When to HODL💎🙌

The mathematically correct answer is always (but also, never to double down)

The Kelly-optimal allocation remains unchanged as new public information adjusts the price, unless new side information alters the perceived edge. A natural consequence is that investors in event contracts (or similar instruments) should not “average down” on their trading ideas or “lock in” profits after contract values increase merely because the price has changed (unless they have additional reason to believe that the contract is oversold / overbought). Along the way, we discuss information cascades, as well as explanations for the value and momentum factor that emerge naturally from models of bounded rationality.

The Kelly Criterion

Readers are probably already familiar with this topic, but here is the Wikipedia link for those who are not. There are a few desirable properties: Asymptotically, the Kelly criterion optimizes the rate of wealth growth, and as such almost surely dominates any other materially different decision criterion over the long run. From a utility perspective, under a constant relative risk aversion (the class of utility functions where investment strategy remains independent of wealth) the utility-optimizing investment strategy is to bet a fraction of the optimal bet in proportion to risk tolerance. These properties make the Kelly Criterion (or fractional Kelly) an attractive framework for decision making under uncertainty.

For fractional odds b (betting $10 on a 2-to-1 odds bet gives you $30 upon win netting a profit of $20) and true probability p of winning, the Kelly bet fraction is

\(f^{*}=p-{\frac {q}{b}}=p-{\frac {1-p}{b}}\)

If the gambler has zero edge (i.e., if b=q/p), then the criterion recommends the gambler bet nothing. If the edge is negative (b<q/p), the formula gives a negative result, indicating that the gambler should take the other side of the bet.

Astute readers may note that the above formula applies to cases of binary outcomes and known probabilities, both of which do not occur naturally in financial markets. For example, while binary exposures can somewhat be found in traditional assets (think stock exposure to litigation outcomes, merger & acquisitions, distressed companies, biotech FDA approvals, and tight option spreads), the payoff of all of these assets are only partially correlated with the event, and all involves a high degree of uncertainty. More importantly, however, is the fact that very rarely are the true probabilities of events actually known; there are only the market-implied / tradable probabilities, and our own “subjective” probabilities.

For the first point, we note that the use of event contracts allows for the most direct exposure to any particular event. For the second point, we note that the Bayesian interpretation of probability allows us to blend two distributions together in a reasonable way.

Bayesian Updating of Posterior Odds

Say you are considering an event contract on your favorite team winning against some other team because it’d make watching the game more entertaining you’ve done your research and built sophisticated betting models which has generates statistically significant alpha. In fact, given information I from your model (and without looking at the market) you expect a probability q of your team winning (call this outcome O) but the market odds are currently pricing a probability of p < q. How should we incorporate our side information into the market? We consider Bayes' theorem, which shows us how to update the fair probability of an outcome O given new information I

\(P(O | I) = \frac{P(I | O) P(O)}{P(I)}\)

\( P(I) = P(I | O) P(O) + P(I | \neg O) P(\neg O). \)

The prior fair fractional odds of the outcome given by the market are

\(\text{Odds}(O) = \frac{P(O)}{P(\neg O)}\)

Doing some algebra, we see that Bayes' theorem gives us an elegant way of updating the posterior odds (probability of O given I over probability of not O given I) using the likelihood ratio (probability of I given O over probability of I given not O) and the prior odds (probability of O over probability of not O)

\(\ln (\text{Posterior Odds} )= \ln (\text{Prior Odds} )+ \ln (\text{likelihood ratio})\)

The likelihood ratio gives a way to evaluate the strength of the information, in particular we note that it is the difference between the posterior and prior odds in log space. The transformation we have done into log odds space means that the impact of the same amount of evidence gets the correct nonlinear adjustment. E.g. if we initially have a 1% chance or a 50% or a 99% chance of an event occurring, an extra bit of information of that event will correctly take us to 2% or 66% or 99.5%.

This should be intuitive if we consider the example of for evaluating the fairness of a coin after observing n flips. Let’s say that we only observe heads in our n flips, then the likelihood ratio is simply 1/(2^n) which decreases quickly with n. I.e. when n is small we don’t really need to adjust our prior, but we do when n gets larger.

A somewhat counter-intuitive framing of this is that deviation of the best guess probability we have from our own private information from the market implied probability doesn’t really affect how much we should adjust our posterior guess; instead, we only care about how likely it was for us to observe our private information if the market implied distribution was correct. E.g. observing n = 1 and n = 10 heads but gives us a “best guess” of the probability of heads being 1, but clearly we we should adjust more on 10 observations versus 1 (in fact, we should probably adjust more on 10000 observations of 60% heads than 1 observation of 100% heads).

