In the previous post on PYT pricing we laid out a model for determining the intrinsic value of PYTs and NYTs, but we did not offer any justifications as to why the market price will follow this model. This post will focus on that.

First of all, there are no risk-free arbitrages available when the price of PYT deviates from the pricing model, because there doesn’t exist a single “correct” price to *deviate from*. The pricing model we offered is parametrized by both the discount factor $\beta$ and the value of the expected future cash flow $Y_\beta(x) = \int_0^\infty \beta^t y(x, t) dt$, both of which depend on the subjective beliefs of the person applying the pricing model, so there isn’t one universally correct “peg” that the price of PYT can be compared against. We could, of course, take the beliefs of a single market participant, which would provide a single pricing model to compare against. Under such a model, one could perform “arbitrages” when the market price deviates from the pricing model by trading in the direction opposite to the deviation; this is, however, not risk-free, due to the simple and obvious fact that nobody knows what the future cash flow will be, the best we can do is make a prediction, and if the actual cash flow value turns out to be different from our predictions (which is usually the case since we’re not mystics who can see the future) then the “arbitrage” can incur a loss, making it less of an arbitrage and more of an informed speculation.

This makes PYTs more similar to options, where each market participants values an option based on their beliefs about future price action and prices it based on its intrinsic value, instead of stablecoins where the price is pegged to a specific value via risk-free arbitrages. Therefore, it doesn’t make sense to say that PYT’s market price is above or below what it *should be* based on the future cash flow, because the market price of PYT *is* what the price *should be* and reflects the average market participant’s beliefs about the future cash flow. If the market price of PYT deviates from what the pricing model predicts, then it simply means you’re using parameters different from the market average; this is just like if you plugged some numbers into Black-Scholes and got a premium different from the current market value, the market isn’t wrong, Black-Scholes isn’t wrong, you’re just using different numbers (volatility in the case of Black-Scholes) than the rest of the market. Of course, market participants can and will use alternative pricing models different from what was described in the previous post (for instance using more advanced discounting methods), but nonetheless the same principle applies: **each market participant determines the fair value of PYTs using a pricing model, based on their individual beliefs, and the market price reflects the average pricing model and beliefs of all market participants**. Now you can see why the question “how will you ensure the market price of PYT follows the pricing model?” makes no sense: it’s putting the cart before the horse.

Of course, there are still things we can say about PYT’s price under certain extreme scenarios regardless of the pricing model. For instance, when the yield rate is known to be zero, PYT and NYT are functionally equivalent, so the market should have the same demand for them, and because their supplies are the same as well, they should have equal prices. This would no longer be true, however, if we broke one of the assumptions: if we burnt some PYT by sending it to an address with unknown private key, such that it will never be accessible by the market, the supply of PYT would become less than that of NYT, meaning PYT should be priced higher.

Instead of risk-free arbitrages, we will use use a more relaxed argument to determine the range that the price is likely to be in. We will try to answer two questions:

- What is the PYT price at or below which a rational actor should always buy PYT?
- What is the PYT price at or above which a rational actor should never buy PYT?

If we can answer these two questions, then we can have a rough bound the price of PYT. This won’t provide an accurate pricing, obviously, but it demonstrates some of the market mechanics behind how PYT changes in price.

## 1. When should you always buy PYT?

Before we go into the analysis, I will make the assumption that the price of PYT should always be above the price of NYT. This makes sense because the only difference between PYT and NYT is that PYT gives its holder cash flow from the generated yield whereas NYT gives its holder nothing, so PYT should be priced higher than NYT. This is equivalent to saying that the price of PYT should always be above 0.5 of the underlying asset.

With that out of the way, we will first make an observation about PYTs: the downside risk of holding PYT is bounded. No matter what your entry price is, in the worst case scenario the price of PYT goes to 0.5. This means that as a PYT holder earns yield, their downside risk decreases over time and eventually becomes zero, after which the holder will always be in profit regardless of what the price of PYT becomes. Furthermore, after the downside risk becomes zero, the PYT holder would be earning yield at a leverage in perpetuity, which is obviously a desirable position to be in.

Therefore, whenever the time-to-zero-risk value (computed based on your belief about future cash flow) is low enough, a rational actor would always buy PYT. To use an extreme example to illustrate this point, say that the price of PYT is 0.55 and the yield rate of the yield-generating vault is 1%/day. You would need around 5.5 days to reach zero-risk, and once you do you would be earning $\frac{1}{0.55} \approx 1.82$ leveraged yield in perpetuity with no downside risk, which is clearly attractive. Not to mention, you would be exposed to the upward movement of PYT’s price, so if you expect the price of PYT will go up you’d have even more reasons to buy PYT. As people buy PYT and push up the price, the time-to-zero-risk value increases, and the potential yield leverage decreases, making it less attractive to buy PYT, until some equilibrium is reached.

## 2. When should you never buy PYT?

Let’s look at an extreme example where the price of PYT is 1. Would a rational person ever buy PYT at this price? The answer is no. To see why, we will compare buying PYT with the alternative of depositing your funds directly into the yield-bearing vault used by the PYT. In both cases, you don’t get any leverage on the yield you earn, but in the case of buying PYT you’re exposed to the downside risk of PYT, namely you could lose money if the price of PYT ever decreases. This makes buying PYT strictly worse than depositing into the vault directly, meaning no rational person would ever buy PYT at the price of 1. A corollary is that you should always buy NYT at the price of 0, since it’s free tokens and there’s nowhere for the price to go but up.

As the price of PYT goes down from 1, the yield leverage increases and the downside risk decreases, making it more attractive to buy PYT, until some equilibrium is reached where taking the downside risk is justified by the yield leverage.