Dark Spirits and Ancient Aliens

Michael Heiser

Joe Rogan and Tucker Carlson mentioned demonic spirits in a recent podcast, and my first thought was that they need to look up Michael Heiser. He was an Old Testament scholar who spent his career examining how various ethereal spirits fit into our world. Heiser died from cancer last year but has a ton of material online, as he did a weekly podcast discussing the bible topics, not limited to his niche focus on angels and demons (see an intro video here). It's too bad he died because he would make for an excellent podcast with Rogan, as he was an excellent communicator and enjoyed investigating UFOs, Ancient Aliens, and Zechariah Hitchens (he was generally skeptical but found it fun; see here). More recently, Tim Chaffey wrote a book on the Nephilim, so perhaps that's an alternative for one of them.

Most intellectuals find the idea of ethereal forces patently absurd, but this is just because they don't understand the assumptions of their naturalistic worldview. Everyone believes in unseen forces and miracles; religious people merely own it. Cosmologists rely on the multiverse's infinite universes to explain various fine-tuning, such as the cosmological constant and the initial entropy of the universe; they also believe in dark energy, dark matter, and inflation, though these forces cannot be observed directly or falsified in any way. Secular origin-of-life researcher Eugene Koonin uses the multiverse to overcome the many improbabilities required. These scientific theories are untestable, differing from pre-scientific theories only by replacing God with hypothetical fields and forces.

Burying miracles into assumptions does not make them any less miraculous. An infinite number of universes probability could easily explain Genesis (with Charleton Heston as God), that we are a hallucinating consciousness imagining life as we know it, or that we live in a simulation designed by an ancient effective altruist. The only ontological difference between religious and naturalistic miracles is that religious people assign their creator not just with personhood but one who defines absolute, objective morality. I find the Biblical virtues stand on their own as optimal praxis for personal and societal flourishing, but simple Bayesian logic implies I should defer my ethical judgments to my creator, given the knowledge disparity. The Bible, however, teaches that my view has been and always will be a minority take under the sun.

Spirit Beings Who Are Not God

If you believe in the New or Old Testament, you believe there is more than God is not the only being in the spiritual realm. There is not only the trinity, but angels Gabriel, Michael, and the fallen angel Lucifer. Hebrews 12:22 mentions' countless thousands of angels,' and Psalm 89 mentions a divine assembly that includes many heavenly spirits, among many other mentions. No Jew or Christian thinks this implies polytheism.

A big problem centers on one of the words for God in the Old Testament. The word elohim is used thousands of times in the Hebrew Bible, and it usually refers to the 'God Most High,' Yahweh, aka God. This has led many to think every use of the word elohim refers to God, which is simply incorrect. First, note that elohim is a word like deer or sheep that can be singular or plural. For example, in Psalm 82, we read.

Elohim presides in the divine assembly; He renders judgment in the midst of the elohim

The word elohim here has two referents: the first for God is singular, and the second is plural because God is 'in the midst of' them. Elohim is used thousands of times in the OT to denote the God (aka Yahweh), but also reference different ethereal beings.

• The members of God’s council (Psa. 82:1, 6)

• Gods and goddesses of other nations (Judg. 11:24; 1 Kgs. 11:33)

• Demons (Hebrew: shedim—Deut. 32:17)

• The deceased Samuel (1 Sam. 28:13)

• Angels or the Angel of God (Gen. 35:7)

Elohim is just a word for spiritual beings, and these are often mentioned in the Old and New Testaments. God is an elohim, but he is also the Most High, as in Exo 15:11: "Who is like you among the elohim, Yahweh?" If He is most high, He has to be higher than something else.

Understanding many other spiritual beings are interacting with God helps us understand the phrase in Genesis 1 where God says, "Let us make man in our image," or when God says, "man has become like one of us in knowing good and evil." Ancient Hebrew did not have the majestic we or trinitarian phrases.1

This is an essential point because the existence of other elohim is commonly overlooked by Christians, and this obscures some profound truths. How else would one make sense of 1 Kings 11, where God is in heaven discussing what to do with Ahab:

Micaiah continued, "Therefore hear the word of the LORD: I saw the LORD sitting on His throne, and all the host of heaven standing by Him on His right and on His left.

And the LORD said, 'Who will entice Ahab to march up and fall at Ramoth-gilead?'

And one suggested this, and another that.

Then a spirit came forward, stood before the LORD, and said, 'I will entice him.'

'By what means?' asked the LORD.

And he replied, 'I will go out and be a lying spirit in the mouths of all his prophets.'

'You will surely entice him and prevail,' said the LORD. 'Go and do it.'

God does not have a personality disorder; he has a council of other ethereal beings, other elohim. He does not need these helpers any more than we need dogs, friends, or children: they bring us joy, but also problems. Humans can empathize because we were made in his image. Many other verses make no sense if God exists in the spiritual world with only his alter egos (the trinity).

Spirits go Bad

Many cannot fathom why a good and all-powerful Yahweh would create beings who become evil, and the answer generally focuses on the importance of free will, which is part of being made in the image of God. There is endless debate on free will, whether one believes in God or not. For example, the existence of evil given a good, all-powerful God is puzzling, but it's also difficult to imagine a world without evil; the word evil would lose its meaning, as well as its antonym, goodness. If they are inseparable, like two sides of a coin, is a world without good and evil better than one with them? One can speculate, but the bottom line is that elohim, like humans, can and do go bad, creating evil spirits that manipulate men.

There are three significant calamities in human history. The first is Adam and Eve's fall in the Garden of Eden, the only one most Christians recognize. However, two others get less attention: the rebellions that led to the flood and tower of Babel incident, which directly involved evil elohim. There are many more texts relating to these latter falls in the Qumran texts than the incident with the apple, highlighting they should get more of our attention.

The backstory for God's decision to flood the earth is mentioned in Gen 6:2-4

"the sons of God saw that the daughters of man were attractive. And they took as their wives any they chose. Then the LORD said, "My Spirit shall not abide in man forever, for he is flesh: his days shall be 120 years."

The Nephilim were on the earth in those days—and afterward as well—when the sons of God had relations with the daughters of men. And they bore them children who became the mighty men of old, men of renown."

The sons of God, fallen elohim, impregnate human females, creating 'mighty men of renown,' the Nephilim, translated as giants in the Septuagint. The Nephilim are referenced in several places, including descriptions of figures like Goliath, the Anakim, King Og of Bashan, and the Canaanite "giants in the land" targeted by Joshua's conquests. Christians often interpret the 'sons of God' who fathered the Nephilim as just good Hebrews, who contrast with the unrighteous' daughters of men.' This makes little sense because their offspring were clearly different, not just in size, but in their capabilities. Why would a sinful-righteous human pairing create supermen? The acceleration of evil created by these half-breeds and their progeny was a travesty so evil Yahweh decided to eliminate most of humanity.

