Pricing Engine | Oracle Papers

Base Model

The pricing kernel is the Black-Scholes model expressed in the SOL numeraire. Since the underlying spot $S$ is quoted in SOL, no FX term enters the valuation. The model values of a vanilla European call $C$ and put $P$ are

C = S\,N(d_1) - K e^{-r\tau} N(d_2), \tag{1}

P = K e^{-r\tau} N(-d_2) - S\,N(-d_1), \tag{2}

with

d_1 = \frac{\ln(S/K) + \left(r + \tfrac{1}{2}\sigma^2\right)\tau}{\sigma\sqrt{\tau}}, \qquad d_2 = d_1 - \sigma\sqrt{\tau}, \tag{3}

where $S$ denotes spot in SOL, $K$ the strike in SOL, $\tau$ the time to expiry in years, $\sigma$ the annualized volatility of log returns, $r$ the continuously compounded risk-free rate in the SOL numeraire, and $N(\cdot)$ the standard normal cumulative distribution function.

Given short tenors and a SOL numeraire, the carry term $r\tau$ is of negligible order relative to total variance $\sigma^2\tau$ . The engine sets $r \approx 0$ , so that $e^{-r\tau} \approx 1$ and $d_{1,2}$ reduce to functions of log-moneyness $\ln(S/K)$ and total variance alone.

Realized Volatility Estimation

The volatility input is anchored to the realized volatility of the underlying. Given spot observations $S_0, S_1, \dots, S_n$ sampled at a fixed interval, define log returns and their sample mean

r_i = \ln\!\left(\frac{S_i}{S_{i-1}}\right), \qquad \bar{r} = \frac{1}{n}\sum_{i=1}^{n} r_i. \tag{4}

The annualized realized volatility estimator is

\sigma_{\text{realized}} = \sqrt{\frac{N}{n-1} \sum_{i=1}^{n} \left(r_i - \bar{r}\right)^2}, \tag{5}

where $n$ denotes the number of returns in the estimation window and $N$ the number of sampling intervals per year, scaling per-interval variance to an annual horizon under the assumption of independent increments. The sampling frequency, window length, and any bounds applied to the estimate are calibrated internally.

Volatility Skew Surface

A flat volatility across strikes is inconsistent with observed order flow and with the empirical return distribution. The engine maps strike and tenor to implied volatility through a second-order parametric surface in log-moneyness $m = \ln(K/S)$ :

\sigma(K, \tau) = \sigma_{\text{ATM}}(\tau) + \beta \cdot m + \gamma \cdot m^2, \tag{6}

where $\sigma_{\text{ATM}}(\tau)$ denotes the at-the-money term structure, $\beta$ the skew coefficient controlling the first-order slope in $m$ , and $\gamma$ the curvature coefficient controlling smile convexity. $\beta$ and $\gamma$ are calibrated to order flow and to the positive skewness of early-stage token returns, such that out-of-the-money calls are quoted at volatilities consistent with the empirical frequency of large positive moves. Their values are proprietary.

Sensitivities

The Greeks are the partial derivatives of model value with respect to its inputs. With $\varphi(\cdot)$ the standard normal density, under the base model:

\Delta_C = \frac{\partial C}{\partial S} = N(d_1), \qquad \Delta_P = \frac{\partial P}{\partial S} = N(d_1) - 1. \tag{7}

Delta ( $\Delta$ ) is the first-order sensitivity of value to spot.

\Gamma = \frac{\partial^2 C}{\partial S^2} = \frac{\varphi(d_1)}{S\,\sigma\sqrt{\tau}}. \tag{8}

Gamma ( $\Gamma$ ) is the second-order sensitivity to spot, identical for calls and puts, and maximal near the money as $\tau \to 0$ .

\nu = \frac{\partial C}{\partial \sigma} = S\,\varphi(d_1)\sqrt{\tau}. \tag{9}

Vega ( $\nu$ ) is the sensitivity to volatility, identical for calls and puts at a common strike.

\Theta_C = \frac{\partial C}{\partial t} = -\frac{S\,\varphi(d_1)\,\sigma}{2\sqrt{\tau}} - rKe^{-r\tau}N(d_2). \tag{10}

Theta ( $\Theta$ ) is the sensitivity to the passage of time. With $r \approx 0$ the second term vanishes and $\Theta_C \lt 0$ for long positions, reflecting the decay of extrinsic value toward expiry.

Model Limitations

Black-Scholes assumes geometric Brownian motion: normally distributed log returns, constant volatility, and continuous sample paths. Early-stage token returns violate each assumption:

Leptokurtosis and positive skewness. The empirical return distribution exhibits excess kurtosis and right skew; multi-sigma positive moves occur at frequencies far exceeding the lognormal law.
Stochastic, clustered volatility. Volatility is non-stationary and clusters around listing, liquidity migration, and order-flow shocks.
Jump discontinuities. Large orders against shallow liquidity induce discontinuous price moves that a pure diffusion cannot represent.

Consequently, the unadjusted kernel systematically underprices out-of-the-money upside on early-stage tokens. Oracle applies proprietary corrections on top of the base model to account for these effects. Equations (1) through (10) define the base kernel only; quoted premiums do not equal the raw Black-Scholes value.