Small Losses, Big Gains: The Complete Convex Investment Path for Ordinary People

"Zhou Yuan" in this article is a fictional character, and all the funds, transactions and company cases related to him are set up for illustrative purposes; the reference materials on which the real market events are based are uniformly listed at the end of the article. This article only discusses investment methods and does not constitute investment advice on any securities, options or crypto assets.

Preface: A 23% Drop and a Narrow Path

Last Tuesday, July 14, IBM handed over a less-than-impressive performance report to the market ahead of schedule.

The company expected adjusted earnings per share in the second quarter to be $2.93, with revenue of approximately $17.2 billion, both below the general expectations of Wall Street earlier. After the news was announced, IBM's stock price once fell by about 23% in early trading. The financial report officially disclosed on July 22 showed that the company's actual revenue for the quarter was $17.162 billion, and the problems that the market had previously worried about were coming true.

A century-old large company lost nearly a quarter of its market value in a single day. What caught the market off guard was not how many customers IBM had lost, but that the growth, order and AI transformation expectations that investors had originally believed in were re-priced by a shareholder letter.

That morning, Zhou Yuan came across this news while scrolling through his phone on the subway to work.

He was thirty-three years old, working as a product manager at a tech company, with a decent income and over 400,000 yuan in savings. Over the past few years, he had bought index funds, made regular investments in tech stocks, and made some money when the market was good. But when he looked at his account as a whole, he clearly understood that relying only on salary, savings and steady returns of a few percentage points each year, it was difficult to truly move up the social ladder.

Assuming he could save 120,000 yuan each year without considering investment returns, it would take more than 20 years to accumulate from 400,000 yuan to 3 million yuan. Compound interest can shorten the time, but the premise of compound interest still lies in the principal, rate of return and long enough waiting period. For ordinary people with limited principal, this path has a clear speed limit.

Zhou Yuan stared at the IBM news for a long time. That 23% drop made him see a completely different return curve.

If someone holds IBM stock, they mainly bear the linear change of the stock price. If the stock falls by 10%, the position will roughly lose 10%; if the stock falls by 23%, the position will roughly lose 23%. If you buy put options at a low cost in advance, when the underlying asset drops far beyond market expectations, the value of the options may rise several times or even more.

Of course, if IBM does not fall, or if the drop happens too late, that option may also go to zero. That's exactly how the former Byte employee who got rich from stock trading a while ago played it.

The same piece of news, two tools, two different curves.

One curve is characterized by the rise and fall following the proportion of principal invested; the other curve is characterized by pre-defined losses, while returns may accelerate several times or even more with market movements. This structure of asymmetric returns and losses is called convexity.

Zhou Yuan realized that although moving up the social ladder cannot be guaranteed by any investment method, for people with limited principal whose income mainly comes from salary, if the account only ever has linear returns, it will be difficult to achieve their goals. The significance of convexity investment is to use a small portion of affordable cost to preserve the possibility of non-linear growth for the account, which can be regarded as a small crack for ordinary people to move up the social ladder through investment.

The quality of life cannot always be bet on the right tail, but an account that changes the slope of wealth cannot be without the right tail either. This is the origin of Zhou Yuan's research on convexity investment.

01 Nine Profitable Trades Could Lose to Nine Losing Trades

A few days later, Zhou Yuan shared his confusion with a friend engaged in quantitative trading, who wrote two types of games on a piece of paper for Zhou Yuan.

The first type: every time you play, there is a 90% chance to earn 1 yuan, but a 10% chance to lose 20 yuan.

The second type: every time you play, there is a 90% chance to lose 1 yuan, but a 10% chance to earn 20 yuan.

Zhou Yuan almost did not hesitate and chose the first one first. A winning rate as high as 90% means you can make money most of the time; the second one requires facing losses frequently, and emotions will definitely fluctuate greatly, which seems unreliable no matter how you look at it.

His friend added a line of formula on the paper: Expected value = Winning rate × Average profit - Losing rate × Average loss

The expected value of the first type is: 90% × 1 - 10% × 20 = -1.1 yuan.

The expected value of the second type is: 10% × 20 - 90% × 1 = 1.1 yuan.

The first game can make people feel good for many consecutive times, but a few losses will take away all the profits; the second game does not look good most of the time, but there is a chance to use one profit to cover all the previous losses.

The indicator most likely to create illusions in investment is the winning rate.

Before 2018, a product with the code name XIV was popular on Wall Street. In essence, it was shorting short-term volatility. When the market was calm, holders could continuously obtain returns, and the account curve looked stable and beautiful. The real structure of this strategy is to exchange frequent small profits for one potential huge loss.