The flood did not fix the problem. A hundred years after the flood, there is the Babel incident, in which humanity attempts to build a tower to heaven, a rebellion based on hubris that offends God. In response, God not only disperses humanity into various nations, but he disinherits them as His people and puts them under the authority of lesser Elohim. This is recalled in Deuteronomy 32.

When the Most High gave the nations their inheritance, when He divided the sons of man, He set the boundaries of the peoples according to the number of the sons of God (Israel). But the LORD's portion is His people, Jacob His allotted inheritance. ~ Deu 32: 8-9

We know this refers to the Babel event because God apportions a table of 70 nations in Genesis 10 from Noah's three sons. In Genesis 11, after the judgment at Babel, each of these nations was assigned to a son of God, a lesser elohim, while one was kept for the Lord.

Historically, most Christians have read Deuteronomy 32 with 'sons of Israel' as opposed to 'sons of God.’ Currently, most have Israel, but many have God, such as the ESV. While that is an accurate translation of the Hebrew Masoretic text used by Jerome, the 'sons of God' phrase was more common among Qumran texts and the Septuagint. This mistranslation led to a lack of interest in its implications, as in the ‘sons of Israel’ interpretation, one assigns rulers like Jacob, while in the ‘sons of God’ view, regional deities. The issue becomes prominent in Psalm 82.

Ps 82:1-2 Elohim presides in the divine assembly; He renders judgment in the midst of the elohim. How long will you judge unjustly and show partiality to the wicked? …

Ps 82: 6-7 I have said, 'You are elohim; you are all sons of the Most High.' But like mortals you will die, and like rulers you will fall.

God is not speaking to Jewish elders as elohim because humans do not sit in God's divine assembly (see Psalm 89 for a description of God's council). Further, it would make no sense to proclaim humans would die like mortals; they would have known that. Thus, Psalm 82 describes God judging the elohim he assigned to rule the nations, as described in Genesis 11 and Deuteronomy 32. You could easily miss it if you did not read 'sons of God' in Deu 32, which explains why this interpretation is relatively new (the Qumran texts that seal the inference were unavailable to Augustine, Calvin, etc.).

To recap, we have

• Gen 6:2-4: Evil spiritual beings come down to earth, defiling women and creating a race of evil, mighty giants, the Nephilim

• Deu 32:8-9: Recounts how God abandons 69 nations to lesser elohim but keeps one to himself that he would inherit through Abram.

• Psalm 82: God condemns 69 elohim for being unrighteous rulers of their nations.

These are the types of dark forces that create problems for humanity.

The Watchers

These verses briefly reflect what is mentioned in The Book of Enoch (aka 1 Enoch) and the Book of Giants, prominent among the Qumran texts. Enoch refers to the sons of heaven as the Watchers. As in Genesis 6, they see the daughters of men, desire them, and decide to come down from heaven and mate with them. Their leader, Shemihazah, knows his plan is sinful, and he does not want to bear the responsibility alone, so the Watchers swear an oath on Mount Hermon, a classic conspiracy. The Watchers teach humans various 'magical' practices such as medicines, metallurgy, and the knowledge of constellations. This knowledge gives people great power but exacerbates their sin and suffering.

Enoch describes how the offspring of the Watchers and women became giants who dominated humanity, 'devouring the labor of all the children of men and men were unable to supply them.' (7:4). 1 Enoch 15:8 refers to the offspring of the giants as demons. These beings are described as spiritual, following their fathers' nature; they do not eat, are not thirsty, and know no obstacles.

One might reject this description of an older form of Ancient Aliens click-bait. However, we can know this is true because listening to spirits is forbidden by God, which would make no sense if it were impossible.

You must not turn to mediums or spiritists; do not seek them out, or you will be defiled by them. I am the LORD your God. ~ Leviticus 19:31

While the Book of Enoch is not canonical, it should still be taken seriously. Both Peter and Jude not only quote from the Book of Enoch, they quote the verses that directly describe the fallen elohim.2 Enoch is also quoted in 3 Maccabees, Ecclesiasticus, and the Book of Jubilees. It is favorably examined by the early Church fathers Justin Martyr, Irenaeus, Clement of Alexandria, Tertullian, and Origen.

A big reason Christians do not revere the Book of Enoch is that influential Christian Augustine ignored it. Augustine initially allied with the Manicheans, who revered The Book of Enoch. Like many who fall away from an ideology, he tried to make as complete a break as possible. Further, Enoch's narrative does not mention human responsibility. It is a determinist view where evil originates from the deeds of the Watchers, in contrast to the Garden of Eden story, where sin emanates solely from our human nature. For Augustine, our most pressing problem is the sin residing in all of us due to the initial fall of man in the Garden of Eden.

Jesus and the Dark Spirits

This view of dark forces explains the cosmic geography mentioned in the New Testament.

To the intent that now unto the principalities and powers in heavenly places ~ Ephesians 3:10

against principalities, against powers, against the rulers of the darkness of this world, against spiritual wickedness in high places. ~ Ephesians 6:12

If you believe in the New Testament, you can't reject demonic forces.

The gospel's good news directly addresses the problem created by demonic forces. Christ enabled those under the lesser elohim's dominion to turn from those gods via faith. The breach caused by the Babel rebellion had been closed; the gap between all humanity and the true God had been bridged.

We see this in Acts, where Luke records Paul's speech in Athens. In talking about God's salvation plan, Paul says:

And God made from one man every nation of humanity to live on all the face of the earth, determining their fixed times and the fixed boundaries of their habitation, to search for God, if perhaps indeed they might feel around for Him and find Him. And indeed He is not far away from each one of us. ~ Acts 17:26-27

Paul clearly alludes to the situation with the nations produced by God's judgment at Babel, as described in Deuteronomy 32. Paul's rationale for his ministry to the Gentiles was that God intended to reclaim the nations to restore the original Edenic vision. Salvation was not only for the physical children of Abraham but for anyone who would believe (Gal. 3:28-29)

In Acts 2, the apostles, filled with the Holy Spirit, begin to speak in various tongues, allowing them to communicate the gospel to people from different nations visiting Jerusalem. Paul describes the disciples speaking in tongues as divided using the same Greek word (diamerizo) from the Septuagint in Deuteronomy 32, 'When the Most High divided the nations, when He scattered humankind, He fixed the boundaries of the nations.' Luke then describes the crowd, composed of Jews from all the nations, as confused, using the same Greek word (suncheo) used in the Septuagint version of the Babel story in Genesis 11: 'Come, let us go down and confuse their language there.' This mirroring highlighted that Jesus and the holy spirit would rectify the disinheritance at Babel and the subsequent oppression by corrupted elohim rulers.

