Thetabruv

Edge

Mon, 01 Jan 0001 00:00:00 +0000

Edge#

Edge is an overused word, so let me start from the definition I work with: an edge is a positive expectation that survives costs, risk and competition, and whose source you can explain. The last clause is the most important: a positive expectation whose origin you don’t know is indistinguishable from luck, and above all you can’t know when it stops existing — whoever makes money without knowing why will sooner or later lose money without knowing why. This page answers three questions in sequence: where the volatility selling edge comes from; why the professionals leave it to us; how to verify that it’s still alive.

Futures

Mon, 01 Jan 0001 00:00:00 +0000

Futures#

The future is the simplest derivative there is: a mutual commitment to exchange the underlying (or its cash value) at a future date, at a price fixed today. No premium to pay, no choice to exercise: just a symmetric obligation. Whoever is long gains if the price rises and loses if it falls; whoever is short, the exact opposite, dollar for dollar. It is the linear instrument of this site’s strategies, and it plays two roles: in the DHCS strategy (DHCS page) it is the delta-hedging instrument; in the TRPS (TRPS page) it is the night sentinel — the hook on which the bot’s overnight watch is armed and, in the last resort, plan B for emergencies — being the only index derivative that trades almost 24 hours a day.

Risk measures

Mon, 01 Jan 0001 00:00:00 +0000

Risk measures#

There’s an uncomfortable fact at the center of this page: the world’s most widely used risk metrics — volatility, Sharpe ratio, Information Ratio — are systematically generous to volatility selling strategies. Not slightly generous: spectacularly so. A well-built short vol strategy posts numbers no traditional manager can dream of, and those numbers are simultaneously true and misleading. Understanding why is the prerequisite for not falling in love with your own track record. I’ll proceed in layers: first the dispersion metrics, then the tail metrics, finally the operational ones.

Setup

Mon, 01 Jan 0001 00:00:00 +0000

Setup (broker, margin, products)#

This page is the system’s bill of materials: what you need, why exactly that, and where the breaking points are. I split it into three layers: the account at the broker, the hardware at home, and the software running on top. Fair warning: it’s the page that will age faster than the rest of the site — the models and prices quoted are valid today — but the selection criteria will hold.

Start here

Mon, 01 Jan 0001 00:00:00 +0000

START HERE#

Welcome to Thetabruv.com.

I use this space to share the theoretical and practical foundations of a few strategies for capturing the volatility risk premium (VRP). I became interested in the subject in 2019, after a decade or so of passive investing, when I started asking myself whether a retail investor could go beyond building a portfolio of ETFs.

Over the years I collected quite a lot of material, and I decided to structure it into this site — for myself, and for the friends and acquaintances I end up discussing it with. It is therefore a living repository rather than a finished work: nobody should be surprised by corrections and re-editions.

Options

Mon, 01 Jan 0001 00:00:00 +0000

Options#

If the future is an obligation, the option is a right. Whoever buys a call acquires the right, not the obligation, to buy the underlying at a predetermined price (the strike) by or on a certain date (the expiry). Whoever buys a put acquires the right to sell on the same terms. Whoever sells the option (the writer) collects a premium upfront and takes on the mirror-image obligation: to suffer exercise whenever it suits the buyer. The asymmetry is all here: the buyer can lose at most the premium, the seller can lose far more. This page builds the minimum vocabulary needed to handle that asymmetry; the next one (Volatility risk premium) will explain why, on average, it is paid more than it should be.

Tail risk

Mon, 01 Jan 0001 00:00:00 +0000

Tail risk#

On October 19, 1987, the S&P 500 lost 20.5% in a single session. With the volatility of the time, under a Gaussian assumption, that was an event of more than twenty standard deviations: something that, if returns were truly normal, shouldn’t happen even once in the life of the universe, multiplied by billions. It happened, and it is the memento mori hanging on the wall of every volatility seller — the day that created the skew (see the Options page), rewrote the models and defined, once and for all, the trade described in these pages. This page is devoted to tails: what they really look like, when and how they arrive, what makes them (partially) manageable and what doesn’t. It is the least pleasant page on the site and the most important.

