I have given a mathematical analysis of the current lower peg of SBD here.
In that post I had just drawn the lowering of the peg from the risk of changing prices during the conversion by hand. Today I am adding this part from a probabilistic evaluation.
The SBD Lower Peg
SBD can be converted for 1$ of steem. But this only works as long as the marketcap of SBD (assuming a price of 1$) is below 10% of that of steem. Otherwise one steem will not be converted to 1$ but less. The formula for this is simple, and combines a flat curve with a linear drop for low prices of steem, seen by the blue line below.
But the kink in the peg produces an asymmetric risk for people doing the conversion. Let us assume that we are well below 10% debt ratio and the peg is flat. As soon as I do the conversion I know that I will have 1$ of steem at the end, no matter the price fluctuations. This is a working peg and I have a non-risky position.
Let us assume the debt ratio is way above 10%. Now we are in the linear part. When I start the peg, I know that rising prices in steem will give me more $ and falling prices less $. I have a risky position, but this risk is symmetric. If I do not expect consistently rising or falling prices, the random fluctuations will sometimes be beneficial and sometimes harmful. Again the peg is working.
But in-between close to the 10% debt ratio the risk is asymmetric. Let us assume that the debt ratio is just below 10%. Now when I convert 1 SBD and the price of steem rises during the 3.5 day conversion period, I still get 1$ (in steem). But when prices fall the peg is broken and I get less than 1$. Overall, integrating over all possible futures I no longer expect to get 1$, but a bit less. Therefore the peg is not efficient and the real price is a little lower.
I have computed the true peg using the following methods. I assume that the price of steem overaged over the next 3.5 days can be described by a normal distribution around its present day value. Using this model I find the variance using data from the last month. I then average over this distribution computing the expected $ generated by a conversion starting today. This is the real lower peg of SBD and whenever the price is below this value we can expect to make money buy buying steem and converting. Any price above and we expect conversions to burn money.
The present Situation
The lower axis is the price of steem and the upper one the minimum guaranteed value of SBD via conversions. The black line shows the current price. The red dashed line the restarting of printing SBD by the chain.
In blue we see the peg using the current data as a function of steem price (assuming no conversion risk). We see that we are well in the linear regime and a SBD is secured at 0.836$. In orange we have the real peg taking into account the asymmetric risk. At the current price-level both pegs are almost identical since it is very unlikely that the peg at 1$ is fully restored within 3.5 days. It is at 0.833$, only 0.003$ lower than the simplified peg.
We see that we need to go well above 0.5$ steem price in order to properly restore the 1$ peg.
The red-dashed line is showing the price from which the chain will restart printing SBD at full rate, changed in HF20! This means that we will print with a broken peg, taking into account the asymmetric risk of conversions. We cannot expect steem to be valued securely at 1$ even in the future when the markets recover. The old restart of printing before HF20 is shown in the green line, showing a strong peg at 1$ as it should be.
Changes introduced in HF20 are broken and should be reverted as soon as possible!
What To Do
The price of SBD currently at 0.822$ is reflecting the actual peg. Note that SBD does not have an upper peg or a mechanism to push the value down to the lower peg. So there is no easy way to benefit from an overvaluation of SBD. In principle the price of SBD can float above the peg for an indefinite time and we have seen that with 6$ SBD over several months.
But at the current time of rather pessimistic market actors there is really no reason why SBD should be above the peg, only misinformation and misevaluation of its price. I you have any SBD, please note that the peg is broken and you hold a risky asset. The current prices make is easy to get rid of it for a rather good amount of steem and currently conversions are also an option, though not very profitable.
Update: I made a small mistake in my computation and have updated the post to give correct information.