Demonic Ancient Aliens

In the Babylonian flood story, divine beings known as Apkallu possessed great knowledge, had sex with women, produced semi-divine offspring, and shared their supernatural knowledge with humanity. In contrast to the Bible, they were hailed as pre-flood cultural heroes, so Babylonian kings claimed to be descended from the Apkallus. To make the connection with Enoch even clearer, Apkallu idols were often buried in Babylonian house foundations for good luck, and were called 'watchers.' The Watchers relates to many mysteries, such as the discovery of bronze and iron or the creation of the pyramids.

The ancient Greeks had their version of this, replacing Apkallu with Titans. These were regional semi-deities with knowledge from the gods that gave them great power. In Hesiod's Theogony, he mentions the titan semi-gods and how Prometheus stole fire from the Gods. Aeschylus's play Prometheus Bound describes the famous punishment for giving humans divine knowledge, as sinful humans would invariably use their greater knowledge to their ultimate detriment, the classic story of hubris.

two takes

A common literary theme is to spin either the good Apkallu or bad Promethean interpretation of human development. For instance, in The Lord of the Rings, the ring has great power but ultimately destroys those who possess it; in The Godfather, Michael wins the war with the five families but loses his soul. In Percy Shelley's Prometheus Unbound and George Bernard Shaw's Back to Methuselah, godlike human intellectual and spiritual development brings humanity's eventual liberation and enlightenment through knowledge and moral improvement.

There is nothing wrong with technological improvement or efficiency. Bezalel is described as having great wisdom and craftsmanship and is promoted by God to create the first Tabernacle and the Ark of the Covenant. Noah was considered uniquely righteous and built a boat that could hold an entire zoo. The problem is succumbing to the temptations created by powerful, dark spiritual powers who know many valuable things that humans do not. Understandably, this power would seduce many. Any elohim who does this is contravening God's plan for their glory, which is evil. Many humans glom onto them, as they would rather rule in hell than serve in heaven, or just not care about the long run and prioritize the ephemeral pleasures of status and its spoils (‘eat, drink, and be merry for tomorrow we die’).

How demons interact with humans is unclear. As humans, we probably cannot exterminate them if we try, but we know we can and should resist them. There are many opportunities to aid and abet evil for a short-term advantage, but it is foolish to gain the whole world to lose your soul. While I doubt anyone can tell if someone is a demon puppet, let alone a demon, both have and do exist. We need not to be naïve and be wary. Discerning good and evil is difficult when dealing with fallen elohim, as they lie and are smarter than us. A simple rule is to speak truth to lies because God is truth. If you must lie to make your point, there's a good chance you are allying with dark forces.

1

see other 'us' language in Gen 1:26, Gen 3:22, Gen 11:7, Isaiah 6:8

2

1 Enoch 19:1 quoted in 2 Peter 2:4

For if God did not spare the angels when they sinned, but cast them into Tartarus

1 Enoch 1:9 quoted in Jude 14-15

It was also about these men that Enoch, in the seventh generation from Adam, prophesied, saying, “Behold, the Lord came with many thousands of His holy ones, to execute judgment upon all, and to convict all the ungodly of all their ungodly deeds which they have done in an ungodly way, and of all the harsh things which ungodly sinners have spoken against Him.

Why Evolution is False

 Recently deceased philosopher Daniel Dennett called Darwin's idea of natural selection the best idea anyone ever had, a universal solvent that melts away all the problems in biology. Dennet had contempt for Christians, coined the term 'brights' for those who shared his worldview, and thought it wise not to respect religion because of its damage to the 'epistemological fabric of society.' Like fellow atheist Richard Dawkins, he never debated biologists, just theologians.

In a 2009 debate, Dennet mentioned he brought his friend, an evolutionary biologist, to a 1997 debate with Michael Behe about his book Darwin's Black Box (1996) because he felt unqualified to address the micro-biology arguments. Dennett described Behe's book as 'hugely disingenuous propaganda, full of telling omissions and misrepresentations,' that was 'neither serious nor quantitative.' Dennet then added he would not waste time evaluating Behe's newest book, The Edge of Evolution (2007).1

Dennett emphasized his approach to understanding the world was rational, reasoned, and evidence-based. Yet, he never directly addressed Behe's arguments and instead stuck to the cowardly pleasure of debating non-scientist theologians with whom he had greater knowledge of biology. He admits he could not evaluate the arguments alone by enlisting a biologist to help him debate Behe. If he could not trust himself to evaluate Behe's argument, a rational approach would be to take a trusted source's opinion as a Bayesian prior, not as a fact so certain that its negation damages the epistemic fabric of society. Unfortunately, many, perhaps most scientists, agree with Dennett and think the only people who don’t believe in evolution are ignorant (e.g., see Geoffrey Miller here, or Dawkins here).

If Dennett had read Behe's Edge of Evolution, he would have seen it as a logical extension of his earlier book, not moving the goalposts. Behe's argument isn't based on parochial microbiology knowledge; it's pretty simple once one gets the gist.

Behe highlighted the edge of evolutionary power using data on malarial resistance to two different antibiotics. For the malaria antibiotic atovaquone, resistance develops spontaneously in every third patient, which, given the number of malaria in a single person, the probability malaria successfully adapts to this threat can be estimated as one in 1e12, or 1e-12. Resistance occurs once every billionth patient for the antibiotic chloroquine, giving an estimated successful mutation probability of 1e-20. This roughly squares the original probability, which led Behe to suggest that at least two mutations were required to develop resistance, as two mutations represent a squaring of the initial probability. This prediction was confirmed a few years later.

Extending this logic, given a base mutation rate of p per base pair per generation (e.g., p ~1e-8 for humans), if n mutations are needed, the probability of that happening scales at pⁿ. Given that new proteins require at least 10 changes to some ancestor (out of an average of 300 aminos), the probability of an existing genome evolving to a new specific protein would be 1e-80. Given that only 1e40 organisms have lived on Earth, this implies that evolution is limited in what it can do (note most evolution we observe, as in dog breeds, just involves changes in allele frequencies).

