# Mark Price

Pricing of options begins by calculating a Mark price for each option. The Mark price is designed to closely approximate the “fair” mid-price of the option and is modeled using the Black-Scholes-Merton (BS) model.

\begin{aligned}&C = SN (d_1) - Ke ^{-rt} N (d_2) \\&\textbf{where:} \\&d_1 = \frac { ln ^ S_K + (r + \frac { \sigma^2_v }{ 2 } ) t }{ \sigma_s \sqrt { t } } \\&\text{and} \\&d_2 = d_1 - \sigma_s \sqrt { t } \\&\textbf{and where:} \\&C = \text{Call option price} \\&S = \text{Current underlying price} \\&K = \text{Strike price} \\&r = \text{Risk-free interest rate} \\&t = \text{Time to maturity} \\&N = \text{A normal distribution} \\\end{aligned}

The BS model takes into account several factors that are known to all market participants - the current market price of the underlying asset, the exercise price of the option, the time to expiration, and the risk-free interest rate. The last input variable into the BS model, is the implied volatility of the underlying asset, which is the market's estimate of the future volatility of the underlying. However, since implied volatility is not known to the SDX, it is derived from an external Volatility and Skew Oracle.

Volatility Oracle

The Volatility Oracle uses the product of BTC/ETH's external market implied volatility (IV) and a linear combination of relative historical volatilities (HV) between the underlying asset and BTC/ETH. The historical volatility is calculated from annualizing the standard deviation of the natural log returns of the hourly closing price over the past N intervals. The external market used as the barometer of implied volatility is Deribit, the leading options exchange with over 80%+ global options volume.

The formula below depicts the method for calculating a 3 day historical volatility:

$\sigma_{3day} = \sqrt{\sum_{i \in 1..72}{\frac{{(X_i - \bar X})^2}{72}}}\\where\\ \bar X = \sum_{i \in 1..72}\frac{X_i}{72} \\ X_i = ln(Price_i / Price_{i-1})$
$HV = \sigma_{3day}*\sqrt{365*24}$

#### Skew Oracle

The BS Model assumes that IV is consistent across all strikes and terms. In practice however, strikes and tenors change the IV, as investors price in liquidity, credit, and other risks.

To approximate these changes in IV, a Skew oracle publishes a mapping of strike prices to an IV multiplier. The IV multiplier is applied to HV to produce a mark IV for each strike price. This mapping is derived by projecting the skew and term structure of BTC and ETH options from Deribit to the new asset, such as SOL. This involves the following steps:

1. Generate a plot of IV Skew against Delta Buckets, by identifying the delta of each strike interval using ATM IV, which establishes the baseline for comparison.

2. Transform Y-Axis of IV Skew by dividing IV over ATM IV to determine an IV multiplier

1. Apply mean average among the BTC and ETH Transformed IV Skews

2. For the new underlying asset, calculate a mapping of strike price to delta of supported intervals, using historical volatility with the BS Model.

3. Find IV multiplier for each delta associated to the option strike price.

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