Understanding Convexity

Written February 17, 2026

IMPORTANT NOTICE: This report is provided for educational purposes only and does not constitute investment advice, a recommendation to buy or sell any security, or a solicitation. All investments involve risk, including possible loss of principal. Past performance is not indicative of future results. Please read the full compliance disclaimer at the end of this document before proceeding.

Introduction

This report provides a mathematical explanation of convexity in investment payoffs. The goal is to illustrate how certain equity proxies can exhibit curved (convex) rather than straight-line (linear) behavior relative to their underlying assets. We use two concrete examples:

1. MicroStrategy (MSTR) vs. Bitcoin in early 2024 — a completed historical case
2. BitMine Immersion (BMNR) vs. Ethereum — a current case study

This is mathematical education about payoff structures. Our objective is to bring clients to the same level of understanding about non-linear payoff dynamics that institutional investors routinely apply in their own analysis.

Part I: Defining Convexity

Linear Relationships

Most investment proxies exhibit linear behavior. If the underlying asset rises 10%, the proxy rises approximately 10%. If the underlying falls 10%, the proxy falls approximately 10%. The relationship between the underlying asset and the proxy forms a straight line. A standard example is an S&P 500 index fund: a 5% gain in the index produces approximately a 5% gain in the fund.

Convex Relationships

A convex payoff exhibits a curved relationship. Small moves in the underlying may produce proportional or muted moves in the proxy. However, larger moves in the underlying can produce disproportionately larger moves in the proxy. The relationship forms an upward-curving line rather than a straight one.

The mathematical intuition: as the underlying asset price increases, each additional unit of increase produces a growing - rather than constant - impact on the proxy's value.

Why Convexity Matters

Convex instruments can appear disappointing or sluggish during quiet markets, then accelerate once certain thresholds are crossed. Understanding this shape helps investors:

• Avoid mistaking normal convex behavior for underperformance
• Recognize when acceleration phases may be beginning
• Size positions appropriately given the non-linear risk/reward profile
• Understand that gains in convex structures can compress into short time windows after long quiet periods — or may never materialize at all

Part II: Historical Example — MicroStrategy (MSTR) vs. Bitcoin

Background

MicroStrategy Incorporated transformed its balance sheet beginning in 2020 by accumulating Bitcoin as a treasury asset. By early 2024, MSTR had become the primary public equity proxy for Bitcoin exposure.

Observed Price Behavior

During this same period, Bitcoin rose from approximately $52,000 to $73,000 — roughly a 40% increase.

If MSTR were a linear Bitcoin proxy: A 40% Bitcoin gain should have produced approximately a 40–60% MSTR gain, accounting for modest operational leverage.

Actual MSTR behavior: A 40% Bitcoin gain produced a 155% MSTR gain — nearly 4x the underlying asset's percentage move.

What Drove the Convexity

Several factors combined to produce this non-linear outcome:

• Balance sheet optics: As Bitcoin rose, MicroStrategy's treasury value increased visibly, making per-share Bitcoin holdings the dominant valuation narrative.
• Multiple expansion: The equity market valued MSTR at a growing premium to net asset value (NAV), reflecting scarcity, liquidity, and institutional access considerations.
• Momentum and positioning: Once MSTR began accelerating, capital flows reinforced the move as investors recognized it as a primary Bitcoin equity vehicle.
• Reflexive feedback: Institutional buyers chasing Bitcoin exposure through MSTR drove the stock higher, which attracted additional buyers, creating a non-linear feedback loop.

The MSTR-to-Bitcoin relationship during this period was not a straight line. It was a curve that steepened as Bitcoin rose — classic convex behavior.

Part III: Current Case Study — BitMine Immersion (BMNR) vs. Ethereum

Portfolio Context

BMNR shares are held in client portfolios with an approximate cost basis of $40 per share. The current price as of February 16, 2026 is approximately $19. BMNR has repositioned itself as an Ethereum treasury company, holding approximately 4.33 million ETH (roughly 3.5–3.6% of total Ethereum supply) plus cash and other assets.