A reasonable criticism is that this argument works for a specific target, such as Behe's example of malaria overcoming an antibiotic. However, the state space of possible unknown proteins is much larger. For example, the average protein has 350 amino acids; with 20 amino acids, that's 20^350 or 1e455 possible permutations. If there are islands of functional proteins within this vast state space, say at only a rate of 1e-55, that leaves 1e400 potential targets. The 'singular target' criticism does not invalidate Behe's assertion, but addressing it would require too much space for this post, so I will just mention it as a defensible criticism.  

However, most reviews of Behe's malaria-based argument centered on the simple assertion that generating n specific mutations scales at pⁿ.

Ken Miller (Nature, 2007):

Behe obtains his probabilities by considering each mutation as an independent event

Sean Carroll (Nature, 2007)

Behe's chief error is minimizing the power of natural selection to act cumulatively.

Jerry Coyne (The New Republic, 2007)

 If it looks impossible, this is only because of Behe's bizarre and unrealistic assumption that for a protein-protein interaction to evolve, all mutations must occur simultaneously, because the step-by-step path is not adaptive.

These criticisms are based on two possible assumptions. One is that if multiple mutations are needed, single mutations encountered along the process of fixing the multiple mutations may each confer a fitness advantage, allowing selection to fix the intermediate cases. While this can be true, it is not for the mutations needed for malaria to develop chloroquine resistance, which needs at least two, and it is undoubtedly not true in general. Indeed, a good fraction of the intermediate steps reduce fitness, some severely (see here or here), which is why a fitness advantage from a two-step mutation does not happen with the same frequency as fitness enhancement that needs one mutation: in the wild, the intermediate mutations are eliminated from the population.

The other assumption would be if the number of indirect paths overwhelms the specific case where sequential mutations occur. There are many indirect paths from one sequence of 300 amino acids into another. The question is their probability, the sum of these successful paths over all possible paths.

Intuitively, multiple specific simultaneous mutations are astronomically improbable and would constitute an impenetrable fitness valley, which is why evolution proponents are emphatic that it is an absurd approximation. Their intuition is that if one needs a handful of specific mutations when one already has 100 or 400 of the needed amino acids in the correct spot, a cumulative process of some sort should be likely, even if the final steps involve neutral mutations that neither help nor hurt fitness. However, none of Behe's critics generated a model to quantify their reasoning, even though they are all scientists in this field.

Model of the Behe Edge of Evolution Argument

Hypothesis: Prob(get n specific mutations sequentially) ~= Prob(get n specific mutations over 1e40 generations)

The model below uses a trinomial lattice to show that if the probability of getting one mutation correct is p, the probability of getting n mutations correct is on the order of pⁿ. Given the small probability of getting one mutation correct, this highlights what Michael Behe calls the edge of evolution: the fitness landscape generates an impenetrable barrier for naturalistic processes. Assuming simultaneous mutations towards the target captures most of the cumulative probability of reaching that target if mutations are neutral until the target is achieved. The other paths drift away at a rate that initially eliminates the benefit of being so close.

We can model this using a variant of Richard Dawkin's Weasel program that he used to demonstrate the power of evolution. It starts with a string of letters and spaces, 27 potential characters in a string of length 28.

neuh utnaswqvzwzsththeouanbm

Dawkins randomly generated strings, fixing characters that matched the target phrase from a Shakespearean play. The application to genetics is straightforward if we think of the phrase as a sequence of nucleotides or amino acids creating a protein.

…

xethinks dt is like a weasek

xethinks dt is like a weasel

xethinks it is like a weasel

methinks it is like a weasel

This algorithm fixes the characters that only make sense at completion. This is like assuming necessary mutations are selected with foresight, which would require some outside designer shepherding the process. Evolutionists countered that Dawkins' weasel program was merely to demonstrate the difference between cumulative selection and single-step selection, but this is disingenuous, as removing forward-looking selection increases the number of steps needed from 100 to 1e39, which would not make for a convincing TV demonstration (see Dawkins’ promoting his weasel here).

Nonetheless, the Weasel program is familiar to many who study evolution and can be used to illustrate my point. Consider the case where we are two characters away from the target sequence.

Start: 2 wrong, 26  correct

XXTHINKS IT IS LIKE A WEASEL

Case 1, closer: 1 wrong, 27 correct.

MXTHINKS IT IS LIKE A WEASEL

XETHINKS IT IS LIKE A WEASEL

Case 2, stasis: 2 wrong, 26 correct. The two mismatched characters can each change into 25 potential targets (27 – 2) that are also wrong, for a total of 2*25. For example, here are three.

YXTHINKS IT IS LIKE A WEASEL

XYTHINKS IT IS LIKE A WEASEL

AXTHINKS IT IS LIKE A WEASEL

Case 3, further: 3 wrong, 25 correct. Each of the remaining matching characters can become unmatched by changing into any of the 27-1 potential characters. The total number of paths is 26 x 26. Here are two.

XXXHINKS IT IS LIKE A WEASEL

XBTHXNKS IT IS LIKE A WEASEL

To generalize the problem, let us define L as the string length, c the number of potential characters at each element in the string, and n the number of needed changes to the string (initial mismatches). The set of potential new strings can be split into three groupings with their distinctive number of paths:

1.      strings that have more characters that match the target:

a.       e.g., n moving from 3 to 2, n possibilities

2.      strings that have the same number of characters matching the target: n*(c – 2) possibilities

a.       e.g., n changing one mismatched amino to another, so staying at 3

3.      strings that have fewer characters that match the target: (L – n)*(c-1) possibilities

a.       e.g., n moving from 3 to 4

The probabilities are the same regardless of which n characters are referenced. For example, the two sequences below are mathematically identical regarding how many paths are towards and away from the target, so they will have the same probabilities of moving towards or away from the target.

XXTHINKS IT IS LIKE A WEASEL = METHINKS IT IS XIKE A WEASLX

This makes the problem simple to model because regardless of where the string starts, the number of cases that need probabilities is at most L+1, as n can range from 0 to L. All mutations are considered equal probability; the number of paths up over the total possible paths {up, same, down} is the probability of moving up. We can, therefore, calculate the probability of moving closer, the same, or further from its target (i.e., n_t+1 < n_t, n_t+1 = n_t, n_t+1 > n_t) for each n <= L. This allows us to model the evolution of the string using a recombining lattice that evolves from an initial node, a standard tool for modeling option values.

At every row of the L+1 nodes in the lattice, we calculate the probabilities of moving up, across, and down via the following formulas.

