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Enhanced Options Pricing
Accurate options pricing relies on sophisticated mathematical models that take into account the parameters of the option as well as volatility measures.

The Issue with Naive AMMs in options

A pitfall of current options pricing is implied volatility. The price of an option is inherently determined by the volatility of its underlying asset. What most current AMMs in options fail to account for is dynamic volatility. As a result, the pricing of options on many platforms often does not accurately represent the true nature of the market, causing price inefficiencies to be exploited by arbitrageurs. This exposes Liquidity Providers (LPs) to considerable losses over time as the AMM struggles to be profitable.

Zeta's Robust Options Pricer

Zeta will be using the Black-Scholes-Merton (BSM) formula for options pricing, a tried and tested mathematical model in TradFi. In order to implement effectively we will be utilising a dynamic volatility calculation that will updated based on real time trade data within our markets. By doing so we will be able to generate a volatility surface that can be fed into our options pricing algorithm. This will be a better method of price discovery that benefits takers and the liquidity providers.
Using an Automated Market Maker will bootstrap liquidity by providing both buy and sell orders for every tradable option in the Zeta-verse.

Zeta's Volatility Adjustment

Volatility is arguably the most important aspect of options pricing, and hence Zeta will keep an active, up-to-date volatility surface in order to provide efficient markets. Our volatility-adjustment algorithm will continuously update the OMM's volatility surface by examining trades that occur on-exchange, incorporating the changing market environment.
The equation below outlines the principles of the adjustment mechanism, which will occur when any trade happens on Zeta Markets. Our model's implied volatility will be "bumped" up or down with every trade, such that options are being marked at where they are truly trading.
The change in our model's implied volatility is a function of the trade size, the implied volatility of the trade, and adjustment parameters that we define below.
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ΔσT,t1−t0=1β(σT,t1−σT,t0)(nn+α)\Delta\sigma_{T, t_1 - t_0} = \frac{1}{\beta}(\sigma_{T, t_1}-\sigma_{T, t_0})(\frac{n}{n+\alpha})
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    α\alpha
    outlines a parameter which is used to adjust the weight that the size of a trade has on volatility (where n is the size of the trade that occurred)
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    β\beta
    outlines a parameter which is used to scale the overall adjustment that occurs due of a trade. A higher beta means smaller changes in implied volatility will occur with each trade.
Effectively, each trade will adjust our volatility surface like the below diagram. If people are buying options, this surface will adjust upward, and vice versa for sales.
Zeta's volatility surface pre-trade (dashed) and post-trade (solid).

Zeta's Delta Hedging

In order to reduce the variance taken on by LPs to the Zeta OMM pool for a particular product, Zeta's pool will provide delta hedging capabilities on its own, or composable platforms. This will serve the function of mitigating delta risk to LPs, in order to isolate the farming yield from the OMM's own volatility trading.
At either of 1) a set time interval ± a verifiable jitter, or 2) the OMM taking on delta risk above its max delta exposure threshold – the OMM will look to rebalance the pool in order to reduce its delta exposure down to its steady-state delta exposure threshold.
Both the maximum and steady-state thresholds will be set as a function of 1) the pool size, 2) the liquidity of the underlying asset, and 3) the volatility of the underlying asset – parameters which will be provided to the smart contract by governors of the Zeta platform.

Impermanent Loss (IL) in an options OMM

Impermanent loss is well explored in the linear AMM space, but still not well explained for options LPs.
Participants in the options OMM are taking on a variety of risks which can be catalogued via the options greeks. Bear in mind that risks are exposures that can cause the OMM pool to win, as well as experience losses.
Delta: Delta is the exposure an LP has to the spot price moving. As options are inherently a derivative instrument, LPs may experience fluctuations in their pool value due to the exposure that pool has to the underlying spot price.
Vega: Vega is the exposure an LP has to the implied volatility (IV) of an option. Because of Zeta's dynamic volatility surface, LPs may experience gains or losses as this surface shifts around.
Theta: Theta is the decay of an option as it loses its time value. LPs may be long or short theta, and thus may either earn yield, or pay yield for optionality throughout the life of the pool. Gamma: Gamma compensates LPs for the movement in the underlying price. Conversely, being short gamma will cause losses if the underlying moves a large amount.
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