Aria: Decentralized Options Pricing with Dual Prediction Markets
Aria is a novel approach for pricing European options with binary options and prediction markets in two assets.
Decentralized Options
Aria is a decentralized protocol for pricing European options. The protocol uses a single pool of liquidity in two assets, backed by binary options and a prediction market AMM, to support a wide range of option types with varying strike prices and maturities.
Here’s how it works:
- Given two assets, a strike price, and an expiry time, create two binary options (a binary call and a binary put) for each asset.
- Create prediction markets for these binary options.
- Replicate the payoffs of European options with an equivalent portfolio consisting solely of binary options.
- Provide liquidity and efficient pricing for European options with an automated market maker (AMM) for these binary options.
By replicating the payoffs of European options with binary options, we’re able to efficiently price and provide liquidity for European call options (long and short) and European put options (long and short).
Other different, more complex financial derivative contracts can be replicated using the same framework.
A Pricing Example With Two Assets
Imagine we have two assets, X and Y. For example, ETH and USD.
The price of ETH/USD represents the units of USD required to obtain 1 ETH at a certain time. We can just as easily quote the price of USD in units of ETH.
For example, if the price of 1 ETH is 2,000 USD, then the price of 1 USD is simply 1/2,000 ETH, or 0.0005 ETH.
The price of USD/ETH represents the units of ETH required to obtain 1 USD at a certain time. This price is the inverse price of ETH in units of USD.
Prices are always expressed in terms of a reference asset. We usually take asset Y to be the reference asset, which corresponds to USD in our example. In practice, it doesn’t matter too much which asset, X or Y, is chosen to represent the reference asset. We’ll use this key principle for pricing binary options, and subsequently, European options and other derivative contracts.
Binary Options
Binary options offer a simple yes-or-no proposition: Will an underlying asset be above or below a certain price at a certain time?
At expiration, a binary option delivers one of two payoffs: either 1 unit of the underlying asset, or nothing at all.
- A binary ETH/USD call delivers 1 ETH if the price of ETH (in units of USD) is greater than a fixed constant K (also known as the “strike” price) at time T. Otherwise, the option expires worthless.
- A binary ETH/USD put delivers 1 ETH if the price of ETH (in units of USD) is less than a fixed constant K at time T. Otherwise, the option expires worthless.
- A binary USD/ETH call delivers 1 USD if the price of USD (in units of ETH) is greater than a fixed constant 1/K at time T. Otherwise, the option expires worthless.
- A binary USD/ETH call delivers 1 USD if the price of USD (in units of ETH) is less than a fixed constant 1/K at time T. Otherwise, the option expires worthless.
The inverse price relates the ETH/USD strike K exactly to its USD/ETH inverse, 1/K. For example, a binary ETH/USD call will deliver 1 ETH if (and only if) the equivalent binary USD/ETH put delivers 1 USD at expiration.
If the price of the underlying asset is exactly the strike price at expiration, a binary option is said to expire “at-the-money” and typically returns the original investment.
For our purposes, we can assume that at-the-money binary options expire worthless, rather than delivering the underlying asset or the original investment, since we’ll be using them to price European options. (At-the-money European options have no intrinsic value at expiration.)
Replicating Assets With Binary Options
First, we notice 1 ETH is equivalent to the following portfolio of binary options:
- Long 1 binary ETH/USD put with strike K, and;
- Long 1 binary ETH/USD call with strike K.
Similarly, 1 USD is equivalent to the following portfolio:
- Long 1 binary USD/ETH call with strike 1/K, and;
- Long 1 binary USD/ETH put with strike 1/K.
A pair of binary options (a binary call and a binary put) with the same underlying asset, strike price, and expiration can be viewed as a single prediction market, since their payoffs sum to exactly 1 unit of the underlying asset for all prices.
Arrow–Debreu Securities
Binary options are also referred to as digital options or Arrow–Debreu securities[1]. Almost any derivative contract can be replicated as a linear combination of Arrow–Debreu securities, and thus as a weighted sum of its state prices. We’ll use this principle to price European options.
Replicating European Option Payoffs With Binary Options
The payoff of a European ETH/USD call option with strike K can be exactly replicated with a portfolio of just two binary options:
- Long 1 binary ETH/USD call with strike K, and;
- Short K binary USD/ETH puts with strike 1/K.
If the price of ETH/USD is greater than K at expiration, we receive 1 ETH and pay K units of USD. Otherwise, the binary options expire worthless. This payoff exactly replicates the payoff of a European call option.
Similarly, the payoff of a European ETH/USD put option with strike K can be exactly replicated with the following portfolio:
- Long K binary USD/ETH calls with strike 1/K, and;
- Short 1 binary ETH/USD put with strike K.
If the price of ETH/USD is less than K at expiration, we receive K units of USD and pay 1 ETH. Otherwise, the binary options expire worthless. This payoff exactly replicates the payoff of a European put option.
Prediction Market AMMs
Binary options, with their simple “asset-or-nothing” payoffs, can be effectively priced with an automated market maker (AMM).
Prediction market AMMs are particularly effective for assets with binary outcomes, for which the event space can be partitioned into a set of two or more distinct outcomes, exactly one of which will occur.
We have binary options who payoffs sum to exactly one unit of either ETH or USD, for all prices, so we can use prediction market AMMs for these binary options to provide liquidity and prices for European options in an efficient manner.
Aria uses a “logarithmic market scoring rule” (LMSR)[2][3] AMM for binary options, but works well with other prediction market AMMs.
Settlement At Expiration
European options are fully collateralized by the pool. In order to ensure European options can always deliver their payoffs, at expiration:
- European ETH/USD call options are settled in units of ETH, and;
- European ETH/USD put options are settled in units of USD.
The payoff of a European ETH/USD call option (in units of ETH) will never exceed 1 ETH (which happens if the price of ETH/USD goes to infinity). If the price of ETH/USD is less than or equal to the strike K, the European ETH/USD call option expires worthless.
Similarly, the payoff of a European ETH/USD put option (in units of USD) will never exceed K units of USD (which happens if the price of ETH/USD goes to zero). If the price of ETH/USD is greater than or equal to the strike K, the European ETH/USD put option expires worthless.
Summary
Aria is a decentralized protocol for European options:
- Aria provides on-chain prices for traditionally “hard-to-price” contracts like European options without a Black–Scholes price oracle or implied volatility calculations.
- Aria enables a single pool of two assets to provide liquidity for European options with varying strike prices and maturities.
References
[1]: https://en.wikipedia.org/wiki/State_prices
[2]: R. Hanson. Combinatorial information market design. Information Systems Frontiers, 5(1):107–119, 2003.
[3]: R. Hanson. Logarithmic market scoring rules for modular combinatorial information aggregation. The Journal of Prediction Markets, 1(1):3–15, 2007.
