Stablecoins, Tokenization, and CBDCs

Instructor: Andreas Park

Case 2: price(1 SC) $<$ 1 FU $\to$ SC cheap

DISCUSSION

backing/collateral is the value of the LUNA/Terra Network
- $\to$ must have value(Terra) $\ge\sum$ SC
when you issue new LUNA tokens
- shift value from current to new owners
- create inflation $\to$ price(Luna) $\searrow$

POTENTIAL PROBLEMS

value(Terra) $<\sum$ SC
- could be pre-drop SC
when SC<FU, LUNA token holders
- know price(LUNA) will drop
- $\to$ pre-emptive selling
- more LUNA issued
- $\to$ more inflation
- $\to$ more price drops
- $\to$ DEATH SPIRAL!!!

So far there has never been an algorithmic stablecoin that did not collapse.
Disclaimer: I am not aware of a mathematical proof that an algo stableconin cannot work.

Case 2: price(1 SC) $<$ 1 FU $\to$ SC cheap

now let's do the maths carefully:
- price UST: $p_s$
- Assume $p_s<1$
- Arbitrage mechanism:
  1. Borrow $
  2. use $ to buy UST
  3. Sell UST for LUNA at $ price $p_l'$
  4. sell LUNA token for $$
  5. Repay $ loan, keep profit

assumptions
- arbitrageur trades max $q_s$ so that $p_s=1$
- $q_s$ UST buy for Luna with price $p_l'$ $\frac{q_s}{p_l'}~\text{LUNA}$
- redeemed UST destroyed, LUNA newly minted
  - $\to$ dilution.
  - $\to$ assume MKT anticipated extra quantity of LUNA $q_s/ p_l'$
assume: TERRA network value unaffected
except: validation costly $\to$ minimum viable price $\underline{p}_l$
network collapses if $p_l<\underline{p}_l$

Case 2: price(1 SC) $<$ 1 FU $\to$ SC cheap

define $Q_d=Q_sp_s$ aggregate UST $-value
arbitrageur buys $q_s$ such that $1=\frac{Q_d+P(q_s)}{Q_s-q_s}$
assume AMM pricing (for simplicity): $q_s=Q_s(1-\sqrt{p_s})$

exchanges these $q_s$ UST for LUNA at dollar-price for LUNA
network value not affected:
$Q_lp_l=(Q_l+q_l)p_l'$
solve this to get $p_l'=p_l-\frac{q_s}{Q_l}.$

Case 2: price(1 SC) $<$ 1 FU $\to$ SC cheap

$p_l'$ cannot drop below $\underline{p}_l$
Maximum feasible quantity forUST: $\overline{q}_s=Q_l(p_l-\underline{p}_l)$
$\overline{q}_s=$ collateral value of aggregate LUNA tokens
- $Q_s\le\overline{q}_s$ : UST is fully collateralized
- $Q_s>\overline{q}_s$ UST is under- collateralized & $\exists$ price deviation that would cause a collapse of LUNA and UST.

user perspective

fractional collateral $\to$ collateralization factor $=$ 150%

\begin{array}{rcl} &&\textsf{maximal amount of DAI in \$}\\\\ &=&\frac{\textsf{\$ equivalent of ETH in escrow}}{\textsf{collateralization factor}}\\\\ &=&\frac{1,500}{150\%}=\$1,000 \end{array}

\begin{array}{rcl} &&\textsf{maximal amount of DAI in \$}\\\\ &=&\frac{\textsf{\$ equivalent of ETH in escrow}}{\textsf{collateralization factor}}\\\\ &=&\frac{1,500}{150\%}=\$1,000 \end{array}

total collateral = $1,500

maximum loan = $1000

overcollateralization = $500

actual loan (example) = $500

buffer = $500