Digital Options

Digital options, also known as binary or bet options, offer a predetermined risk and reward for the holder. Digital calls and puts are minted as Spear and Shield tokens, respectively.

The Game is a win-or-lose battle, where bullas and beras combat with Spears and Shields, for prizes claimable at expiry. -- anon🤿

Divergence v1 offers European-style digital options, each backed by an ERC-20 token collateral. They are exercised at expiry, paying a fixed amount or nothing at all.

What are Digital Options?

Digital Call

• Premium costs less than 1 collateral

• Buyer has the right to claim 1 collateral, if the underlying price settles at or above the strike.

• Otherwise, it expires worthless.

Digital Put

• Premium costs less than 1 collateral

• Buyer has the right to claim 1 collateral, if the underlying price settles below the strike.

• Otherwise, it expires worthless.

How Much Does it Cost to Long or Short a Digital Option?

Position	Call	Put
Long	Premium	Premium
Short	1 - Premium	1 - Premium

Risk & Reward for a Digital Option

Settlement	Long Call	Short Call	Long Put	Short Put
Price ≥ Strike	Gain 1 - Premium	Lose 1 - Premium	Lose Premium	Gain Premium
Price < Strike	Lose Premium	Gain Premium	Gain 1 - Premium	Lose 1 - Premium

How to trade Digital Options?

When using Divergence v1 smart contracts, one can come across:

• ⚔️ Battle an AMM pool for digital options

• 🏟️ Arena deployer and admin of Battle

• Spear a European digital call option token

• Shield a European digital put option token

Each Battle mints Spear and Shield, ERC-20-compliant options tokens of a specific underlying, strike price, collateral, and expiry. A Battle's contract address is not re-used after expiry. A new Battle can be created for the following expiry.

Suppose a digital call costs 0.01 DAI. Its underlying is ETHUSD, and its strike is $2,000. If ETH settles above or equal $2,000, the option pays 1 DAI. Otherwise, it expires worthless. The buyer of this call earns 1 - 0.01 = 0.99 DAI less fees, i.e., a 99x return.

The v1 interface is one of the many ways one may interact with the protocol. The following pages provide concise, step-by-step guides on how to trade digital options:

Long Options

Short Options

How to Price Digital Options?

Simply put, each and every individual user of the protocol decides the price, as they buy and sell tokenized digital options. The protocol does not use a theoretical pricing model to arbitrarily set option prices. The price for an option is simply the amount of collateral paid divided by the number of options received. This model-free approach enables real-time price discovery:

• There's not a one-size-fits-all theoretical price model that works at all times.

• Commonly used pricing models typically require discretionary inputs and real-time adjustments. The costs of these adjustments need not be imposed on all participants. Instead, the calculations can be done off-chain by individuals, before a transaction.

Spear and Shield tokens are priced between 0.01 and 0.99 collateral units. Their prices suggest market expectations about the direction of the underlying price.

Suppose you pay 40 USDC to long 100 Spear. You spend 40/100 = 0.4 USDC for a spear. You are estimating a 40% chance that the underlying will settle above the strike price.

For more theoretical and practical guides to digital options pricing, see the reference section.

What is Put-Call Parity?

A crucial underpinning of Divergence v1's design is the price parity of digital options:

$$Digital Call Price + Digital Put Price = 1$$

In other words, at maturity, the probability of a call being profitable and the probability of a put being profitable add up to 100%. One can expect the payoff for a long or short digital option to be:

A Long Spear is a Short Shield. | A Long Shield is a Short Spear.

• Long Digital Call = Short Digital Put

Suppose Spear/Shield trades at .40/.60.

• A trader who longs a Spear at .40 expects to receive a 1.0 payout, which includes the initial investment of .40 plus .60 profit if the underlying settles at or above the strike.

• A trader who shorts a Shield at .60 expects to keep the collected premium as profit under the same condition.

• Long Digital Put = Short Digital call

Suppose Spear/Shield trades at .40/.60.

• A trader who longs a Shield at .60 expects to receive a 1.0 payout, which includes the initial investment of .60 plus .40 profit if the underlying settles below the strike.

• A trader who shorts a Spear at .40 expects to keep the collected premium as profit under the same condition.

Put-Call Parity Explained

Why Digital Options?

Digital options can be used to replicate the payoff structure of any financial asset. They are useful as a hedge against sudden, large price movements in illiquid underlying assets.

Binary Calls Explained

Binary Puts Explained

Cash-or-Nothing, Asset-or-Nothing or ???

Spears and Shields can be considered as:

• Cash-or-Nothing when pool collateral is a stablecoin;

• Asset-or-nothing when the underlying asset is used as collateral;

In some cases, a pool may be funded with a collateral asset that is different from the underlying asset, and in this case, the payout is a correlated or uncorrelated asset.

Asset-or-Nothing & Cash-or-Nothing Calls Explained

Asset-or-Nothing & Cash-or-Nothing Puts Explained

Composability of Digital Options

Digital options are the basic components of many structured products. One can use a basket of digital options, or combine digital options with other financial products to compose defined risk strategies. They can be a versatile, powerful tool for hedging a DeFi portfolio.

Digital Options are also the building blocks of standard options. As noted in above video explainers, Asset-or-Nothing and Cash-or-Nothing options can replicate vanilla options:

• Standard European call = Asset-or-Nothing call - Cash-or-Nothing call where the cash-or-nothing call payoff equals the strike value

• Standard European put = Cash-or-Nothing put - Asset-or Nothing put

  where the cash-or-nothing put payoff equals the strike value

Therefore they also present opportunities to arbitrage or hedge against standard options available in DeFi and beyond.🤿