This is a long semi-technical 🧵  on Tail Risk Hedging. It's unusual, in the sense that I am posting it as I am trying to figure stuff out out of frustration. I find OPP (other people's posts) about Tail Risk Hedging a bit vague or confusing (exception: @bennpeifert).

1/ These are some short on tail risk hedging (TRH). They are an attempt to understand, for an outsiders' point of view, how tail risk hedging, could be of value. In other terms, I am trying to "steelman" to the best of my understanding.

2/

Let X[i] iid random variables. These are the PnL of the units (PMs, stocks, ur mom, whatevah). And N > 0 a rv taking integer values, independent of X[i]. So you have total PnL equal to

Y = Sum_{i=1}^N X[i]

E(Y) = E(N)*E(X[1])

Just condition on N
1/

https://twitter.com/__paleologo/status/1918017112724971561

And Var(Y) is given by the Blackwell-Girshick formula () which is another conditioning argument.

2/en.m.wikipedia.org/wiki/Blackwell…

I realized that my book is three weeks away from publication and I have down zilch in the past five months to promote it. So read on. A 🧵.

The first thing you have to know about my book is that it will make all your dreams come true.
1/ The second thing to know is that it is damn expensive. But hey, it's already discounted and it has not even been released! What the heck Wiley. Your pricing algo is a mystery.
2/

A few days ago, I gave a talk at Columbia's conference on "The Future of Portfolio Management". Title: "Simpler (portfolio) heuristics that make us smarter". Below is the link to the slides. A short intro.
1/ While I was writing my book, I discarded a lot of stuff that was too cute, too hard, to simple, etc. This talk is some of that discarded material, at the intersection of cute and simple.
2/

"The Unintended Consequences of Rebalancing" by Harvey, Mazzoleni, and Melone. This paper has even a press release. A couple of notes. (HT: Joe M. Link at the end)
1/ First, this is the quintessential "outsider" paper: academics (no slight intended) "looking into" a strategy run by traders. That's ok, although it is a few decades late and not very precise. This paper is the equivalent of seasons 7&8 of Game of Thrones.
2/

So much chatter about index rebalance this week. In the words of an old colleague, “many [candidate] portfolio managers can talk a great game about index rebal, but very few can do it well” (I know you are reading this). A short 🧵
1/ Rather than talking a good game myself (could/won’t), here is a curated list of references. But first: it’s really hard to find good material on the web! Everyone talks a *bad* about index rebal.
2/

Basic hedge fund economics 🧵. N.B.:
1. for educational purposes only and I will keep it super-simple
2. ofc, personal views
3. No soliciting! I am not coming to defend hedge funds, just to explain some basics. And I am not going to answer replies on the Current Thing.

1/ Yet (on Current Thing) if you read an article about HFs (*anything*, really), you should be equipped with some sound mental model to evaluate it and critique it in your head. I hope this helps.
2/

A short 🧵 about 📉s and #️⃣ ratios.

People say you can't eat Sharpe. But losses can eat you, and to a first approximation, Sharpe can help control them. But what is the link between losses and Sharpe? Read on.

0/ Say that your strategy has annualized Sharpe Ratio s and normal returns. you deploy $sigma dollar of volatility. At the end of the year, you would like your PnL to be above -L*sigma with high probability, say 1-p. What is the condition that s should satisfy?

"Sunday morning brings the dawn in/ It's just a restless feeling by my side" according to the Velvet Underground.

According to me, it's time to compare LLM models in financial applications. A simple problem, with comments, solved (with my comments) in:

o1
o1-mini
o3
4o
3 Sonnet
DeepSeek R1

A semi-long 🧵 1/n It was inspired by this straightforward question on Kelly betting.

A gambler has a 10% edge (60% win) and starts with $100 in a game of winning the amount risked or losing it. He uses the Kelly fraction to bet. What's the probability (3% error) that in the course of playing 400 games the equity will drop to $10 (stopped out)?

You should try to solve it yourself. Code it, or have it coded by an LLM and check code's correctness. That is not the interesting part to me. Here is the source: x.com/mikeharrisNY/s…

2/n

A meandering 🧵 on heavy tails.