1.      Prob(closer): n/(L*(c-1))

2.      Prob(same): n*(c-2)/(L*(c-1))

3.      Prob(farther: (L-n)*(c-1) /(L*(c-1))

Figure 1 below shows a case with five rows representing a string of 4 characters. The columns represent generations defined by a change in one of the spaces in the sequence (a mutation). The bottom row of nodes reflects zero matches, and the top level is a complete match. In this case, the initial node on the left implies 2 mismatches out of 4. If the new mutation flips a mismatched position to the target, it is closer to the target and thus moves up one level, just below the top row; if it stays two away by having an incorrect letter changed to a different incorrect letter, it moves horizontally; if a correct letter changes to an incorrect letter, it moves down.

Figure 1

A complete match is a success, and we assume that once the sequence reaches the target, it does not regress (it is then subject to selection and fixates in the population). Thus, there is no path downward from the top row. While this is not realistic in practice, it inflates the total probability of reaching the target and does not affect the gist of my argument. The point is to highlight the relative importance of the direct path as opposed to the actual probability.

In Figure 2 below, we have a snip from a worksheet modeling the case where the starting string has two mismatched characters from the target sequence of 'methinks...' For the first mutation, there are 676 changes away from our target, 50 that would maintain the same distance, and 2 that move towards the target, giving probabilities of 92.8%, 6.9%, and 0.3% to the node branches. At the next upward node in step 1, the probability of going up again and reaching the target is 1/728 or 0.14%, so the probability of reaching the target in two mutations is 0.27% * 0.14%, or 3.77e-6.

Figure 2

The 'nodeProb' row represents the probability of reaching the target on that mutation, while the 'cum prob' row represents the cumulative sum of those nodes, given we assume reaching the target allows the organism to thrive with its new protein function.

The node probabilities asymptotically decline, so taking the hundredth 'nodeProb' as equal to all subsequent nodeProbs generates a conservative estimate of the number of generations (G) needed to double the direct path probability.

2*nodeProb(2) = cumProb(100) + G*nodeProb(100)

G = (2*nodeProb(2) - cumProb(100) )/nodeProb(100)

For this example

G = (2*3.77e-6 – 4.23e-6)/3.00e-37 = 1e31

This implies at least 1e31 generations are needed for the cumulative probability to be twice the direct probability. As estimates in this field have logarithmic standard errors (e.g., 1e-10 to 1e-11, as opposed to 1.0e-10 to 1.1e-10), a direct path probability within a factor of 2 of the cumulative case over 1e31 paths is about the same.

Modeling this with a lattice allows us to see the relative importance of the direct path because everything reaches the target over an infinite amount of time, so the probability of reaching the target is the same regardless of the starting point. With a lattice approach, we can model the finite case that is still large enough (e.g., 1e40) and see the relative probabilities.

The probability of a direct path is approximately equal to a simultaneous path because, if we assume mutations obey a Poisson process, the probability of a simultaneous mutation is the same as sequential mutations, just one probability times the other. For example, the malarial parasite cycles through several generations during an infection, and the resistant parasite could acquire the necessary mutations in generations 3 and 5 or simultaneously in generation 3 (both would have the same probability).

Thus, Behe's hypothesis that the probability of reaching a target n mutations away scales with pⁿ is a reasonable estimate when intermediate steps are neutral.

The worksheet contains the Weasel program applied to 2 and 3 characters away from the target. It can be generalized straightforwardly for an arbitrary n of needed mutations, 20 amino acids, and protein of length L, as shown in the worksheet ‘12awayAminos.’ Looking at the worksheet '2away' in the Excel workbook, the probability of reaching the target in a direct sequence is the probability of hitting the first top node. All the probabilities here are relative because this model assumes a mutation along an existing string of amino acids. This occurs only at a rate of 1e-8 per nucleotide per generation in humans. So, the extension to amino acid changes needs an extra adjustment (nucleotide changes do not necessarily change amino acids). The purpose of this post is just to show the relative probability of the direct and indirect paths, which is independent of the absolute probability.

There are an estimated 35 million single nucleotide differences and 90 Mb of insertions and deletions, creating 700 new genes in chimps and humans that were not present in our last common ancestor. That's at least 100 functional mutations that must be fixed every generation over the past 6 million years. Genetic drift can generate the 100 newly fixed mutations, but this drift is random. The probability that a random amino acid sequence would do something helpful is astronomically improbable (estimates range from 1e-11 to 1e-77, a range that highlights the bizarre nature of evolution debates), which creates a rift within the evolution community between those who emphasize drift vs. those who emphasize natural selection.2 This debate highlights that there are rational reasons to reject both; each side thinks the other side’s mechanism cannot work, and I agree.

In Dawkin's Mount Improbable, he admits the de novo case is effectively impossible. He suggests indirect paths involving gene duplication, as the base structure provided by an existing gene would give the scaffolding needed for protein stability. Then, one must merely nuance a few segments to some unknown function. The above shows that if one needs a modest number of specific amino acids (e.g., a dozen), the probability of reaching such a target will be greater than 1 in 1e100. For this copy-and-refine process to work, it requires a protein space with a dense set of targets—islands of functionality in the protein state space—which is possible but eminently debatable.

1

See the 2009 debate between Dennett and Christian philosopher Alvin Plantinga, especially around 64 minutes in.

2

The 1e-11 estimate applies to an 80-amino string converting from weak to strongly binding to ATP, which is understandable given this is the most common binding protein machine in a cell, as it powers everything. Further, this was studied in vitro, which is 1e10 less likely to work in vivo. The 1e-77 estimate is for beta-lactamase, an enzyme produced in bacteria with a specific function, unlike binding to ATP which is common. Other estimates include 1e-65 for cytochrome c, 1e-63 for the bacteriophage lambda repressor, and 1e-70 for a certain phage protein. 

How to Eliminate Impermanent Loss

 Generally, markets are efficient in that it isn't easy to make above-average returns day-trading, and most mutual funds underperform market-weighted ETFs. Yet historically, in various applications, options have been underpriced for decades. For example, asset return distributions were known to have fatter tails than the lognormal distribution back in the 1960s (see Benoit Mandelbrot ('62) or Eugene Fama (' 65)). Most option market makers, however, applied the basic Black-Scholes with a single volatility parameter, which underpriced out-of-the-money options. On a single day, October 19, 1987, the stock market fell 17%, which, using a Garch volatility estimate, was a 7.9 stdev event. From a Bayesian perspective, that did not imply a miracle but, rather, a misspecified model. Option market-makers soon adjusted their models for out-of-the-money puts and calls, ending decades of underpricing (20 years before NN Taleb's Black Swan introduced the concept of fat tails to the masses as if it were 1986).

