Fix wrong math regarding confidence interval of laplace distribution (#89)

Co-authored-by: Christian Koopmann <[email protected]>

Author: chrisCryptoCom

pythnet-price-feeds/best-practices.md

@@ -71,11 +71,11 @@ This analogy suggests two simple solutions to races:
 
 At every point in time, Pyth publishes both a price and a confidence interval for each product. For example, Pyth may publish the current price of bitcoin as $50000 ± $10. Pyth publishes a confidence interval because, in real markets, there is _no one single price for a product_. For example, at any given time, bitcoin trades at different prices at different venues around the world. While these prices are typically similar, they can diverge for a number of reasons, such as when a cryptocurrency exchange blocks withdrawals on an asset. If this happens, prices diverge because arbitrageurs can no longer bring prices across exchanges into line. Alternatively, prices on different venues can differ simply because an asset is highly volatile at a particular point in time. At such times, bid/ask spreads tend to be wider, and trades on different markets at around the same time tend to occur at a wider range of prices.
 
-Pyth represents these possibly-different prices by giving its users a _probability distribution over price_ instead of just a single price. Pyth models the price according to a Laplace distribution centered on the Pyth aggregate price with a standard deviation equal to the confidence interval (the scale parameter b of the Laplace distribution is equal to the square root of 2 times the standard deviation). The Laplace distribution contains ~95% of the probability mass within ~4.24 standard deviations (~3 times the scale parameter). If markets are behaving normally, then the confidence interval will be tight -- typically much less than 1% of the price -- and the Laplace distribution will be highly peaked. However, at unusual times, the confidence interval can widen out dramatically.
+Pyth represents these possibly-different prices by giving its users a _probability distribution over price_ instead of just a single price. Pyth models the price according to a Laplace distribution centered on the Pyth aggregate price with a standard deviation equal to the confidence interval (the scale parameter b of the Laplace distribution is equal to the standard deviation divided by the square root of 2). The Laplace distribution contains ~95% of the probability mass within ~2.12 standard deviations (~3 times the scale parameter). If markets are behaving normally, then the confidence interval will be tight -- typically much less than 1% of the price -- and the Laplace distribution will be highly peaked. However, at unusual times, the confidence interval can widen out dramatically.
 
-When consuming Pyth prices, we recommend using the confidence interval to protect your users from these unusual market conditions. The simplest way to do so is to use Pyth's confidence interval to compute a _range_ in which the true price (probably) lies. You obtain this range by adding and subtracting a multiple of the confidence interval to the Pyth price; the bigger the multiple, the more likely the price lies within that range. We recommend considering a multiple of 4.24, which as mentioned above gives you a 95% probability that the true price is within the range (assuming Laplace distribution estimates are correct). Then, _select the most conservative price within that range for every action._ In other words, your protocol should minimize state changes during times of large price uncertainty.
+When consuming Pyth prices, we recommend using the confidence interval to protect your users from these unusual market conditions. The simplest way to do so is to use Pyth's confidence interval to compute a _range_ in which the true price (probably) lies. You obtain this range by adding and subtracting a multiple of the confidence interval to the Pyth price; the bigger the multiple, the more likely the price lies within that range. We recommend considering a multiple of 2.12, which as mentioned above gives you a 95% probability that the true price is within the range (assuming Laplace distribution estimates are correct). Then, _select the most conservative price within that range for every action._ In other words, your protocol should minimize state changes during times of large price uncertainty.
 
-This principle is common sense. Imagine that you are lending money to a friend, and your friend pledges a bitcoin as collateral. Also imagine that Pyth says the bitcoin price is $50000 +- $1000. (Note that $1000 is an unusually large confidence interval for bitcoin; the confidence interval is typically \~$50 dollars). You therefore calculate that the true price is between $45760 and $54240 using the multiply by 3 rule from above. When originating the loan, you would value the bitcoin at $45760. The lower price is conservative in this instance because it limits the amount of borrowing that is possible while the price is uncertain. On the other hand, once the loan has been issued, you would value the bitcoin at $54240. The higher price is conservative, as it prevents you from liquidating your friend purely due to increased price uncertainty.
+This principle is common sense. Imagine that you are lending money to a friend, and your friend pledges a bitcoin as collateral. Also imagine that Pyth says the bitcoin price is $50000 +- $1000. (Note that $1000 is an unusually large confidence interval for bitcoin; the confidence interval is typically \~$50 dollars). You therefore calculate that the true price is between $47880 and $52120 using the multiply by 3 rule from above. When originating the loan, you would value the bitcoin at $47880. The lower price is conservative in this instance because it limits the amount of borrowing that is possible while the price is uncertain. On the other hand, once the loan has been issued, you would value the bitcoin at $52120. The higher price is conservative, as it prevents you from liquidating your friend purely due to increased price uncertainty.
 
