ManualMath

Methodology

Every number on the desk is computed from on-chain state with the math below. Each formula is typeset exactly as it runs, with a pointer to the source file it lives in, so you can read the model and then read the code.

SVI Surface

deepskew fits each expiry with the raw Stochastic-Volatility-Inspired parameterization of Gatheral and Jacquier (2014). Five numbers ( $a, b, ρ, m, σ$ ) define total variance $w (k)$ as a function of log-moneyness $k = ln (K / F)$ , and IV follows by annualizing over time-to-expiry $T$ in years.

Total variancesrc/lib/svi.ts

w (k) = a + b (ρ (k - m) + (k - m)^{2} + σ^{2})

w (k) = a + b (ρ (k - m) + (k - m)^{2} + σ^{2})

The on-chain SVI smile in total-variance space, the object every downstream metric reads off.

Implied volatilitysrc/lib/svi.ts

IV (k) = w (k) / T, k = ln (K / F)

IV (k) = w (k) / T, k = ln (K / F)

Annualized implied vol at strike

K

, recovered from the total variance and the time-to-expiry.

The fixed-point integers from the indexer are decoded into floats in decodeSvi; the analytic first and second derivatives $w^{'} (k)$ and $w^{''} (k)$ feed the arbitrage and density math below.

Arbitrage-Free Checks

A surface is shown green only when it leaves no risk-free money on the table. Two conditions are sampled across the log-moneyness grid: a butterfly check on each slice and a calendar check across the term structure.

Durrleman butterfly conditionsrc/lib/svi.ts

g (k) = (1 - \frac{k w ^{'}}{2 w})^{2} - \frac{w ^{'2}}{4} (\frac{1}{w} + \frac{1}{4}) + \frac{w ^{''}}{2}

g (k) = (1 - \frac{k w ^{'}}{2 w})^{2} - \frac{w ^{'2}}{4} (\frac{1}{w} + \frac{1}{4}) + \frac{w ^{''}}{2}

The slice is butterfly-arbitrage-free if and only if

g (k) \geq 0

for every sampled

k

; one negative point flags an impossible negative implied density.

Calendar conditionsrc/lib/svi.ts

w_{T_{2}} (k) \geq w_{T_{1}} (k) (T_{2} > T_{1})

w_{T_{2}} (k) \geq w_{T_{1}} (k) (T_{2} > T_{1})

Total variance must be non-decreasing in

T

at fixed

k

; a longer-dated tenor priced cheaper in variance than a nearer one is a calendar arbitrage.

checkButterfly scans $g (k)$ over the grid and returns the minimum and any violations; checkCalendar sorts the slices by $T$ and counts crossings between adjacent tenors.

Digital Pricing

The keystone of the desk: an SVI smile is a continuum of binary prices. The model-fair value of an UP binary at strike $K$ (paying 1 above $K$ , 0 below) is the risk-neutral probability the underlying finishes above it, a pure function of $k$ and the params. Every on-chain mint is scored against this.

Up-binary fair valuesrc/lib/svi.ts

P (S_{T} > K) = Φ (d_{2}), d_{2} = - \frac{k}{w} - \frac{w}{2}

P (S_{T} > K) = Φ (d_{2}), d_{2} = - \frac{k}{w} - \frac{w}{2}

Here

w = σ^{2} T

is total variance and

Φ

is the standard normal CDF, so no separate

T

is needed. The down side is

1 - Φ (d_{2})

Call delta (for delta strikes)src/lib/svi.ts

Δ_{c} = Φ (d_{1}), d_{1} = - \frac{k}{w} + \frac{w}{2}

Δ_{c} = Φ (d_{1}), d_{1} = - \frac{k}{w} + \frac{w}{2}

The Black-Scholes-style call delta under the smile’s local variance, inverted by bisection to locate the 25-delta strikes used by the skew metrics.

Risk-Neutral Density

One SVI slice implies a full distribution of where the market prices the underlying ending at expiry, recovered via Breeden-Litzenberger as presented by Gatheral in The Volatility Surface. It is the whole shape, not a single number.

Implied density of the log-returnsrc/lib/svi.ts

q (k) = \frac{g ( k )}{2 π w ( k )} e^{- d_{-} (k)^{2} /2}, d_{-} = - \frac{k}{w} - \frac{w}{2}

q (k) = \frac{g ( k )}{2 π w ( k )} e^{- d_{-} (k)^{2} /2}, d_{-} = - \frac{k}{w} - \frac{w}{2}

The density of

x = ln (S_{T} / F)

in terms of Durrleman’s

g (k)

, sampled across the grid and normalized numerically to integrate to 1.

