Volatility, fair premium and weekly recalibration

In one paragraph

The shape of the premium curve never changes, but its height is reset once a week. At the start of each week, the realized volatility of the market's official price over a recent window is fed into the Black-Scholes formula to obtain the theoretical fair premium of a "one-week floor"; that value is multiplied by a safety factor greater than 1 and rounded up to give the start of the curve, and the kink and the end follow at fixed multiples. The new three points apply only to later purchases; protection already bought keeps its rate. The inputs, the formula and the results are published every week, so anyone can recompute them. This page explains what realized volatility is, where the fair premium comes from, why a safety factor is added, what happens in calm and in stressed weeks, and what this model cannot do.

What realized volatility is

Volatility measures how violently a price moves. PUTGO uses realized volatility (also called historical volatility): take the daily returns of the official price over the look-back window, compute their standard deviation, and scale it to an annual figure, written σ. The length of the look-back window is on Markets and parameters.

One way to get a feel for σ: an annualized volatility of 50% means, roughly, that the asset's "typical" move over a year is about 50%. To shrink that to one week, scale by the square root of time:

\[ \sigma_{\text{one week}} \approx \sigma \times \sqrt{\tfrac{7}{365}} \approx 0.138\,\sigma \]

In words: the typical move over one week is about 14% of the annualized volatility. With σ at 50% the typical weekly move is about 6.9%; with σ at 120% it is about 16.6%. The floor sits at 95% or 90% of the current price, so the chance of falling through it within a week is plainly tied to this number, and that is how volatility enters the premium.

Why realized volatility rather than the "market's expectation", implied volatility? Because implied volatility has to be read from an active options market, and there is no such price onchain to rely on, whereas the official price series is visible to everyone. Realized volatility is simple, public and reproducible; the cost is that it only looks backwards, which the last section discusses.

Fair premium: the Black-Scholes value of a one-week floor

Seen through finance, a PUTGO protection is a right, held for one week, to sell at the floor price, judged by a single price at expiry. Such a right has a standard theoretical value, given by Black-Scholes. PUTGO applies it in the simplest possible setting:

• the current price is set to 1, so the floor is the floor ratio itself (NVDA 0.95, UNI 0.90);
• the term is the nominal tenor of 7 days, that is 7/365 of a year;
• the interest rate is set to zero;
• the volatility is the σ of the previous section.

The result is a unitless ratio: the theoretical value of the floor as a fraction of the asset's value, which is exactly the unit of a "rate per 7 days".

The formula

\[ \text{fair} = K\,N(-d_2) - N(-d_1),\qquad d_1 = \frac{\ln(1/K) + \tfrac{1}{2}\sigma^2 T}{\sigma\sqrt{T}},\qquad d_2 = d_1 - \sigma\sqrt{T} \]

where \(K\) is the floor ratio, \(T = 7/365\), and \(N(\cdot)\) is the standard normal distribution function. In words: the fair premium is "what the pool pays at the floor when the expiry price is below it" minus "the value of the asset the pool receives", summed over all possible expiry prices weighted by their probability. It captures both the probability of falling through the floor and how far below it the price lands on average.

Setting the current price to 1 means the result is independent of the price level. Whether NVDA trades at 220 dollars or 22, the rate is the same as long as the floor ratio and the volatility are the same. The interest rate is set to zero because a week is too short for interest to matter, and because the premium is paid in the asset while delivery is paid in USDG; the two sides are not the same money in the first place.

The table gives illustrative figures (current price 1, term 7 days, interest rate zero):

Market	Floor ratio	σ = 30%	σ = 50%	σ = 80%	σ = 120%
NVDA	95%	0.21%	0.90%	2.26%	4.26%
UNI	90%	0.01%	0.18%	0.96%	2.51%

Two patterns are worth remembering. First, at the same volatility a lower floor means a lower fair premium: the UNI floor sits at 90%, so the price has to fall twice as far to reach it. Second, the fair premium is not linear in volatility: for NVDA, going from σ 30% to 80% multiplies the premium by more than ten, not by under three. A 5% fall within a week is a rare event at σ 30% and a routine one at σ 80%.