 The same principle would apply if you wrote a derivative contract. If someone wants to open a derivative contract with you, you would value their collateral at the lower price. However, if you were deciding whether someone's margin limits were violated, you would value their collateral at the higher price. If a contract needs to be settled at a price, you could take approaches such as the following:
 

solana-price-feeds/best-practices.md

@@ -71,11 +71,11 @@ This analogy suggests two simple solutions to races:
 
 At every point in time, Pyth publishes both a price and a confidence interval for each product. For example, Pyth may publish the current price of bitcoin as $50000 ± $10. Pyth publishes a confidence interval because, in real markets, there is _no one single price for a product_. For example, at any given time, bitcoin trades at different prices at different venues around the world. While these prices are typically similar, they can diverge for a number of reasons, such as when a cryptocurrency exchange blocks withdrawals on an asset. If this happens, prices diverge because arbitrageurs can no longer bring prices across exchanges into line. Alternatively, prices on different venues can differ simply because an asset is highly volatile at a particular point in time. At such times, bid/ask spreads tend to be wider, and trades on different markets at around the same time tend to occur at a wider range of prices.
 
-Pyth represents these possibly-different prices by giving its users a _probability distribution over price_ instead of just a single price. Pyth models the price according to a Laplace distribution centered on the Pyth aggregate price with a standard deviation equal to the confidence interval (the scale parameter b of the Laplace distribution is equal to the square root of 2 times the standard deviation). The Laplace distribution contains ~95% of the probability mass within ~4.24 standard deviations (~3 times the scale parameter). If markets are behaving normally, then the confidence interval will be tight -- typically much less than 1% of the price -- and the Laplace distribution will be highly peaked. However, at unusual times, the confidence interval can widen out dramatically.
+Pyth represents these possibly-different prices by giving its users a _probability distribution over price_ instead of just a single price. Pyth models the price according to a Laplace distribution centered on the Pyth aggregate price with a standard deviation equal to the confidence interval (the scale parameter b of the Laplace distribution is equal to the standard deviation divided by the square root of 2). The Laplace distribution contains ~95% of the probability mass within ~2.12 standard deviations (~3 times the scale parameter). If markets are behaving normally, then the confidence interval will be tight -- typically much less than 1% of the price -- and the Laplace distribution will be highly peaked. However, at unusual times, the confidence interval can widen out dramatically.
 
-When consuming Pyth prices, we recommend using the confidence interval to protect your users from these unusual market conditions. The simplest way to do so is to use Pyth's confidence interval to compute a _range_ in which the true price (probably) lies. You obtain this range by adding and subtracting a multiple of the confidence interval to the Pyth price; the bigger the multiple, the more likely the price lies within that range. We recommend considering a multiple of 4.24, which as mentioned above gives you a 95% probability that the true price is within the range (assuming Laplace distribution estimates are correct). Then, _select the most conservative price within that range for every action._ In other words, your protocol should minimize state changes during times of large price uncertainty.
+When consuming Pyth prices, we recommend using the confidence interval to protect your users from these unusual market conditions. The simplest way to do so is to use Pyth's confidence interval to compute a _range_ in which the true price (probably) lies. You obtain this range by adding and subtracting a multiple of the confidence interval to the Pyth price; the bigger the multiple, the more likely the price lies within that range. We recommend considering a multiple of 2.12, which as mentioned above gives you a 95% probability that the true price is within the range (assuming Laplace distribution estimates are correct). Then, _select the most conservative price within that range for every action._ In other words, your protocol should minimize state changes during times of large price uncertainty.
 
-This principle is common sense. Imagine that you are lending money to a friend, and your friend pledges a bitcoin as collateral. Also imagine that Pyth says the bitcoin price is $50000 +- $1000. (Note that $1000 is an unusually large confidence interval for bitcoin; the confidence interval is typically \~$50 dollars). You therefore calculate that the true price is between $45760 and $54240 using the multiply by 3 rule from above. When originating the loan, you would value the bitcoin at $45760. The lower price is conservative in this instance because it limits the amount of borrowing that is possible while the price is uncertain. On the other hand, once the loan has been issued, you would value the bitcoin at $54240. The higher price is conservative, as it prevents you from liquidating your friend purely due to increased price uncertainty.
+This principle is common sense. Imagine that you are lending money to a friend, and your friend pledges a bitcoin as collateral. Also imagine that Pyth says the bitcoin price is $50000 +- $1000. (Note that $1000 is an unusually large confidence interval for bitcoin; the confidence interval is typically \~$50 dollars). You therefore calculate that the true price is between $47880 and $52120 using the multiply by 3 rule from above. When originating the loan, you would value the bitcoin at $47880. The lower price is conservative in this instance because it limits the amount of borrowing that is possible while the price is uncertain. On the other hand, once the loan has been issued, you would value the bitcoin at $52120. The higher price is conservative, as it prevents you from liquidating your friend purely due to increased price uncertainty.
 
 The same principle would apply if you wrote a derivative contract. If someone wants to open a derivative contract with you, you would value their collateral at the lower price. However, if you were deciding whether someone's margin limits were violated, you would value their collateral at the higher price. If a contract needs to be settled at a price, you could take approaches such as the following: