---
tags: dAMM
---

# Option based market making


A formal equivalence exists between posting sell/buy orders and selling calls/puts. This equivalence gets practical application once market making is optimized over distant price intervals.

## Equivalence between a short call and an ask

Consider a market maker operating in the ETH/USDC market. Posting a **limit sell order** at price $S$ (in USDC) of 1 ETH until date $T$ while earning the return rate $r$ during the period gives, with $w$ terminal wealth:

- $p<S$: $w = p+r$
- $p\geq S$: $w = S+r$

Selling **a (European) call** of 1 ETH at strike price $S$ for a premium $\pi$ at expiry date $T$, covered by 1 ETH gives:

- $p<S$: $w = p+\pi$
- $p\geq S$: $w = p-(p-S)+\pi =S+\pi$

A covered call is equivalent to a sell order if the the former strategy follows the additional step:

- at expiry date $T$, if the option is exercized ($p\geq S$), convert the remaining ETH in USDC

$S/p$ ETH are converted so that the strategy ends up with $S$ USDC (plus the option premium) and is immune from ETH price fluctuation after expiry date.


In the figure, the MM has $S$ USDC at date $T$. At the same date $T$, the MM selling a call ends up with $S/p < 1$ ETH, that he immediately sells for $S$ USDC.

## Equivalence between a short put and a bid

Posting a **limit order buy** at price $S$ of 1 ETH until date $T$ gives:

- $p>S$: $w = S+r$
- $p\leq S$: $w = p+r$

At price $S$, the strategy exchanges $S$ USDC for 1 ETH.

Selling **a put** of 1 ETH at strike price $S$ for a premium $\pi$ at expiry date $T$, covered by $S$ USDC gives:

- $p>S$: $w = S+\pi$
- $p\leq S$: $w = S-(S-p)+\pi = p+\pi$

This is equivalent to a limit order buy if the the put strategy follows the additional step:

- at expiry date $T$, if the option is exercized ($p\leq S$), convert the remaining $p$ USDC in ETH

so that the strategy has eventually 1 ETH and is immune to subsequent ETH price fluctuation after expiry date. Note also that the $S$ USDC provides enough collateral even in the worst case scenario $p=0$.


In the figure, the MM bought 1 ETH at price $S$. As the price continues to decrease, his ETH is worth $p$ USDC at date $T$. At the same date, the MM selling a put wouyld have $S$ USDC, that he immediately converts to ETH.

## Combining puts and calls


## Option-based vs lending-based market making

The first strategy deposits either 1 ETH in case of a call or $S$ USDC in case of a put and earns the option premium. The second strategy deposits the same amount in a lending protocol, either in the ETH pool or in the USDC pool to earn the lending interest rate.

The main benefit of the option strategy over the lending strategy is the yield associated with writing call and put options. As an illustration, those are the APR taken from Dopex (on Arbitrum) on 30 April 2022 for one month covered calls (ETH current price $2800):

| Strike | 3,200 | 3,600 | 4,000 | 4,500 |
| :------ | -----: | -----: | -----: | -----:|
| APR | 115% | 42% | 16% | 7% |

Aside from the yield, the two strategies dot not bring the same result when the price goes back and forth until option expiry.

Consider for intance a market making strategy consisting in selling at $3000 and buying at $2500. If the price goes from $2800 to $3100, then back to $2400 and eventually returns to $2800, the lending strategy sells at $3000 and buy back at $2500 whereas the option strategy only pockets the premium from the call strategy at $3000 and the put strategy at $2500.

For every period, the option strategy can buy or sell at different strike levels but cannot both buy and sell, whereas the lending strategy can do both an arbitrary number of times. However, the number of executed bids and asks during a given period will be limited the shorter the option period (a week or a month) and the broader the price intervals (e.g. $2500, $3000, ...).