Current Market Conditions

As of February 17, 2026, Ethereum is trading around the $2,000 level, which technical analysts describe as a historically oversold zone. Key technical levels include the $2,000 zone as a near-term pivot, $1,900 as a recently tested capitulation band, and $1,750 as deeper support in the current correction structure.

A classic technical bottoming pattern typically involves a breakout above resistance with strong volume, followed by a pullback on lower volume that confirms the former resistance as new support, followed by continuation higher. This pattern — breakout, retest, continuation — is observed across asset classes and is based on price structure and volume analysis.

Structural Characteristics That May Produce Convex Behavior

Treasury Scaling. Every $1,000 move in Ethereum price changes BMNR's treasury value by approximately $4.3 billion. As Ethereum rises, NAV per share increases, and the gap between current stock price and marked treasury value shifts. This scaling effect is structural to any entity holding a large concentrated position in a volatile underlying asset.

Staking Revenue Growth. BMNR has approximately 2.87 million ETH staked, generating roughly $374 million annually in staking rewards at current Ethereum prices. Because rewards are denominated in ETH, nominal staking revenue rises proportionally with ETH price. At higher price levels, the business model may be evaluated less as a speculative holding and more as a cash-generating infrastructure platform, potentially allowing earnings multiples to be applied on top of treasury value.

Institutional Positioning. Q4 2025 13F filings showed significant institutional accumulation in BMNR:

These filings are public record. They indicate that professional investors with substantial resources committed significant capital to BMNR at prices within the recent trading range. We present this data factually; institutional ownership does not guarantee future price appreciation, and institutions can and do exit positions.

Potential Regulatory and Narrative Catalysts. If U.S. crypto regulation clarifies — for example, through Clarity Act passage or a framework agreement between banks and crypto companies — and Ethereum gains traction as institutional settlement infrastructure, the market may reassess BMNR's role from a volatile crypto stock to a scarce equity proxy for a meaningful share of Ethereum supply. Such a reassessment, if it occurs, could produce non-linear repricing similar to the MSTR-Bitcoin dynamic. However, regulatory clarity is not guaranteed and could take forms that are less favorable than anticipated.

Treasury Value Sensitivity

To illustrate how BMNR's treasury value changes across a range of Ethereum prices, the following table shows both upside and downside scenarios. This is arithmetic, not projection.

This table is deliberately symmetrical. Ethereum can decline further from current levels, and such a decline would reduce BMNR's treasury value accordingly. A position in BMNR carries meaningful downside exposure to Ethereum price.

Part IV: Risks and Limitations

Understanding convexity also means understanding the conditions under which it fails to express or works against a position. The following risks are material:

• Ethereum price risk: BMNR's value is heavily concentrated in a single volatile asset. Ethereum could decline substantially from current levels due to macroeconomic conditions, competitive pressure from other blockchain protocols, or loss of developer and institutional interest. A sustained decline in ETH would reduce BMNR's treasury value proportionally.
• Dormant convexity: The convex payoff curve can remain flat indefinitely. If Ethereum stagnates, regulatory clarity fails to materialize, or BMNR's execution disappoints, the acceleration phase described in this report may never occur. Owning a convex structure does not guarantee it will express.
• Liquidity and concentration risk: BMNR is a smaller-capitalization equity with concentrated institutional ownership. In a downturn, the same dynamics that can produce rapid upside repricing can also produce sharp downside moves if large holders reduce positions simultaneously.
• Regulatory risk: The regulatory environment for cryptocurrency remains uncertain. Legislation could take forms that restrict staking, impose punitive taxation, or limit the corporate treasury model that BMNR relies on.
• Dilution risk: Companies pursuing treasury accumulation strategies may issue additional equity to fund purchases, diluting existing shareholders. Any assessment of per-share value must account for potential future share issuance.
• Premium/discount to NAV: The market price of BMNR can trade at a significant premium or discount to the marked value of its treasury holdings. There is no guarantee that the market will value BMNR at or above NAV at any given time.
• Execution risk: BMNR's management must execute its strategy effectively. Operational missteps, poor capital allocation, or governance failures could impair value independent of Ethereum's price trajectory.