On February 5, 2018, volatility suddenly soared. The indicative closing value of XIV on the previous trading day was still $108.2681, but the final redemption price was only $5.99. Many small profits accumulated in the past were swallowed up by a tail event in a very short period of time.

This is negative convexity: it is easy to make money in normal times, but once a black swan comes, all previous profits together with the principal may be returned to the market.

Many seemingly robust strategies have similar structures. Using high leverage to collect capital fees, selling options in markets with insufficient liquidity, taking credit risks for a few basis points of interest, or continuously adding positions to reduce costs over a long period of time, these strategies may work most of the time, but once a black swan event occurs, the loss may far exceed the long-term accumulated returns, and even trigger margin calls or the principal going to zero.

The second game is not as easy as it seems on paper either.

If a transaction only has a 10% probability of generating large returns, the probability of losing ten times in a row is 0.9 to the tenth power, which is about 34.9%. Even if the expected value of this strategy is positive, investors still have a probability of more than one-third that the first ten attempts will end in losses.

This means that investing in convexity cannot only focus on the "odds". The account must also survive until the day the right tail appears.

If Zhou Yuan bets half of his principal on the second game, as long as he makes two consecutive mistakes, he will be close to bankruptcy in reality.

Therefore, there are two iron laws for investing in convexity: the loss of a single transaction must be small, and before large returns arrive, one must have the ability to repeat attempts. A big win is not a reward for courage, but the result of retaining many opportunities for asymmetric odds that finally come to fruition.

02 Searching for Tenfold Opportunities but Falling into the "Convexity Illusion" First

After understanding the formula, the first thing Zhou Yuan did was to create a "tenfold candidate list".

The list includes cheap out-of-the-money options, small-cap AI companies, newly listed cutting-edge tech enterprises, and Web3 tokens recommended by friends. They all have one thing in common: the story is grand enough, and price fluctuations are violent enough. As long as you bet on the right one, the book return may be astonishing.

He marked all these underlying assets as "convexity opportunities". Until a week later, he began to doubt this list.

On July 13, 2026, General Fusion landed on NASDAQ through a reverse merger, becoming the first publicly listed nuclear fusion company. On the first trading day, its stock price once rose by about 40% from $12.85. The imagination space for nuclear fusion has almost no upper limit. Once technological and commercial breakthroughs are achieved, the valuation will skyrocket.

The report also mentioned that the company had previously faced financing pressure, and new technological milestones might not be available until 2028 or later. The right tail of nuclear fusion may be huge, but before the right tail arrives, the company still has to face practical problems such as R&D investment, financing, dilution, and technological failure.

Zhou Yuan found that a story with huge imagination space does not mean that buying stocks naturally comes with good convexity.

A company's stock price may rise tenfold, but it may also raise funds continuously before the tenfold story comes true, diluting the equity of original shareholders; a small-cap token may rise a hundredfold, but it may also quickly go to zero due to liquidity depletion, team selling, or contract loopholes; although the loss of an option is only the premium, if the probability has been fully priced by the implied volatility, it may still be a transaction with poor odds.

Convexity is a return structure, not buying an underlying asset that sounds exciting with a good story.

Small market capitalization is not convexity, volatility is not, and leverage is not either. Alpha and convexity are not the same thing either.

Alpha is "market mispricing", while convexity is "if the judgment is correct, the return can be huge; if the judgment is wrong, the loss can be limited". A fairly priced option has convexity value, but may not necessarily be Alpha; a severely undervalued traditional company may be Alpha, but its return structure is not convex.

Zhou Yuan re-examined the candidate list and gradually divided the sources of convexity into several categories.

Contract structure: Buying options, warrants or credit default swaps, the loss is mainly the cost paid in advance, while the return may accelerate with the change of the underlying asset.

Enterprise operation: A company has basically completed its fixed cost investment, the marginal cost of new revenue is very low, a 20% increase in revenue may lead to a doubling or even more of profits.

Event revaluation: New drug approval, major lawsuit victory, debt restructuring, asset sale or regulatory changes may allow a value originally ignored by the market to be incorporated into the price in a short period of time.

Network effect and reflexivity: User growth brings more developers, more developers bring better products, and better products in turn attract more users. This flywheel reinforcement may occur in platforms, payment networks and some Web3 protocols.

Zhou Yuan found that many items on the list can only answer "there is huge space in the future", but cannot answer "how the value of the held assets will rise". Although some projects may rise tenfold, there is no clear bottom line for their decline, and "small loss" is just investors' wishful thinking.

Only when the cost of failure is affordable, the path to success is explainable, and there may be deviations in market pricing, does it make sense for convexity investment.