Tail risk protection selling (TRPS)

Mon, 01 Jan 0001 00:00:00 +0000

Tail risk protection selling — TRPS#

The first strategy is the simplest to describe and the hardest to stick to: every day, at the end of the session, sell puts on the S&P 500 expiring the next day, so far out of the money that they almost never get touched, collecting tiny premiums multiplied by moderate leverage and by 252 repetitions a year. It’s a strategy publicly documented since 2011 by a fifteen-year retail track record — the only case I know of that survived 2018, 2020, 2022 and 2025; the sources are on the Resources page — and one I call tail risk protection selling: selling other people protection from the tails. I’ll cover its architecture, the parameters with plausible ranges, the real numbers and the weak points, in that order.

The TRPS bot

Mon, 01 Jan 0001 00:00:00 +0000

The TRPS bot (Tail Risk Protection Selling)#

The TRPS bot is conceptually simple: the strategy on the TRPS page is already a list of rules with times attached, and the code does nothing but execute it. The hard part isn’t the trading logic — a hundred lines or so — but everything around it: reconciliation, handling broker errors, behavior in degenerate cases. I’ll describe it the way the bot lives it: a state machine moving through the New York day, where every transition has explicit preconditions and every failure has a safe destination (Principle 2 on the Execution page). Times are in the America/New_York timezone, naturally, and the exchange holiday calendar is the first thing the bot consults on waking: on market holidays it doesn’t transition at all, and on half-days (early closes) the evening times shift accordingly.

Delta-hedged convexity selling (DHCS)

Mon, 01 Jan 0001 00:00:00 +0000

Delta-hedged convexity selling — DHCS#

The second strategy starts from a question TRPS never asks itself: if the product for sale is convexity, why keep the delta attached to it? A sold put is a package of two exposures: a directional bet (the positive delta: you profit if the index rises) and a volatility bet (short gamma and vega, long theta: you profit if the world moves less than what was priced). The first exposure you already own, in abundance, in the collateral portfolio: that’s the ERP. The second is the reason you’re here: the VRP. Delta-hedged convexity selling surgically separates the two and keeps only the second: it sells the option and simultaneously sells futures to neutralize the delta, rebalancing the hedge every day. It’s the strategy from the Israelov and Tummala paper (Which Index Options Should You Sell?) that you’ve already met twice — for the STAR methodology on the Risk measures page and for the critique of TRPS on the TRPS page — and which here I finally assemble in full.

Ergodicity

Mon, 01 Jan 0001 00:00:00 +0000

Ergodicity#

This is both the most abstract page on the site and the one on which every concrete number depends. The question it answers sounds like a riddle: how can a bet with positive expectancy ruin almost everyone who plays it? The answer lies in a word borrowed from statistical physics — ergodicity — and in the distinction, simple and vertiginous, between the average computed across scenarios and the average computed across time.

The DHCS bot

Mon, 01 Jan 0001 00:00:00 +0000

The DHCS bot (Delta-Hedged Convexity Selling)#

If the TRPS bot lives on fifteen-minute windows, the DHCS bot has a single appointment a day — but the numbers have to balance to the decimal. The strategy from the DHCS page — front-month ATM/−1σ put, daily delta-hedging with Micro futures, monthly roll — translates into an architecture shorter and more relaxed than the one described in The TRPS bot, and the difference between the two bots is itself instructive: it replicates, at the software level, the comparison between the two strategies on the TRPS vs DHCS page. Here too, all times are in America/New_York.

Volatility risk premium

Mon, 01 Jan 0001 00:00:00 +0000

Volatility risk premium#

Here we are at the protagonist. The volatility risk premium (VRP) is the systematic gap between the volatility implied in option prices and the volatility the underlying then actually realizes. In the language of the Derivatives page: it is the distance between the Q world and the P world, measured on variance. In the insurer’s language: it is the difference between the premium collected and the average claim paid out. If it is positive and persistent, selling options is a trade with positive expectancy. This page gathers the evidence, explains the causes and — above all — makes clear exactly what the premium is the price of.