A similar reasoning applies in markets, where the deviation of our own fair value estimate from the market price isn’t the key factor in determining how much we should adjust our view—instead, what matters much more is how likely it was that our private signal hasn’t already been incorporated into the price / how likely our private signal is given the price. There is obviously no rigorous way to determine this, so in practice we may consider heuristics i.e. we think about why others haven’t traded on a perceived edge and why we are the people in the best position to take advantage of said edge; if we can’t give a reason, then perhaps we shouldn’t trade.

Invariance of Kelly Sizing to New Information

The logit function transforms probabilities to log odds and the logistic function turns log odds into probabilities; we define them as follows

\(x = \lambda(p) = \log \frac{p}{1-p} \ \ \ \ \ p = \sigma(x) = \frac{1}{1+e^{-x}}\)

We now go back to the example of our event contract on the game; we’ve made our bet according to rational Kelly principles, and now the game is on. Assume that our initial price / probability prior to the arrival of new information (the game being played) is p₀, and the odds offered is b. We have positive side information in the form of a likelihood ratio greater than 1, which we use to size our initial bet of size

\(f^*_0 = \frac{\sigma(\lambda(\frac{p_0}{b}) + \log(\ell_0)) \times\frac{b}{p_0}-1}{\frac{b}{p_0}-1} = \frac{p_0(\ell_0-1)}{p_0(\ell_0-1)+b}\)

Now, new information arrives (i.e. the game is being played) and the market adjusts its implied probability with a price moving from p₀ to p₁ in real time. The position, as a fraction of our total wealth, is now

\(f_1 = \frac{f_0^* \frac{p_1}{p_0}} {(1-f_0^*)+f_0^* \frac{p_1}{p_0}} = \frac{ p_1 (\ell_0-1) }{p_1(\ell_0-1)+b}\)

But, per previous reasoning, this is precisely the fraction we’d want if we bet our Kelly sizing (assuming that our likelihood ratio is the same over time). Indeed, our optimal Kelly fraction after the price change naturally emerges if we had bet our optimal Kelly fraction before the price change, and in fact we should do nothing to change the size of this position (no doubling down, no selling, just hold). In other words, the Kelly sizing is invariant to the arrival of new information, so long as that new information does not change the original piece of side information that caused us to put on the bet.

This should be an intuitive result if we think of our bet as the optimal infusion of information into market prices; any positive expectancy was made at the time of trade, and so unless we see new private information unincorporated into the market, the correct thing to do is nothing.

Now, our analysis above assumes that our likelihood ratio doesn’t change over time; but if our initial private information was created under uncertainty then surely it should itself be subject to an updating process with new public information. If the market price has increased then this is confirmatory evidence on our bullish side information, and perhaps we should increase our likelihood ratio since we are now more confident that our side information was correct. This means that we should in fact buy more as the price increases! A similar argument shows that we should sell when the price decreases i.e. the optimal strategy when our private information is subject to uncertainty is to momentum trade price moments (maybe?).

Information Cascades

The discerning reader may note that while it may be “rational” for any one individual to momentum trade the contract, it is problematic when everyone does so.

In particular, consider a group of market participants acting sequentially. A person's signal telling them to buy is denoted as H (H for higher price) and a signal telling them to sell is L (L for lower price). The model assumes that when the correct decision is to buy, individuals will be more likely to see an H, and vice versa.

The first agent determines whether to buy or sell solely based on their own signal. However, the second agent then considers both the first agent's decision (observe via price impact) and his own signal, again in a rational fashion. In general, the nth agent considers the decisions of the previous n-1 agents, and his own signal. He makes a decision based on Bayesian reasoning to determine the most rational choice

\({\displaystyle P(A|{\text{Previous}},{\text{Personal signal}})={\frac {pq^{a}(1-q)^{b}}{pq^{a}(1-q)^{b}+(1-p)(1-q)^{a}q^{b}}}}\)

Where a is the number of buys in the previous set plus the agent's own signal, and b is the number of sells (a + b = n). WLOG, as more traders observe buys, the likelihood of buying grows non-linearly, eventually overpowering any private information.

As such, while it may be “rational” for any one individual to momentum trade the contract by adjusting the likelihood ratio of their private information so as to trade with the direction of the price, the result of many individuals doing this may be a price that is fundamentally dislocated from anyone’s original fair estimate. This phenomenon describes an information cascade, where individuals make decisions based on the actions of those who came before them, despite their own private information, leading to a chain reaction of similar choices.

In contrast to herd behavior, information cascades can occur even when we assume rationality. Specifically, the model above assumes: (1) Bounded rationality, where agents will always make rational decisions based on the information they can observe, but the information they observe may not be complete or correct; (2) Incomplete information, where agents do not have complete knowledge of the world around them (which would allow them to make the correct decision in any and all situations), and in particular do not have knowledge of the private information of agents which precede them; i.e. the current agent makes a decision based only on observable action; (3) behavior of all previous agents is known by the current agent.

