Convertible Liquidity

Liquidity providers sell calls and puts within custom price ranges. They can either naked short calls or puts, or sell previously bought calls or puts.

How Liquidity Provision Works on Divergence v1

In a Divergence v1 pool, liquidity providers can concentrate their liquidity within price intervals where they anticipate buying interest. An LP can have many individualized liquidity positions per pool. For each liquidity position, a non-fungible token is minted.

The price curve of Divergence v1 is segmented with ticks, using concentrated liquidity functions of Uniswap v3. The key difference is that v1 pools track the relative price between calls and puts. Calls or puts have a price range of [0.01, 0.99], and their relative price is between [0.01/0.99, 0.99/0.01]. Within this price range, LPs can choose lower and upper price bounds when initiating a liquidity position. Both calls and puts can be sold from a position.

Suppose the current call price is 0.28. Liquidity provider Alice selects a price range of [0.30, 0.40] to short calls. This implies that her position has a put price range of [1-0.40, 1-0.30], or [0.60, 0.70]. The position boundaries are initiated at ticks near a put/call price range of [0.60/0.40, 0.70/0.30]. After the call price rises to 0.40, Alice's position is fully crossed. The current put price is 1-0.40 = 0.60. If the next trader buys puts, the price will cross back Alice's position.

LPs earn premiums and transaction fees from selling options. Liquidity is active when the price moves within a range. Premiums are recorded and allocated proportionally to active liquidity, just like transaction fees. Once the price exits a liquidity range, liquidity is deactivated and no longer earns premiums and fees. When removing liquidity, LPs finalize their short interest and discontinue option market making. They can collect premiums and fees by either squaring off their open shorts, or settling positions at expiry. Using just one asset as seed collateral, a liquidity position avoids impermanent loss due to changing asset ratios.

Impermanent Optionality

A liquidity position can have unlimited gross shorts in both calls and puts within its price range. For the same price interval, a short call and a short put settle each other, regardless of the outcome. This enables auto-risk reduction for a liquidity position. An LP has seller obligations only for net shorts in either calls or puts. This short exposure is not permanent until liquidity is removed.

Chad provides liquidity for a single call option. Alice buys a call, fully crossing his position. Then Bob buys a put, undoing Chad's short call exposure. Alice buys another call and Bob buys another put. They trade back and forth until Alice ends up with 1 million calls and Bob has 1 million puts. If the underlying price settles below the strike, Alice’s premium ends up with Bob, who receives 1 million collateral. Otherwise, Bob’s premium goes to Alice, who gets 1 million collateral. Chad has no options exposure. He collects fees for all the trades within his liquidity range.

The chosen price bounds enable a position to either sell more calls than puts or sell more puts than calls. When the price exits the range, a position stops selling options, capping the amount of net shorts it accumulates. After offsetting call and put sales, a liquidity position accumulates a predefined amount of net shorts in calls or puts. This ensures that a position’s seed liquidity can pay off the accumulated net shorts, in case they expire profitably. Therefore the maximum loss LPs can expect is restricted to the seed liquidity amounts.

Suppose Alice buys 1 million calls, and Bob gets 999,999 puts in the above scenario. LP Chad's short exposure maxes out at 1,000,000 - 999,999 = 1 call.

Liquidity Types

A liquidity position can be minted using either collateral or prior-purchased options. LPs can passively convert different types of liquidity to take on or off options risk. It is important to note that a position's options exposure is not final until its liquidity is removed.

• Liquidity Positions Minted with Collateral

• Similar to limit-sell orders to naked short calls or puts.

• An alternative to market buying digital calls or puts, according to the put-call parity

  • If one is bullish, shorting a put is an alternative to buying a call.

  • If one is bearish, a short call works similarly as a long put.

• If the price crosses back a position, the LP's open shorts in calls or puts are neutralized.

• Liquidity positions Minted with Either Call or Put Options

• Similar to limit-close orders for existing longs.

• When a position is seeded with calls (puts), its net shorts in calls (puts) neutralize the long.

• Act like one seeded with collateral when the price crosses back. The premium proceeds are effectively used as liquidity to sell more options, which reverses the close order.

Seed Deposit

When a position is minted using collateral, the required seed deposit is the expected gain for the largest net shorts in calls or puts it can have. For a liquidity position intended to sell 1 call (or 1 put) at the price of N collateral (0.01≤ N ≤ 0.99), the required collateral deposit is 1 - N.

Suppose Chad wants to sell 1 call at the effective price of 0.7 DAI. The required collateral deposit is 1 - 0.7 = 0.3 DAI. For the same liquidity range, 1 put is priced at 0.3 DAI. If he wishes to sell a put, 1-0.3 = 0.7 DAI of seed liquidity is required.

When seeding prior-purchased options to close longs, calls or puts must be provided to a liquidity range above the current call (or put) price.

Reserved for Settlement

Before expiry, liquidity can be withdrawn to close short exposure, while collaterals must be reserved in the pool as an obligation to cover profitable options upon expiration. These reserved collaterals correspond to the larger short interest, whether on calls or puts. Once collateral is reserved for settlement, any unused liquidity, including both collateral and options, is returned.

$$C_{obligation} = Max (Short Interest on Call, Short Interest on Put)$$

Where the open shorts in options are zero unless the sold amounts are larger than seeded

• Short Interest on Call = calls sold - calls seeded

• Short Interest on Put = puts sold - puts seeded

Examples:

A liquidity position is minted using 10 calls. It sold 6 calls and 3 puts. The position has 0 short interest on call, as the calls sold are less than the calls seeded. However, it has 3 short interest on put. So, the pool reserves 3 collateral and returns 10-6=4 calls, when liquidity is removed. After settlement, if puts expire worthless, 3 collateral can be withdrawn by the LP. The returned 4 calls can be exercised for 4 collateral in payoff. If the puts expire in the money, the reserved 3 collateral goes to the put holders.

A liquidity position is minted using 10 collateral. It sold 5 calls and 3 puts.

The position has 5 short interest on call and 3 short interest on put. So, the pool reserves 5 collateral (which would include the collected option premiums) upon liquidity removal. If the calls expire worthless, the LP can withdraw 2 collateral from the expired net shorts. Otherwise, the full reserved amount is paid to the call holders.