# Components

The Risk Protocol (TRP) uses its proprietary Risk Engine to construct tokenized products with specific risk/return profiles. The Risk Engine consists of two parts.

* **Part 1** is the TRP volatility forecasting model.
* **Part 2** is the option pricing engine, which includes a SABR stochastic volatility model along with an implied volatility skew model and option pricing functions.

TRP employs its volatility forecasting model to determine the strike prices of options used in its products and to price tokens in real time. Extensive research by TRP into crypto volatility (refer to "The Nature of the Beast") led to the selection of the 2-component GARCH-GJR (Glosten, Jagannathan, Runkle 1993) model for forecasting Bitcoin and Ethereum volatility. This model leverages historical return data to estimate the underlying volatility term structure.

The 2-component GARCH-GJR forecast is crucial for pricing options within the Risk Protocol’s tokenized products. All TRP products include options generated using TRP's proprietary SMART mechanism. In most cases, these options do not have listed or actively traded counterparts. As a result, TRP lacks access to market prices for these options, making it necessary to estimate option values in real time to accurately value tokens.

TRP is committed to providing investors with a high-quality, unbiased estimate of the value of its tokenized products. To achieve this, options must be priced in real time and incorporated into token price calculations. The options used in Risk On/Risk Off products include an out-of-the-money call and an out-of-the-money knockout barrier put with a cash rebate. Fortunately, closed-form solutions exist for pricing these options, though they involve simplifying assumptions. One of the most unrealistic assumptions in commonly used pricing models is that volatility remains constant throughout the life of the option. This assumption is clearly flawed. The SABR model is a stochastic volatility framework that adjusts volatility in traditional pricing models to account for non-constant volatility. A key advantage of SABR is that it produces a full volatility surface, which includes both an implied volatility skew and a volatility term structure. TRP uses SABR to derive stochastic volatility-adjusted implied volatilities for all strikes and expirations.

The primary input to SABR is the instantaneous volatility estimate produced by TRP’s 2-component GARCH-GJR forecast. Additional inputs include volatility of volatility (denoted as xi), a beta parameter that adjusts for different underlying price return distributions, and rho, which is the correlation between asset price returns and instantaneous volatility.

Option pricing models are employed to estimate TRP options. For standard vanilla options, TRP uses the Black 76 variant of the widely known Black-Scholes pricing model. Inputs for this model include the current underlying crypto price, the option’s strike price, a risk-free interest rate, income from the underlying cryptocurrency, time to expiration, option type (put or call), and the volatility estimate from the 2-component GARCH-GJR model adjusted by SABR. For the down-and-out knockout barrier put with a cash rebate, TRP uses formulas developed by Merton, Reiner, and Rubinstein, as detailed in Haug (2007). Inputs for this model include the underlying crypto price, strike price, risk-free interest rate, knockout barrier, cash rebate (in case of a knockout), time to expiration, income from the underlying crypto, instantaneous volatility of the underlying asset, and option type (put or call).