Safety factor, kink multiple and cap multiple

The fair premium is the anchor of the curve, not the curve itself. The three points written each week are:

\[ r_0 = \lceil \text{fair} \times \text{safety factor} \rceil,\qquad r_1 = \text{kink multiple} \times r_0,\qquad r_2 = \text{cap multiple} \times r_0 \]

In words: the start equals the fair premium times the safety factor, rounded up to one basis point (0.01%); the kink and the end are fixed multiples of the start. The current values of the three parameters are on Markets and parameters.

Why the start is above the fair premium. The fair premium is the model's expected value under idealized assumptions, while depositors carry real-world risk:

• the tails of the price distribution are fatter than the model assumes; gaps and single-day crashes are more common than a normal model predicts;
• the actual length of a protection is 7 to 8 days, whereas the model uses 7 (see Settlement);
• volatility is computed from the past, and next week's may be higher;
• depositors' USDG is reserved for the life of the protection, which is itself an opportunity cost.

The safety factor folds these things, which cannot be modeled precisely, into one multiplier. It is not a "profit margin"; it is a buffer against model error. It rounds up, in the same direction as the premium and the reserve.

Why the kink and the end follow at multiples. The three points rise and fall with volatility in fixed proportion, so the shape of the curve is the same in any week: before the kink, each additional percentage point of utilization adds a fixed fraction of the start; the slope of the scarcity range is also a fixed multiple of the start. How utilization maps to the rate is explained in How the premium is priced.

Illustration: one recalibration

Computed with illustrative parameters of safety factor 1.2, kink multiple 2 and cap multiple 5 (current values on Markets and parameters):

• NVDA, floor 95%, σ = 50%: fair premium ≈ 90 basis points, start = ⌈90 × 1.2⌉ = 108 basis points, that is 1.08%; kink 2.16%; end 5.40%.
• UNI, floor 90%, σ = 120%: fair premium ≈ 251 basis points, start ≈ 302 basis points, about 3.0%; kink about 6.0%; end about 15.1%.

The first week uses the initial curve on Markets and parameters; every week after that is recalibrated as described here.

The recalibration process

flowchart LR
    A[Official price series<br/>over the look-back window] --> B[Daily returns]
    B --> C[Annualized volatility σ]
    C --> D[Black-Scholes<br/>fair premium]
    D --> E[× safety factor<br/>rounded up → start]
    E --> F[× kink multiple → kink<br/>× cap multiple → end]
    F --> G[Written to the market's curve<br/>applies to new purchases only]
    A -. published .-> P[Inputs, formula, results]
    G -. published .-> P

Once a week, computed separately for each market. NVDA and UNI have different volatilities and different floor ratios, so their curves differ; their pools are separate as well, see Money Pool economics.

Why only new purchases are affected

Your rate is locked at the moment you buy and written into your position; no later recalibration changes it. The reasons are direct:

• the premium was already deducted in the asset at purchase and the protected amount already fixed; the identity "deposit = premium + protected amount" cannot be rewritten after the fact;
• the pool has already reserved USDG at your floor, and depositors have already started receiving your premium second by second;
• if rates could be adjusted afterwards, neither buyers nor depositors could know their terms at the time of the trade.

So a new curve changes only "what the next protection costs". Protection already in force keeps releasing premium at its original rate until expiry. Depositors' income therefore has inertia: in a week when the new curve becomes cheaper, the premium from older protection in the pool still flows in at the old rate.

Why the inputs are published

The recalibration is computed offchain and the result is written onchain. So that "computed offchain" does not become "black box", three things are published together every week: the official price series used over the look-back window, the calculation formula (including the annualization convention), and the resulting volatility and three points. Anyone can recompute from the same data and arrive at the same three points. If some week's three points cannot be reproduced from the published data, that is a problem that deserves to be pointed out.