I usually recommend "Fundamentals of Heavy Tails" by Nair, @AdamWierman and @BertZwart1 as an introduction to Heavy Tailed phenomena:

1/8 However, for an even more concise (free) introduction, Newman is very good.

2/8

arxiv.org/abs/cond-mat/0…

As a retail investor (we all are) this piqued my interest. Maybe I should invest in Oakmark rather than my vanilla tax-efficient SP500? A short 🧵. N.B.: this is not investment advice. It's just to show how tools empower a retail investor to circumvent issues that would have made their life harder, just five years ago. 1/ If you search Bill Nygren, you'll find that he's managed Oakmark Select since 1996. Another search, and you land on . Below is the chart. Problem: the are no returns. 2/ oakmark.com/our-funds/oakm…

CUSUM is an algorithm for change-point detection. It's kind of magical. Here is a financial application, which is very important in itself, to showcase its magic.

A short informal 🧵, to be rewritten in a more formal technical note. Ready, let's go.

1/ Imagine you have a strategy a stream of daily (or, better, intraday) returns r[t].

One of the very basic problems in life is to know when your strategy's performance is degrading. Solving this can make or break you.

2/

I have put some more thought in the math of "optimal" diversification, aka "Dan's math" (as per @tracyalloway; which sounds a bit like "Lorenzo's oil", minus Susan Sarandon). A short 🧵. (1/n)

https://twitter.com/__paleologo/status/1843277776364691501

Let's recap the common wisdom:
- diversification improves Sharpe like sqrt(n) for uncorrelated strategies
- but for pairwise correlation rho *and identical sharpes* the aggregate Sharpe peaks at (single strategy sharpe)/sqrt(rho)
- so there's a max number of strategies you want to run.
In Dan's own words [verbatim]:

"If the correlation was exactly zero, then the more people you add, essentially the more your Sharpe ratio increases — it increases with the square root of n essentially. If there's correlation, however, there's like a maximum limit of how much your aggregated Sharpe ratio can be, right?"

And that's what the consensus believes.

(2/n)

I ran a mini-test for ChatGPT o-1. Since yesterday I said that Jensen's inequality underlies so much of everything (), I asked a questions that boils down to Jensen's. A 🧵. 1/

https://x.com/__paleologo/status/1834359510762197442

Here is the question. I remember reading something like this on X. It's a suitable question for a quant interview. 2/

I just listened to @NateSilver538 's interview by @tylercowen . I was reminded of some podcasts on rational decision making that stayed with me. Also, I am reading about retail trading, and it's related. What follows is a very incomplete list.. 🧵 (1/8) OK, Silver & Cohen. Was fun. (2/8)open.spotify.com/episode/3IMBHo…

There's an article by @jasonzweigwsj on "Why Your Fund Manager Can’t Beat Today’s Stock Market" that I find interesting because it poses important questions (but I question the analysis). So a longish🧵👇 (1/n) wsj.com/finance/invest… The question: The CBOE DSPX Dispersion Index is up and the 3MCBOE implied Correlation is down. The intuition is that, if stock returns are less correlated, there are more independent bets to be made, and if you have skill, more money to be made too. (2/n)

A few thousand people have started following this account, so I thought I'd provide a quick intro.

My name is Giuseppe Paleologo. I effectively exist on twitter thanks to this tweet by @ByrneHobart, which outed me in sort of a funny way. (1/4)

https://x.com/ByrneHobart/status/1745149539676217358

So I am non-compete now. LinkedIn profile notwithstanding, I have not been forcibly unemployed that long for hedge fund standards.
And I happily worked for a long time in the math dept. of IBM Research.

All my recent tweeting is an interstitial activity in between jobs. (2/4)

There is a paper that I would strongly recommend: "Optimal Turnover, Liquidity, and Autocorrelation" by Baldacci, Benveniste, and Ritter.  🧵 (1/5) papers.ssrn.com/sol3/papers.cf… I will post more about the paper, because a) there is a lot to unpack in terms of applications and interpretation; b) the proof is a bit elliptic, and so I will expand it.
Meanwhile, take a read. (2./5)

TL;DR: