Learning from DeFi: Are AMMs better?

Preliminaries & Some Motivation

Liquidity Supply and Demand in an Automated Market Maker

Returns to liquidity providers

Similar to Lehar and Parlour (2023), Barbon & Ranaldo (2022).

(incremental) adverse selection loss when the return is $R$

fees earned

on informed

fees earned

on balanced flow

for reference:

If the asset price drops by 10% the incremental loss for liquidity providers is 13 basis points on their deposit
- $\to$ total loss=-10.13%
If the asset price rises by 10%, the liquidity provider gains 12 basis points less on the deposit
- $\to$ total gain =9.88%

For fixed balanced volume $V$ & fee $F$ :

Larger pool size $\to$ smaller shares of the fees
$\to$ LP expected return $\searrow$ in pool size
Competitive liquidty provision:
- $\to$ find the upper-bound on pool size above which LPs lose money
- we characterize this by $\bar{\alpha}$ - fraction of the asset's market cap to be deposited to the pool

Liquidity Demander's Decision & (optimal) AMM Fees

Better off with AMM relative to traditional market if
$\text{AMM price impact} +\text{AMM fee} \le \text{bid-ask spread}.$

Two opposing forces when fee $F\nearrow$
1. more liquidity provision
  $\to$ lower price impact
2. more fees to pay

Result: there exists an optimal (min trading costs) fee $>0$

Similar to Hasbrouck, Riviera, Saleh (2023)

Background on Data

some volume may be intermediated
with AMMs: no need for intermediation
$\to$ intermediated volume could disappear
$\to$ use volume/2
Some caveats, e.g.
- arbitrageur volumes
- larger volume if AMM has lower trading costs

CRSP for shares outstanding
WRDS-computed statistics for
- quoted spreads (results similar for effective)
- volume
- open-to-close returns
- average trade sizes, VWAP
Time horizon: 2014 - March 2022
Exclude "tick pilot" period (Oct 2016-Oct 2018)
All common stocks (not ETFs) (~7550).
Explicitly not cutting by price or size
All "boundless" numbers are winsorized at 99%.

Approach 1: "ad hoc"
- "one-day-back" look
- take yesterday's return and volume when deciding on liquidity provision in AMM
Approach 2: estimate historical return distribution

Deposit Requirements

Our approach: measure liquidity provision in % of market cap
Share-based liquidity provision is not a problem: the shares are just sitting at brokerages.
But: AMM requires an off-setting cash amount: $c =a\cdot p(0)$ .
Cash is not free:
- at 6% annual rate, must pay 2bps per day.
- Would need to add to fees
But: do we need "all that cash"?
No.

(hand-waving argument)
2nd gen AMMs have liquidity provision "bands": specify price range for which one supplies liquidity
Here: specify range for $R\in(\underline{R},\overline{R})$
- Outside range: don't trade.
- Inside range: "full" liquidity with constant product formula.
Implication: only need cash and shares to satisfy
in-range liquidity demand.

$\Rightarrow$ Need about 5% of the value of the shares deposited -- not 100% -- to cover up to a 10% return decline

Deposit Requirements

For return $R$ , the following number of shares change hands: $q=a\cdot(1-\sqrt{R^{-1}}).$
Fraction of share deposit used $\frac{q}{a}=1-\sqrt{R^{-1}}.$
Fraction of cash used $\frac{\Delta c ("R")}{c}=\frac{1-\sqrt{R^{-1}}}{\sqrt{R^{-1}}}.$
Example for $R=.9$ (max allowed price drop $=10\%$ ) $\frac{\Delta c ("R")}{c}=-5\%.$
$\Rightarrow$ "real" cash requirements $\not=$ deposits

$\Rightarrow$ Need about 5% of the value of the shares deposited -- not 100% -- to cover up to a 10% return decline

Learning from DeFi: Would Automated
Market Makers Improve Equity Trading?