Last updated on 2022.5.13
This paper introduces a new type of derivative, Stablecoin Depeg Swap, adapted from everlasting options. It is for the holders (taking the long side) to hedge their risk of stablecoin depegging and the sellers (taking the short side) to earn the funding fees by taking that risk. In terms of derivative structure, it’s almost the same as a one-strike everlasting put option. However, the pricing of the SDS would be quite different. This paper introduces a CDS-style pricing model for SDS.
Let’s take an already-depegged stablecoin, UST, as example.
1 Stablecoin Depeg Swap (SDS) of UST is associated with 1 UST (i.e., notional = 1UST). The price of UST is $p$. The funding fee that 1 long pays 1 short per funding period (denoted as $T$, to be determined) is:
$$ Funding = Mark - [1-p]^+ $$
Here we use $[.]^+$ to represent the floor-at-0 function. The derivative is structured to hedge the risk of $p<1$ (depeg downward), while it is implicitly assumed that the case of $p>1$ (depeg upward) could be easily reverted and hence causes no problem.
From the structuring perspective, SDS is essentially an everlasting put option of the stablecoin price with the strike at 1.
From the long side’s perspective, this is how it works: when UST depegs from USD, $p$ moves below 1, $Mark$ would move up by around $(1-p)$. 1 long position of SDS entered before the depegging will have a profit of $(1-p)$, approximately compensating the loss of 1 UST.
If traded on an orderbook-based exchange, the definition above is all that is needed for the product specification. Mark would be determined by exchange trading. In that sense, providing SDS on an exchange is almost the same as providing everlasting put options, except different margin rules might be applied.
However, for SDS to function properly, the market-makers would need to be able to fairly price it. Also, if AMM is adopted as the trading venue, then a built-in pricing algorithm would be needed. Therefore, SDS would be useless if we don’t know how to price it. The next section will introduce a pricing model and explains why the pricing is quite different from that of the general options.
While the derivative structure of SDS is the same as an everlasting put option, the pricing would be quite different. This is because the stochastic process of a stablecoin price would be nothing like geometric Brownian motion or any of its derived forms. Specifically, a stablecoin’s price would stay at around 1 until it depegs. And when an actual depeg takes place, it is usually very hard to get back to pegging. Put simply, there are two states for a stablecoin: pegged and depegged. It stays in the pegged state until switching to the depegged status. This is largely similar to a bond, which fluctuates around the non-defaulting prices until it defaults. Therefore, it’s reasonable to price SDS similarly to CDS. We will explain such a pricing model in this section.
The theoretically fair price of Mark could be calculated with the framework of funding-fee-based perpetual derivatives. The pricing of such derivatives for discrete funding was explained in the original paper introducing Everlasting Options by Dave White et al. And the pricing of such derivatives for continuous funding was explained in these extending works of continuously funded everlasting options and power perps. Here we introduce the pricing of SDS for the continuous funding mode, which is much simpler to handle for continuous models of depegging events.
With the framework of pricing continuously-funded funding-fee-based derivatives, one such derivative is a portfolio of a continuous spectrum of the expiring derivatives with the same parameters. That is, a continuously-funded SDS with funding period $T$ is equivalent to a portfolio of "regular" SDS expiring at $t\in(0,\infty)$ with an exponentially decaying weight density:
$$ w(t)=\frac{1}{T}e^{-t/T} $$
Assuming the fair value of an SDS expiring at time $t$ is $P(t)$, the fair price of the perpetual SDS is：
$$ P=\int_0^\infty P(t)w(t)dt=\int_0^\infty \frac{P(t)}{T}e^{-t/T}dt $$
Now we only need to derive the fair price of $P(t)$.