03 Michael Burry Did Not Short the Real Estate Market, But Found the Market Settlement Time

After the candidate list was shortened, Zhou Yuan encountered a new problem.

He knew what to look for, but he did not know how a convexity clue could move from investment research reports to real prices. Many judgments eventually proved to be correct, but before they were realized, people might wait too long, pay too high a cost, or exit the position due to adverse market fluctuations.

He dug out the previous story of Michael Burry shorting the US subprime mortgage.

The protagonist of the story, Burry, saw the real estate bubble earlier than others, and also found the possible time for the bubble to burst.

When Burry studied the underlying loans, he found that more and more mortgages were issued to borrowers with insufficient income and credit. The rise in house prices was clearly not supported by wage growth, and a large number of floating-rate loans might be repriced in two years. Because after the initial preferential interest rate expires, the significantly increased monthly payment will lead to many credit defaults.

According to the report of the US Financial Crisis Inquiry Commission, by mid-2005, Burry had established short exposures to billions of dollars in mortgage-backed securities and related financial company bonds through credit default swaps. He paid relatively limited and continuously occurring premiums in exchange for huge potential returns when the real estate credit system collapsed.

The complete logic of this transaction is a continuous transmission chain:

The continuous decline of lending standards is the accumulated fuel.

The reset of two-year floating interest rates is a catalyst whose timing can be estimated.

The rise in borrower overdue and default rates is the ignition condition for risks to enter the cash flow.

Mortgage-backed securities, CDOs and financial institution leverage will amplify losses layer by layer.

The widening of credit spreads, the impairment of securities and the losses of financial institutions are the confirmation signals that the logic begins to be realized.

If residents' income rises significantly, house prices continue to rise rapidly, or refinancing channels are always unobstructed, Burry's judgment may fail.

Rather than predicting the future, Burry was looking for a settlement process that would eventually be verified by cash flow. This is the most easily missing link for many ordinary investors when studying convexity.

A company has a large market space, but it does not know when the next order will come; a certain technology may change the world, but it does not know how much more money will be burned for commercialization; a stock is undervalued, but it does not know what force will prompt other investors to reprice it. The fact of undervaluation exists, but the market price has no obligation to recognize the fact immediately.

According to Michael Burry's story, a tradable convexity clue usually goes through five stages.

The first is the Hypothesis Stage: Investors see possible mispricing, but the evidence is not yet complete. The odds are the best at this stage, but the probability of failure is also the highest.

Then comes the Hard Evidence Stage: Orders, users, revenue, regulatory documents, clinical data or industrial chain changes begin to support the judgment. The upside space starts to narrow slightly, but the probability of success increases significantly.

Next is the Ignition Stage: Financial reports, approvals, policies, product launches or cash flow changes make the logic that was previously ignored by no one enter the vision of more people in the market.

Then comes the Diffusion Stage: Analysts raise expectations, institutions increase positions, short sellers cover, option Gamma and liquidity jointly drive the market to accelerate. At this time, it is easiest to make large profits, and it is also easiest to mistake the realized returns for the remaining convexity that still exists.

Finally, the Odds Decay Stage: The story is widely spread, and the price has incorporated a lot of optimistic expectations. Even if the fundamentals continue to improve, the remaining returns may no longer be sufficient to compensate for the risks.

The enlightenment of Burry is not to let everyone look for the next crisis, but to understand that truly sustainable convexity investment must find facts, timing and investment tools at the same time.

It is not enough to just find a potentially correct conclusion. You also need to know what catalyst will force the market to recognize it, when it will be recognized, and whether your position can survive until that day.

04 Convexity Is Not Only Hidden in Options, But Also in Profit Statements and Transaction Terms

After studying Burry, Zhou Yuan was overly obsessed with options for a period of time. He felt that only when the maximum loss was limited to the premium and the return could grow several times, could it be worthy of the name "convexity".

Until a gathering of old colleagues, he shifted his sight from options back to the company itself.

A friend working at a software company mentioned that the company had been investing in R&D and sales teams for the past three years, with high expenses but insignificant profits. Recently, the product has gradually matured, and there is no need to add more R&D personnel, but customer growth has not stagnated. If the renewal rate and average revenue per customer continue to increase next year, revenue growth may soon be converted into profits.

To figure out the accounts clearly, Zhou Yuan assumed that his friend's company had an annual revenue of 100 million yuan, a gross profit margin of 80%, and a gross profit of 80 million yuan. The total expenses of R&D, sales and management were 70 million yuan, and the operating profit was only 10 million yuan.

Assuming that the revenue increases by 50% in the second year to 150 million yuan, the gross profit will correspondingly increase