This is also part of what Markets and parameters means when it promises that "parameter changes are updated on this page and announced": the three points are parameters, and every change to them has a traceable basis.

A calm week and a stressed week

Looking at both situations together shows how the volatility anchor and the utilization curve work with each other.

Calm weekStressed weekA sudden turn

The price has barely moved over the look-back window and σ has fallen to a low level. After recalibration the start is lower and the whole curve shifts down: protection is cheaper. Cheap protection attracts more purchases, utilization rises, and the rate rises along the curve. Depositors receive less premium per protection, but if the number of protections grows, total income need not fall. The real risk depositors carry is also lower at this time: when the price is flat, the chance of falling through the floor is small.

A sharp fall occurred within the look-back window and σ has risen. After recalibration the start is higher and the whole curve shifts up: protection is dearer, reflecting the greater chance of falling through the floor and the higher compensation depositors require. At the same time, demand for protection tends to rise in turbulent periods, utilization climbs, and the two effects stack, so the rate can be far above that of a calm week. Protection already bought is unaffected and is held to expiry at the lower rate locked at purchase.

The moment that deserves the most attention is the abrupt switch from calm to stress. Volatility is computed over the whole look-back window; a single-day crash is one point in that window and takes several days to lift σ noticeably, and recalibration then waits for the next week. Until then the height of the curve still reflects "the calm of the past". The only thing that can react immediately in that interval is utilization: demand floods in, utilization rises, the rate climbs along the curve, into the scarcity range and possibly to refusal at 100%. This is why PUTGO does not price on volatility alone, but on two layers stacked: the weekly anchored height, plus instantaneous scarcity pricing.

Limits of the model

To be honest about it, this model has clear boundaries:

• It looks backwards. Realized volatility is a backward-looking statistic and contains no expectation of future events (earnings, regulation, macro). The safety factor is a blunt buffer and does not replace expectations.
• It assumes a continuous, symmetric price distribution. Black-Scholes has no gaps. During hours when the official price stops updating, an asset can jump past the floor straight to a lower level, and the model knows nothing of this.
• It uses 7 days. The actual protection period is between 7 and 8 days; the extra hours are covered only indirectly, by the safety factor.
• It recalibrates once a week. The lag inside the window plus the weekly cadence means the curve's height reacts to sudden events with a delay of at least several days.
• The sample in the look-back window depends on how often the official price updates. If the official price updates less often at certain hours, the daily-return series reflects that, and the effect can be seen in the published input data.

Who bears the consequences of these limits? Depositors. Their USDG may be converted into the asset at the floor, and the asset's market price at that moment may be below the floor; see the risk section of Earn. The premium and the safety factor compensate for that risk; they do not remove it. The safety factor, the multiples and the look-back window are adjustable parameters, and changes to them are announced on Markets and parameters.

Key takeaways

• The shape of the curve is fixed; its height is reset weekly: start = fair premium × safety factor, rounded up; the kink and the end are fixed multiples of the start.
• The fair premium is the Black-Scholes value of a "one-week floor at the floor ratio" with the current price set to 1, a term of 7/365 of a year, an interest rate of zero, and the annualized realized volatility of the official price.
• A lower floor means a lower fair premium; the fair premium rises faster than linearly with volatility.
• The safety factor covers fat tails, gaps, the actual 7-to-8-day term and the backward-looking nature of volatility; it is a buffer against model error, not a profit margin.
• A new curve affects only later purchases; the rate of protection already bought is written into the position and never changes.
• The input data, the formula and the results are published every week, and anyone can recompute them.
• The volatility anchor updates once a week and reacts slowly; utilization pricing reacts instantly; the two layers stack.
• The model looks backwards, assumes no gaps and recalibrates only once a week; depositors bear the consequences of these limits, and the premium and the safety factor are compensation, not removal.