Another example is convertible bonds, which are regular bonds that are convertible into stock at a fixed price, a bond plus an equity call option. The optionality was underpriced for decades. In the 1990s, a hedge fund that merely bought these bonds hedged the option's equity delta and bond duration and generated 2.0 Sharpes, which is exceptional for a zero-beta strategy. Eventually, in the early 2000s, investment bankers broke these up into bonds and options, allowing them to be priced separately. This isolation highlighted the option underpricing, and this market inefficiency went away. One could add Korean FX options that were underpriced in the 90s, and I'm sure there were many other cases.

So, there is precedent for a lot of money locked up in money-losing short convexity positions, as with liquidity providers (LPs) on automatic maker makers (AMMs). However, LP losses explain why AMM volume is still below its 2021 peak; transformative technologies do not stagnate for three years early in their existence unless something is really wrong. It's like how multi-level marketing scams like Amway have a viable business model relying on a steady stream of new dupes; they can persist, but it's not a growth industry.

The Target

The way to eliminate AMM LP convexity costs centers on this formula:

Impermanent Loss (IL) = LPvalue(pt) – HODL(pt)

LPvalue(p_t) is the value of the LP position at the new price, p_t, and the 'hold on for dear life' (HODL) portfolio is the initial deposit of LP tokens valued at the new price. It turns out that this difference is not an arbitrary comparison but instead captures a fundamental aspect of the LP position, the cost of negative convexity in the LP position. The expected value of the IL is the LP's convexity cost, which is called theta decay in option markets.

The LP convexity cost also equals the expected value decay of a perfectly hedged LP position (minus fees). Hedging is often thought of as reducing IL, and it does reduce the variance of the IL, eliminating extreme LP losses. However, this does not affect the mean IL, which, over time, equals the expected value. Lowering a cost's variance reduces required capital, which is why hedging is a good thing, but it will not solve the IL’s main problem, its mean, which is generally larger than fees for significant capital-efficient pools.

If we expand the above formula, we can rearrange variables to get the following identical equation as a function of pool token quantity changes and the current price (see here for a derivation).

IL = USDchange + pt×ETHchange

I am using a stablecoin USD and the crypto ETH as my two tokens because it makes it easier to intuit, though this generalizes to any pair of tokens (I like to think of prices in terms of dollars, not token A, but that's just me). The duration used to calculate the token changes implies the same LP hedging frequency. 24 hours is a pretty good frequency as hedge errors cancel out over a year via the law of large numbers, but anything between 6 hours and 1 week will generate similar numbers. If we can set ETHchange=0, USDchange will be near zero, and we will effectively eliminate IL.

An Extreme Base Case

One way to eliminate IL is to have a single LP who can trade for free while everyone else trades for a fee. Whenever the AMM price differs from the true price by less than the fee, only the LP can profit from arbitrage trading. A simple way to intuit this is to imagine if an AMM was created and not publicized, so it has no traders outside of the creator who assumes the role of LP and arbitrageur. With no other traders, he is trading with himself in a closed system. If his accounts are netted, he could both set the price on his contract efficiently and hedge his IL; nothing changes except the price. His net position in the tokens is static: ETHchange=0, IL=0.

It's helpful to think of traders as being two types: arbitrage and noise. Arbitrage traders are trying to make instant money off minor price discrepancies between exchanges; noise traders are like white noise, mean zero net demand that flows in like Brownian motion. A market price equilibrates supply and demand, so the volume that nets out is, by definition, noise. Noise traders are motivated by individual liquidity shocks, such as when a trader needs money to pay taxes, or the various random buy and sell signals generated by zero-alpha trading strategies.

If the AMM's base trading fee was 15 bps, and the LP could trade for free, the LP could turn loose an automated trading bot based on the Binance/CME/Coinbase price, trading whenever the mispricing exceeded 10 bps. Over time, the LP/arbitrager will be responsible for all the AMM's net price changes. With the AMM at the equilibrium price, immune to arbitrage by non-LP traders, the other trades will be white noise, net zero demand, by definition.

The figure below shows how the LP's ETH position for an AMM restricted range from $1300 to 1700 changes. The Pool ETH change implies traders are accumulating an offsetting amount; if the pool lost 2.5 ETH, traders gained 2.5 ETH. This net trader accumulation is arbitrage trading because it is not random, which is mean-zero over time. Gross trading will be greater due to noise trading, and in a healthy exchange, the noise traders will compensate the LP for his predictable IL over time.

If the monopolist zero-fee LP were the arbitrageur, at any price his net ETH position would be the same; the sum of those lines, his net position on the contract, would be horizontal. With a constant net ETH position, his impermanent loss would be zero. This works because, unlike traditional markets, the option seller is not reacting to prices but setting them. The latency inherent in any decentralized AMM implies the price setter does not need any alpha, just an API to the lower latency centralized exchanges.

The LP would still have exposure to this net ETH position, but this is simple to hedge off the AMM with futures, as it would not need frequent adjusting.

If we add a separate margin account on the contract that held the LP's trades (with himself), they will net to a constant, his initial token position. The monopolist LP's net position on the contract would be represented as in the table below (on average).

Single LP with Exclusive Arbitrage Trading Access

Thus, if the LP could trade for free, allowing him to dominate arbitrage, and he had a separate trading account on the contract, he could eliminate his IL without moving tokens on and off the blockchain or making other trades on different exchanges.

Multiple LPs

An AMM with only one LP would not work. An extensive set of LPs is needed to avoid centralization, which presents an attack surface for regulators and hackers; it is also needed to give the AMM unlimited scale. However, now we must consider LP competition. If we gave all the LPs free trading, a power law distribution of trading efficiency would invariably reveal a few dominant LPs monopolizing the arbitrage trading, shutting the slower LPs out and leaving them all exposed to IL as before.

Fortunately, the AMM provides a simple mechanism to allow all the LPs to arbitrage the price and hedge their positions simultaneously. This is because, on an AMM, the price is only changed by trades, so given a specific liquidity amount, a price change implies a specific change in ETH and vice versa. This makes it feasible to apply a rule using the LP's net ETH position on the contract to see if they qualify for the free discount.

Consider the following framework. Each LP gets a trading account in addition to their LP position. The key point is these accounts are netted, so like above, the LPs can retain a constant net position as the price moves significantly.

This can be adjusted to give them capital efficiency without changing the implications.[see footnote below] For my purpose in this post, this is irrelevant.

