# Perpetual Option Development Update

Firstly, we apologize for the lack of communication with the community on the perpetual option sides. As the believer of non-oracle based decentralized derivatives, we work hard on finding and refining our models. (Read here for our non-oracle argument) The difficulty was beyond our estimation The roadmap was delayed. We faced criticism and doubt on the validity of Antimatter's perpetual options. Facing difficulties, we actively seek help from math professionals including some DeFi founders for help. With the help of two PhDs from Oxford university, we made some good breakthroughs and we would love to give the community update.

Antimatter Mathematical model has been updated several times since its first appearance.

Antimatter was, at first, conceived as a type of option (tokens) to show whether a specific asset is in an upward or downward trend by using some financial parameters. The first model we were working on tried to define a function F by the call and put option volume(x,y respectively) and their ratio. We let the pricing of call and put tokens as partial derivatives of a function F that is expressed in terms of token volume. The first function F is defined as

To make the entire system work, we require that one needs to provide two types of assets to generate a token: one stable coin and one another type of asset. Here we let the price of the asset be dependent on x, y, which is a mistake we fixed later. It turned out this model does not work well because the system does not stabilize. We tried models such as

but in vain.

Defining F generally turned out to be a dead end. Therefore, we constructed the model from the investors’ perspective. We realized that when call and put tokens are in a certain ratio, no matter how the quantity of tokens varies, the prices should remain the same. This gives as a homogeneous condition that F (kx, ky) = kF (x, y). With this in mind and a switch to polar coordinates, we arrived at the following expressions:

Remember that one needs to provide two types of assets. Here U is the quantity of stable coin, E is the quantity of the asset, θ = y/x, and Ek , Uk are undetermined coefficients. With this we fixed several problems, we have F(θ) = Uu + Ee, where u is always $1 because U is stable coin such as US dollars and e is always given by the outside source, such as the price of ETH. It fixed a problem that the price of the asset depends on x and y, which is implausible. Another merit is that trend showing comes as a byproduct. The entire goal of our model shifts to providing a platform on which one can bet on the future trend of a type of asset.

Our next goal is to determine these coefficients. By using infinite sum, we can get the best approximation of the exact F. We have already tried k = 1 and chosen some coefficients. The results improved significantly and we believe that with calculus of variation and other techniques, we will work out soon.

Despite the mathematical progress we have made, the logistics of our platform are updated as well. Previously, the system has three parts: Creator of the options, market makers who generate and redeem call and put tokens, and the investors who trade tokens in the secondary market such as Uniswap. Now the entire system has only two parts. People who buy the tokens are treated the same as those who generate a token. People who sell the tokens are effectively the same as those who redeem the token. By simplifying three stages into two, we make the system more concise and arbitraging easier. In addition, you will be able to use smart routing to generate and redeem calls and put tokens through a single currency such as ETH, USDT. This avoids the liquidity issue with call and put tokens on Uniswap. More importantly, as the builder of the system, we find this way more stabilizing under arbitraging activity.

We will improve our communication with our community. As one of the only few players working on non-oracle-based derivatives, we believe in our ideas and we will work hard to reach our goal.

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**Website:** https://antimatter.finance/

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