-cohort ORIA grid (4 moneyness buckets, 6 expiry buckets) and updates every 10 minutes for all six coins. The parabola is visible in every cycle, not just the May 7 snapshot shown here. The derivation applies to any underlying, not just BTC. For assets with lower spot prices, expect visible asymmetry from the linear term.