Capital efficiency

Mon, 01 Jan 0001 00:00:00 +0000

Capital efficiency#

We have established that the VRP exists (Volatility risk premium page). The next question is how much it pays, and here comes the first cold shower: in absolute terms, little. A deep OTM 1DTE put sells for about 15 cents, that is, $15 per contract. Repeated 252 times a year on a notional of roughly $730,000 (one SPX contract with the index at 7,300), that makes 15 × 252 / 730,000 ≈ 0.5% per year on notional. Half a percentage point. If collecting it required parking the entire notional in cash, the game would not be worth the candle: a T-bill pays more with no tail risk. The entire viability of volatility selling therefore hinges on a seemingly bookkeeping question: how much capital is actually needed to support the position. This is the subject of capital efficiency, and it is the point where many forum discussions go off the rails.

TRPS vs DHCS

Mon, 01 Jan 0001 00:00:00 +0000

TRPS vs DHCS#

You’ve made it to the end, and I can finally set the two strategies side by side. An honest premise: there is no winner, and anyone selling you one hasn’t understood the problem. TRPS and DHCS draw from the same source — the VRP of the Volatility risk premium page — but they collect it with almost opposite architectures, and the choice depends on which risk you’d rather carry, which routine you’re able to sustain, and what’s in the rest of your portfolio. I’ll proceed by comparisons, then give my synthesis.

Resources

Mon, 01 Jan 0001 00:00:00 +0000

RESOURCES#

This is the site’s annotated bibliography: the sources I cite throughout the pages and the material I built the site with are all in here. Where a PDF is distributed freely and legally by its authors or institutions (AQR, NBER, arXiv, university sites), I keep a local copy on this site next to the original link — papers have a bad habit of vanishing from the web. Books, videos and paywalled articles are only linked to their source. As with everything else on this site, the Disclaimers page applies: this is study material, not an invitation to trade.

Glossary

Mon, 01 Jan 0001 00:00:00 +0000

GLOSSARY#

The terms this site uses constantly, defined in a few sentences and with no claim to completeness: each entry points, where one exists, to the page that treats it properly. If a term goes missing while you read, this page is the place to come back to.

1DTE#

One day to expiration: an option with one day of life left. It’s the home turf of TRPS: tiny vega, a daily strike reset and 252 near-independent bets a year — which is what makes the seller’s statistics legitimate.

About me

Mon, 01 Jan 0001 00:00:00 +0000

ABOUT ME#

I was born in a village deep in the Italian countryside, long enough ago that my earliest memories go back to the magical nights of Totò Schillaci at Italia ‘90. I was good at maths in school, so I was told I should get an engineering degree, which I did. Once I entered the workforce, I ran into the inconvenience of having to invest the savings sitting idle in my bank account, and I made two discoveries. The first was the world of passive investing, through the books of Bogle, Malkiel and Bernstein. The second was the early FIRE movement in its most radical incarnations, Early Retirement Extreme (ERE) and Mr Money Mustache (MMM). After reaching my long-coveted FU money, I softened my positions and I am keeping my corporate job while waiting for something to happen.

Disclaimers

Mon, 01 Jan 0001 00:00:00 +0000

DISCLAIMERS#

This page exists because the rest of the site deals with instruments that can wipe out an account, and being clear about what this site is — and above all what it is not — is no formality: it is part of the same intellectual honesty I try to apply to risk.

This is not financial advice#

All content on this site is for educational and informational purposes only. It does not constitute financial advice, investment advice, a personalized recommendation, a solicitation of investment, or an offer to buy or sell any financial instrument. I am not a licensed financial advisor, and I know nothing about your financial situation, your goals or your risk tolerance: nothing you read here can take them into account.