In the earlier example with one LP, his margin account's net change is calculated using the same liquidity as the pool because, with one LP, his liquidity is the pool's liquidity. This implied the LP's trades would always exactly offset his pool token changes. With many LPs, this is not so. A trade is against the entire pool, which is necessarily greater than any one LP. Thus, if LP(i) trades in a pool with many LPs, it will change LP(i)’s individual net position.

netETHchg(i) = (totLiq - liq(i))/totLiq *ETHtrade

As totLiq > liq(i), each trade changes the LP's net position, unlike when the monopolist LP trades with himself. For example, if an LP owns 10% of the liquidity, buying 1.0 token increases his margin position by 1.0 but decreases his pool position by only 0.1.

Assume LPs can only trade for free in the following conditions:

Buy for free only if

PoolEth(i) + MarginEth(i) - initEthDeposit(i) < + 0.01

Sell for free only if

PoolEth(i) + MarginEth(i) - initETHDeposit(i) >  - 0.01

If we assume the LP hedged his initial ETH deposit off-chain, the rules basically say if an LP is net net long, he cannot buy for free; if the LP is net net short, he cannot sell for free [presuming she hedged elsewhere initially, her ‘net net' position is her net position minus her initial deposit, as we assume it is hedged].

To see how this works, assume we have two LPs, Alice and Bob, with equal amounts of liquidity. Both Alice and Bob start with pool ETH positions of 990, and they have zero positions in their margin accounts. The trading fee is 0.5%, so all the price changes in this example are only profitable opportunities for Alice and Bob.

LP Pool, Margin, and Net ETH Balances

Assume the price initially rises by 0.41%, generating a arbitrage opportunity for the LPs. Alice wins the race and arbitrages this price discrepancy, setting the AMM price at its new equilibrium level of $3,313. She bought 4 ETH, and her pool position declined by 2 for a new ‘net net’ ETH position of +2. LP Bob just sees his pool position decline and has a new net net ETH position of -2; Bob is short, and Alice is long.

Alice cannot arb the AMM if the price rises again due to the rule preventing long LPs from buying at zero fee. If the price rises, Bob will win the arb race (by default), and LP’s Bob and Alice are each flat again. In the table above, Alice buys in the rows where she has a light blue box, and Bob buys in the rows where he has a light orange box. They can only buy when their net net position is flat or negative when prices are rising. This prevents Alice or Bob from dominating arbitrage.

If the price immediately went back down after Alice initially arbitraged the pool, only Alice could arbitrage the pool price. This is because in period 1, Alice is long, and Bob is short, so Bob cannot sell for free, while Alice can. When she does sell, she restores the initial net position for both herself and Bob. Alice’s extra trading is benign.

The LPs will experience net token position changes for single trades or small price movements. Over longer price movements, however, each individual LP's net token change would be insignificant, and assuming the LP hedged, their net net position would be insignificant.

There is no reason to pay arbs to set AMM prices, as the latency in decentralized blockchains implies arbitrage requires no special expertise, unlike the price discovery on the lowest latency exchanges. External arbitrageurs generate a deadweight loss for high-latency decentralized AMMs, so there is no trade-off with price efficiency.

A specialist with fast APIs and redundant arbitrage bot programs distributed in the cloud could manage a bot for a set of LPs. If the contract allowed LPs to whitelist another address that could do one thing on his account: trade for zero fee. The rule would prevent this arb vault manager from overtrading some accounts at the expense of others, as either it would single out one account for trades that cancel out, like in the case of Alice buying then selling above, or quickly find that her accounts that have traded first in one direction can no longer trade in that direction. One could cap trade size so the arb vault manager does not max out a trade on an LP to generate a profit for the other LPs. Competition among such specialists would allow passive LPs to get most of the benefit from arbitrage/hedging without creating and running their own arb bots.

It wouldn't work on coins not traded on major centralized exchanges because it presumes the true price is common knowledge. The next Shina Ibu will have to first trade on a conventional AMM. Yet, that's a minor amount of AMM volume.

Alternatives

As the equilibrium price is presumed to be on CEXes, and an equilibrium price implies net zero demand trading, one could use an oracle to update the AMM's last price to this price just as an LP arbitrage trade would. Over time, the LP net token changes would not be explicitly tied to price changes such that LP position values had negative convexity (i.e., linear in the square root of price). The main problem for oracle-based AMMs is centralization. This creates an attack surface for hackers. It also creates an attack surface for regulators, as the SEC knows where American Chainlink CEO Sergey Nazarov, for example, lives. Once they figure out how to do it—it took regulators several years to shut down InTrade—they will prevent his company from supporting anything that competes with regulated industries like finance and sports betting because regulation’s quid pro quo is protection from competition. Another problem is incentives, in that with more players with different actions, the state space complexity increases exponentially, so any solution will be more costly and less efficient. Arbitrageurs with more skin in the game would invest more time exploiting the oracle than the oracle would defending itself, especially as it requires a collective decision-making process and will be much slower to respond.

A decentralized limit order book (LOB) could avoid IL like an oracle-based AMM by allowing the LPs to move their resting limit orders costlessly to the latest CEX price, but there would be several problems. First, on-chain LOBs aspire to look and feel like the trading interfaces we are used to on centralized exchanges, so they emphasize low latency. Low latency leads to quasi, if not complete, centralization, an attack surface, and, more importantly, insiders who will get the equivalent of co-located server access. As third parties do not audit these ‘decentralized’ exchanges, the LOB leaders or a lower-level dev could sell undisclosed privileged access, and the costs would be virtually zero. It would take a lot of faith to presume they are immune to this subterfuge. Secondly, it would require many messages to cancel and replace resting limit orders. Unlike my mechanism, where one account rectifies the mispricing, on an LOB each LP account must be adjusted separately. On a CEX, these cancel-and-replace orders are costless, but if the blockchain is marginally decentralized, it will cost a few cents, and zero is a significant price point. Lastly, no one has presented a mechanism to incent a manager to oversee a set of LPs who wish to outsource their active management coherently. The LPs would still compete for favorable queue positioning, making it needlessly complicated and shutting out potential passive LPs.

The auction approach proposed by Maollemi et al. lets people bid to be the zero-fee trader. In that approach, the LPs would sell their right to fees to an arbitrageur who would get privileged zero-fee trading rights over some future period. The bidder pays a lump sum to the LPs, and the trading fees for that period go to the bid winner (meaning his trading fees return to him, so he trades for free). Assuming risk-neutrality and zero capital costs or other expenses, the arb would pay the expected arbitrage profit, which equals the expected LP convexity cost. They would also pay for the expected noise trader volume. Considering the arb bidder would be exposed to risk in that actual volatility and noise trader volume will be much lower than expected on occasion, he would pay considerably less than the expected value of the arbitrage opportunity and noise trading fees. The arb would have to hedge his trades on a different exchange, requiring extra collateral, and have to move tokens on and off the blockchain between those two exchanges (always losing one and gaining another). Lastly, frequent novel auctions can be manipulated, which means they will, especially at first.