Density non-negativitysrc/lib/svi.ts

q (k) \geq 0 ⟺ g (k) \geq 0

q (k) \geq 0 ⟺ g (k) \geq 0

Because

g (k)

appears in the numerator, a valid probability density never going negative is exactly the butterfly condition on the smile.

Skew Metrics

The headline directional-skew gauges, in vol points. The 25-delta call and put strikes are solved off the smile by inverting $N (d_{1})$ , then their IVs are differenced.

25-delta risk reversalsrc/lib/svi.ts

RR_{25} = IV (25Δ call) - IV (25Δ put)

RR_{25} = IV (25Δ call) - IV (25Δ put)

How much richer 25-delta calls are than 25-delta puts; negative is puts bid, the normal BTC crash-fear regime.

25-delta butterflysrc/lib/svi.ts

BF_{25} = \frac{1}{2} (IV_{c} + IV_{p}) - IV_{atm}

BF_{25} = \frac{1}{2} (IV_{c} + IV_{p}) - IV_{atm}

How far the 25-delta wings sit above ATM, a measure of how convex (fat-tailed) the smile is.

Forward (between-expiry) volsrc/lib/svi.ts

σ_{fwd} = \frac{w _{2} - w _{1}}{T _{2} - T _{1}}

σ_{fwd} = \frac{w _{2} - w _{1}}{T _{2} - T _{1}}

The vol priced for the window between two tenors, from their ATM total variance

w = w (0)

; blank when variance is not increasing (calendar arbitrage in the window).

Implied movesrc/lib/svi.ts

\pm σ_{ATM} T

\pm σ_{ATM} T

The 1-sigma move the market prices into this expiry, the straddle-width breakeven, as a fraction of the forward.

Vault

The PLP vault is the counterparty to every position. The open book is reconstructed per (oracle, strike, side) from minted minus redeemed flow, marked at the model-fair $Φ (d_{2})$ , then aggregated into edge, concentration, and a stress scenario.

Vault edge per fillsrc/lib/analytics.ts

edge = paid - Φ (d_{2}), d_{2} = - \frac{k}{w} - \frac{w}{2}

edge = paid - Φ (d_{2}), d_{2} = - \frac{k}{w} - \frac{w}{2}

What the taker paid minus model fair, per fill, using each fill’s own oracle SVI and forward; positive means the vault overcharged.

MtM concentration (HHI)src/lib/analytics.ts

HHI = \sum_{i} s_{i}^{2}, s_{i} = \frac{liability _{i}}{\sum liability}

HHI = \sum_{i} s_{i}^{2}, s_{i} = \frac{liability _{i}}{\sum liability}

The Herfindahl-Hirschman index of marked liability across expiries:

\geq 0.5

is concentrated,

\leq 0.25

diversified.

Breach sigmasrc/lib/analytics.ts

σ_{breach} = min {∣ n ∣ : vaultValue (n) \leq 0}

σ_{breach} = min {∣ n ∣ : vaultValue (n) \leq 0}

The worst-direction shock the vault survives before LP equity is wiped. The forward is shocked to

F e^{n σ_{T}}

per oracle with the smile held fixed, and the change in marked liability is applied to the live vault value.

CautionReconstruction caveat

The book is rebuilt from the indexer flow window and marked at the client’s

Φ (d_{2})

, while the contract uses its cached matrix MtM. The two should track, but the reconstruction is an estimate bounded by the flow window. See Vault.

Vol-Risk-Premium

What option sellers earn for bearing risk, the LP’s edge. The desk reads it two ways: against an external reference, and as realized dollars harvested on-chain.

Vol-risk-premiumsrc/lib/cross-venue.ts

VRP = σ_{implied} - σ_{realized}

VRP = σ_{implied} - σ_{realized}

Implied vol minus trailing realized vol (Binance hourly closes, annualized). Positive means options are rich, the LP’s edge.

Realized vol-risk-premium capturedsrc/lib/analytics.ts

\sum_{i} (paid_{i} - fair_{i}) \cdot q_{i}

\sum_{i} (paid_{i} - fair_{i}) \cdot q_{i}

Cumulative signed dollar edge across the fill window, positive when the book was sold above model fair. The on-chain counterpart to the VRP spread above.

The cross-venue reference is the only out-of-protocol data the terminal touches: Deribit’s BTC DVOL index for implied and Binance BTCUSDT klines for realized. Both send open CORS and are called straight from the browser.

Implied reference: Deribit BTC DVOL, hourly, last 7 days.
Realized: annualized stdev of Binance hourly log returns over a rolling window.