There are some problems with vanilla options on Ethereum and more broadly for some market participants. The markets for options are stratified across two dimensions, strike and expiry. Markets must constantly be opened up and closed based on price movements and time movements. This fragments liquidity and makes it costly for market makers to update their offers. These problems also exist in traditional markets but to a lesser degree due to the fact that there are no gas costs.
Many people trade options to get a convex payoff function with respect to the underlying asset. This means that for a small upfront investment the investor is able to get a large gain if the price increases(decreases) for a call(put). They also have limited liability. This represents the single biggest difference between leveraged products and convex products. When you trade with leverage it is entirely possible that the asset reaches your price target in your specified timeframe but a bad candle hits your stop loss and you are forced out of your position or worse yet, liquidated. This does not happen when a trader buys a convex product like an option.
There is a better way to get convexity than options that is entirely unexplored. Within DeFi an entire system of synthetic assets was born. These are projects such as Maker and Synthetix. Generically speaking through various mechanisms these platforms provide a token whose price tracks some value provided by an oracle. It is important to understand that the value provided by the oracle (now called an index) does not need to be the price of something.
There is a very intriguing family of derivatives that can be created based on this fact. Consider the family of nonlinear functions f(x) = C x^N. When this function is applied to the price of something you can end up with something that looks like N times leveraged exposure.
The formula for the linear PnL of the derivative as a function of the linear PnL of the asset over a given time period is given below.
PnL Derivative = (PnL Asset + 1)^N - 1
What is remarkable about this is that this formula applies across time and price. This creates fungible convex exposure. The convexity parameter N may be as high as is practical considering collateral requirements. Here is a simple table of PnL calculations for example if the current price of the asset is 32.
price x x^-10 x^-4 x^-2 x^-1 x^2 x^4 x^10
25 -22% 1081% 168% 64% 28% -39% -63% -92%
26 -19% 698% 129% 51% 23% -34% -56% -87%
27 -16% 447% 97% 40% 19% -29% -49% -82%
28 -13% 280% 71% 31% 14% -23% -41% -74%
29 -9% 168% 48% 22% 10% -18% -33% -63%
30 -6% 91% 29% 14% 7% -12% -23% -48%
31 -3% 37% 14% 7% 3% -6% -12% -27%
32 0% 0% 0% 0% 0% 0% 0% 0%
33 3% -26% -12% -6% -3% 6% 13% 36%
34 6% -45% -22% -11% -6% 13% 27% 83%
35 9% -59% -30% -16% -9% 20% 43% 145%
36 13% -69% -38% -21% -11% 27% 60% 225%
37 16% -77% -44% -25% -14% 34% 79% 327%
38 19% -82% -50% -29% -16% 41% 99% 458%
39 22% -86% -55% -33% -18% 49% 121% 623%
These synthetic assets can provide long and short exposure with predictable time and price invariant PnL characteristics, high degrees of convexity and limited liability to the purchaser. These derivatives are relatively easy for retail traders to understand also especially when compared to options. For practical purposes the constant C can be chosen such that when trading launches on the index the initial index price is something convenient like 1000.
This asset fits in fairly easily to an existing options market making operation. The biggest problem here is that there is no funding mechanism built in. To address that problem, the index must be adjusted continuously to the benefit of the minter. The value of the synthetic asset decays in a continuous and simple way such that the minter gains value via the index.
The previously proposed index was f(x) = C x^N. The problem with using this as the payoff is that introducing We need to include a discount term on the index such that the market maker has the incentive to hedge. The new index is
f(x) = C x^N exp(\rho t)
\rho = -1/2 \sigma^2 N (N-1) - r (N-1) + q N
With the conventional definitions of \sigma, r and q from the Black Scholes framework.
\sigmais the volatility of the assetris the risk free rateqis the continuously compounded rate of dividends for the asset
Consider a table of values for typical vol rates in crypto.
σ / N -10 -4 -2 -1 1 2 4 10
50% 2750% 500% 150% 50% 0% 50% 300% 2250%
65% 3575% 650% 195% 65% 0% 65% 390% 2925%
80% 4400% 800% 240% 80% 0% 80% 480% 3600%
95% 5225% 950% 285% 95% 0% 95% 570% 4275%
110% 6050% 1100% 330% 110% 0% 110% 660% 4950%
125% 6875% 1250% 375% 125% 0% 125% 750% 5625%
This illustrates a current problem with the system. The index rapidly approaches zero for higher leverages. It may be preferable to have these payments not be done through the index and passed as a yield from the owner of the asset to the minter.
In dollar terms, if the minter creates $10MM of the N=2 token, sells it, buys back after 6 months and redeems and have the volatility at 80% the amount paid in hedging fees to the minter is $3.3MM.
The problem with introducing an asset with a price that tracks f(x) = C x^N is that you introduce arbitrage into the market. To eliminate the arbitrage, we must adjust the drift such that f(S_t) = E [ f(S_T) ] . Using black scholes gives us a starting point for the analysis in closed form. The value of \rho defined above is the value that admits no arbitrage.
Valuing this derivative in the Black Scholes framework and providing some greeks.
value = C S^N
delta = dv/dS = C N S^{N-1}
gamma = d^2 v / d S^2 = C N(N-1) S^{N-2}
vega = dv/d\sigma = 0 Note the very interesting characteristic that the vega is 0. This means that the value of the option today is entirely independent of volatility. This product is pure convex long or short exposure to the underlying asset.
Market makers have been dominated by large firms until the advent of DeFi. DeFi allows anyone to provide liquidity and make the gains that the large market makers would have made. That is what the power claim protocol does as well.
- A pool of liquidity denominated in the base token (say ETH) is formed.
- LPs deposit ETH and get LP tokens
- At any time people that want to buy a power claim can buy one from the protocol at face value
CS^N(subject to liquidity) they receive a position NFT - When that NFT comes back, the owner of the NFT gets
CS^N exp(-\rho t)for it from the pool. - At any time, subject to liquidity, the LP may withdraw and receive their proportion of assets in the pool.
- The solvency of the protocol determines available liquidity, the protocol is over-collateralized.
Unfortunately making these leveraged tokens fungible is probably not something that is realistic. The index price will continuously decrease and eventually run out of precision within just a few months to years. Thus, the protocol creates non-fungible leverage tokens that track their fees separately on their own index. Thankfully the LP tokens are liquid.
The protocol uses a bonding curve for \rho over \delta such that the protocol attempts to be delta neutral. There are two different bonding curves, one for N > 1 and one for N <= 1 we will call them long fee and short fee. They are monotone such that the long fee becomes increasingly high as delta grows etc. See the table below
+delta -delta
long high low
short low high