Conclusion

The signature AMM is the ETH-USDC 5 bps pool on the Ethereum mainchain. Over the past 12 months, it has generated about $44MM in LP fee revenue and experienced $54MM in convexity costs. Yet even if LPs were, in aggregate, making a profit, the convexity cost is still significant and unnecessary. Given that the entire Uniswap AMM market is 4x the above pool, and there are other AMM protocols, that's several hundred million dollars a year wasted annually. The good news is this can be eliminated, propelling AMMs out of their doldrums and securing a long-term solution to an essential blockchain mechanism: swapping coins.

Most crypto people do not intuitively understand an LP's convexity costs, but they are not some new speculative theory (e.g., hedge fund sniping). Gamma, convexity costs, and theta decay have been analyzed empirically and theoretically for the past 50 years. They are the unavoidable consequence of convex payouts based on an exogenous underlying stochastic price. Constant product AMMs that link trade amounts to price changes, combined with the latency, allow the derivative owner (LP) to both hedge and set the underlying price simultaneously.

I haven't met anyone who understands this because they would be excited if they did. It's not often you find ways of saving hundreds of millions of dollars a year. My friends in academia or tradFi don't have much interest, let alone knowledge, of AMMs. My acquaintances in crypto, even those actively building AMMs, don't understand convexity beyond the technical definition, so they do not think it is a big deal. Fear of convexity costs is the beginning of AMM wisdom.

I wrote my dissertation in 1994 on low-vol stocks, noting they generated abnormal returns because they generate a slight premium to the market at 30% less risk (see here for a history of low-vol investing). I took a job as a bank risk manager but was busy pitching the idea to various asset managers, including several at the major investment banks. They didn't reject what I was saying but were always eager to know if well-known people or institutions in this field were on board. They weren't, and no one thought a fund offering virtually the same return as the SP500, regardless of risk, was compelling. I was out by the time low-vol investing started to grow. The responses I get from crypto people to this idea are similar.

It's not an abstract argument, just MBA-level finance. My best hope for this approach is that in a few years, LLMs will pick it up as they scrape the web for data, and via pure logic, over a couple of years, ChatGPT6 will use it as an answer for 'How do I remove impermanent loss?' There is pleasure in just being right about something important. 

footnote

In a levered AMM, the initial pool amounts are multiples of the initial deposit. This implies the LP starts with debits in his margin account for both tokens. The process works as follows:

Take an initial ETH deposit, ETH₀. Given the initial price, p₀, and assuming a leverage of 20x, apply the following liquidity to that LP

liquidity = 20*sqrt(p₀)*ETH₀

Given that liquidity, the initial USD deposit, also levered 20 times, is calculated:

Initial USD deposit = liquidity*sqrt(p₀)/20

The objective of monitoring the LP’s net position for free trading is the same as above. Here, the initial margin account is negative, and the LP is susceptible to insolvency. However, a more pressing concern is whether the LP will be insolvent in one of the tokens, even though their total position value is positive. A pool where all the LPs are solvent, but the LPs have zero of one of the tokens would prevent purchases of that token. The solution is to allow liquidation based also on the minimum net position for the LP in the two tokens.

If the LP actively hedges its position, its net position will be constant in both tokens, so LPs would only be subject to liquidation if they exposed themselves to IL out of negligence. 

Spurious High Frequency Autocorrelation

 A curious aspect of high-frequency data is that most come from centralized limit order books (CLOBs) where the bid-ask spread makes the data look negatively autocorrelated as trades are randomly made at the bid and the ask.

The returns driving this pattern are well below transaction costs, so they do not generate an arbitrage opportunity. However, one might be tempted to use the high-frequency data to estimate variance for pricing options or convexity costs (aka impermanent loss, loss versus rebalancing). This is a problem because the 1-minute Gemini returns generate a variance estimate 40% higher than one derived from daily data.

Variance grows linearly over time; volatility grows with the square root of time. Thus for a standard stochastic process, the variance should be the same when divided by the frequency. If the return horizon is measured in minutes, the variance of the 5-minute return should be half of the variance of the 10-minute return, etc. Variance(ret(M minutes))/M should be constant for all M. It’s helpful to divide the data by a standard, which in my case is the variance of the 1-day return over this sample (1440 minutes), so we can clearly identify those frequencies where variance is over and under-estimated. If the ratio is above 1.0, this implies mean-reversion at this frequency (negative autocorrelation), and if below 1.0, this implies momentum (positive autocorrelation).

Here is the ratio of the m-minute return variance divided by m for various minutes, normalized by dividing by the 1-day return. It asymptotes to 1.0 at 300 minutes.

I added the ETH-USDC 5 and 30 bp pools, and we can see the effect of stasis created by the fee and lower transaction volume. The one-minute return variance ratios for AMMs are well below 1.0, implying momentum—positive autocorrelation—at that frequency. Again, the effect driving this is well below 5 basis points, so it’s not a pattern one can make money off.

A common and easy way to calculate variance is to grab the latest trade and update an exponentially weighted moving average. This would generate a variance estimate at an even higher frequency than once per minute. The perils of this are clear when many important data are linear in variance, such as the expected convexity cost, and 40% is big.

As a refresher, I show how the common concept for impermanent loss equaling a function linear in variance because it’s not obvious. We start with the original IL definition

This is measured as

The following AMM formulas can substitute liquidity and prices into the above equation.

plugging these in, we can derive the following formula.

This ‘difference in the square roots squared’ function calculates a realized IL over a period (from 0 to 1). If one estimated this using daily data over several months, it would equal the expected IL via the law of large numbers.

One can estimate the expected IL using the expected variance and gamma of the position. The key nonlinear function on the AMM LP position is the square root of price, which has negative convexity. This implies a simple variance adjustment to the current square root to get the expected future square root.

Note that in the above, the variance is for the time period from 0 to t, so if it’s an annualized variance applied to daily data, you would divide it by 365. Returning to the above IL formulated as a function of square roots, we can substitute for E[sqrt(p₁)] to see how this equals the ‘LVR’ equation with the variance.

Using (p₁/p₀ - 1)² for the variance will generate the same formula as the difference in square roots squared if you use the same prices. We can’t measure expected returns, but average returns equal expected returns over time, and that’s all we have. One chooses some frequency to estimate the IL, regardless of the method. Just be careful not to use returns under a 300-minute horizon, as these will bias your estimate.