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Introduction to Derivatives

Mark Pricing

Mark pricing for futures and options are both done within Zetaβs internal risk engine, and are refreshed on a per-second basis.

Mark pricing for options

A key component of pricing options is volatility - this measures how much an underlying asset moves. This is critical in options trading as it helps project how likely an option is to be in or out of the money. The Zeta Protocol maintains its own volatility surface for each expiry. For each trade that occurs between 5d and 60d the volatility surfaces are adjusted as there is new information that must be factored into pricing.

Zeta uses the Black-76 model to price its options, this is a model based on the Nobel prize winning Black-Scholes-Merton model.

The model has the following inputs:

**Future Price**: Is calculated using the underlying price provided by the oracle, the risk free rate and the time till expiry.**Volatility**: Of the underlying asset is calculated by the volatility surface that Zeta maintains (mentioned above).**Interest Rate**: Is returned from the risk free rate curve which is stored by Zeta.**Strike**: Is the price predefined in the contract for which to buy or sell the underlying - this a parameter defined in the options contract.**Time to Expiry**: Calculated from the expiry time - which is a parameter defined in the options contract.

Volatility Surface Calibration

- An internal volatility surface is calibrated to each options trade on the platform occurring between 5d and 60d.
- Volatility smiles are independent per tenor, and values are the total implied black-scholes volatility between the current time and the expiry T.
- Volatility surface defined by a 5 point moneyness (F/K) surface, with nodes at 0.5/0.75/1.0/1.25/1.5.
- Surface is linearly interpolated, with volatilities <0.5 and >1.5 taking the values at 0.5 / 1.5.
- On every trade, the volatility adjustment amount is approximated as:

- The volatility adjustment is applied to the two closest nodes at which the trade occurred, with only 1 adjustment applied if the trade is >1.5 or <0.5. For new expiries, the new volatility surface to be initiated is a clone of the volatility surface of the expiry preceding it.

In order to translate the spot price into a future price (at expiry) an interest rate curve is used that is maintained natively by the Zeta Protocol. Every time a futures trade occurs the interest rate curve is adjusted as the system now has new information about the market that should be factored into pricing.

The following inputs are used to determine the futures pricing:

- Spot Price: This is returned from the oracle feed
- Time to Expiry: Calculated from the expiry time - which is a parameter defined in the futures contract
- Interest Rate: Is returned from the risk free rate curve which is stored & maintained by the Zeta Protocol

Interest Rate Calibration

- An interest rate curve is fitted to each futures trade on the platform.
- Interest rate curve is defined as the total interest rate per expiry that futures are listed on. Interest rates are independent per tenor.
- On every trade, the interest rate adjustment amount is approximated as:

- The interest rate adjustment is applied to the interest rate directly corresponding to the tenor of the future.
- For new expiries, the new interest rate value to be initiated is determined by the interest rate of the expiry preceding it.

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Mark pricing for options

Mark pricing for futures