From the above assumptions, one can show a few resulting conditions: (1) Cascades will always occur; in the simple model, the likelihood of a cascade occurring increases towards 1 as the number of people making decisions increases towards infinity. (2) Cascades can be incorrect – because agents make decisions with both bounded rationality and probabilistic knowledge of the initial truth, the incorrect behavior may cascade through the system. (3) Cascades can be based on little information – mathematically, a cascade of an infinite length can occur based only on the decision of a single person. (4) Cascades are fragile – agents with weak probabilistic knowledge considering opinions from other agents who are making decisions based on actual information can be dissuaded from a choice rather easily.

The concept applies more generally than our model of trading above; information cascades occur generally in real-world contexts where individuals make decisions sequentially, observing the choices of others rather than relying solely on their own private information. In financial markets, they contribute to speculative bubbles and price volatility as traders follow prior market movements rather than independent analysis. This phenomenon can also be seen in product adoption (i.e. “real” markets), where consumers purchase new technologies or brands based on observed popularity (social media influencers) rather than intrinsic quality. Another example is that in real estate, sellers may strategically price homes higher initially to avoid triggering negative cascades that could signal low quality. Indeed, if you are vacationing in a foreign city, and observe two restaurants where one is crowded and one is not, you may be tempted to go into the crowded one (if you can get a table).

Value and Momentum

In our original trading model, we assume that as prices move, we update our likelihood ratio in the same direction, and leading to momentum trading. However, a “value” investor who is much more sure about their fundamental predictions may, at some point, update their likelihood ratio in the opposite direction, interpreting extreme price moves not as confirmation that their fair value estimate is wrong, but rather as evidence of irrational market forces such as information cascades, liquidity squeezes, or forced liquidations.

From this perspective, if prices deviate too far from fundamental fair value, the very nature of how they got there becomes suspect—such extreme moves are unlikely to be the result of purely rational adjustments to fundamentals but instead reflect market inefficiencies. This reasoning leads the “value” trader to double down when prices move against them at the extremes, as they perceive an increasing discrepancy between price and intrinsic value. Unlike the “momentum” trader, who follows the direction of price movement as confirmatory evidence, the value trader assumes that the larger the deviation, the more likely it is driven by non-fundamental distortions—prompting them to increase their position rather than capitulate.

This divergence in belief updating between “momentum” and “value” traders relates directly to the well-documented momentum and value factors observed in financial markets. In general, momentum traders interpret price moves as confirmatory signals, leading them to buy assets that are rising and sell those that are falling. Conversely, investors who trade against large deviations from intrinsic value are exposed to the value factor. As prices move far from their estimated fair value, value investors update their beliefs in the opposite direction, interpreting extreme price dislocations as evidence that non-fundamental factors are at play.

As a side note, the value and momentum factors are NOT captured by typical value / momentum ETFs or mutual funds, as by definition the factors must be constructed such that they are orthogonal to systematic market risk. I.e. the factors are constructed such that they are long the top x% of stocks by some proprietary value / momentum measurement and short the bottom x% in a market-neutral way. Moreover, the factors are themselves not directly tradable since they don’t take into account trading costs; both value and momentum experiences high turnover and empirically a significant amount (though not all) of the factor premia disappears after trading and management costs. Finally, factor premiums have decreased in the current millennium as they’ve become more crowded and well understood.

In times of strong trend persistence, momentum strategies tend to dominate; however, when trends become overextended, value tends to outperform. The intuition provided by our simple model also helps to explain the different empirical skews of the value and momentum factors and their negative correlation. Indeed, momentum trades tend to crash downwards (often very sharply and very violently) after prolonged trends; while value factor underperformance can persist for long periods and then crash upwards when cheap stocks get bought and expensive ones get sold. The two factors should be negative correlated and persist across all markets - we observe this empirically; in the paper Value and Momentum Everywhere form AQR

We find consistent value and momentum return premia across eight diverse markets and asset classes, and a strong common factor structure among their returns. Value and momentum … are negatively correlated with each other, both within and across asset classes. Our results indicate the presence of common global risks that we characterize with a three-factor model. Global funding liquidity risk is a partial source of these patterns, which are identifiable only when examining value and momentum jointly across markets.

Intuitively, I think many investors readily accept the existence of value and momentum risk factors in equities; however, when it comes to currencies, interest rates, and commodities, many struggle to internalize why these same factors should exist (what do you mean we should buy more yen because it’s been going up all year?)

Our example with event contracts helps clarify why value and momentum can exist everywhere; i.e. in general if traders are boundedly rational and update their beliefs imperfectly, we can get momentum effects where traders chase trends based on past price movements, reinforcing existing price action. At the same time, extreme price moves can trigger value-based reversions, where traders recognize that mispricings have become too large to be justified by fundamentals alone, leading to a correction.

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