Part V: The Nature of Convex Acceleration

During quiet or declining markets, convex instruments often lag. The payoff curve is flat in the lower price region, and investors may become frustrated with apparent underperformance. This is structurally normal — it is how convex payoffs behave before conditions align.

When the underlying asset enters an acceleration phase and catalysts converge, the curve steepens. Additional moves in the underlying can produce disproportionately larger moves in the proxy, and price action can compress into days or weeks rather than grinding over months. Observers who model relationships linearly may perceive such moves as irrational, but they are consistent with the mathematics of convex payoff structures.

The MSTR example illustrates this concretely: a 155% gain in 27 days during a period when Bitcoin gained approximately 40%. For investors anchored to linear expectations, that acceleration appeared euphoric. Modeled as a convex Bitcoin proxy, it was a predictable consequence of balance sheet optics, multiple expansion, and institutional flow dynamics converging simultaneously.

Whether BMNR follows a similar pattern depends on conditions that are uncertain: Ethereum price trajectory, regulatory developments, institutional behavior, and market sentiment. The structural characteristics are present; whether and when they express is not something we can predict.

Part VI: Summary

Convexity describes investment payoffs that curve upward rather than tracking linearly. MicroStrategy's 155% gain during Bitcoin's 40% rise in early 2024 illustrates convex behavior in practice.

BMNR exhibits structural characteristics that could produce similar non-linear responses to Ethereum price movements: a large ETH treasury (4.33M ETH, approximately 3.5–3.6% of supply), staking revenue that scales with ETH price, significant institutional accumulation in Q4 2025, and potential regulatory catalysts. It also carries material risks including concentration in a single volatile asset, liquidity constraints, regulatory uncertainty, and the possibility that convex acceleration never materializes.

Understanding convexity helps investors recognize that dramatic acceleration phases — if they occur — are mathematical features of certain structures, not anomalies. It also helps investors understand that convex structures can remain dormant for extended periods and can produce meaningful losses if the underlying asset declines.

We size positions to survive if the curve never triggers, while maintaining exposure to asymmetric upside if it does. The shape of the curve is mathematics. The probability of the curve triggering is judgment. Neither is certainty.

References

[1] Bitcoin historical price data, February–March 2024. Bitcoin rose from approximately $52,000 (February 16, 2024) to $73,000 (March 15, 2024), representing approximately 40% gain.

[2] MicroStrategy (MSTR) historical stock prices. February 16, 2024 close: $69.96. March 15, 2024 close: $178.24. Gain: 154.7%. Source: Yahoo Finance, Stock Analysis.

[3] Bitmine Immersion Technologies press release, February 8, 2026. "BMNR Announces ETH Holdings Reach 4.326 Million Tokens."

[4] Bitmine Immersion Technologies disclosure, February 2026. Staking position: 2.87M ETH staked generating approximately $374M annually. Combined crypto, cash, moonshots valued at ~$10B at current prices.

[5] BlackRock 13F-HR filing, February 12, 2026, reporting Q4 2025 holdings as of December 31, 2025. BlackRock increased BMNR position by 5,642,590 shares (+165.6%) to 9,049,912 shares.

[6] Institutional 13F filings Q4 2025 summary. Major increases: Morgan Stanley +2.5M shares (+25.8%), ARK +2.0M shares (+26.6%), Vanguard +6.0M shares (+254.8%), Goldman Sachs +7.0M shares (+920.6%). Data compiled from SEC EDGAR filings February 2026.

[7] Ave Maria School of Law, CLARITY Act analysis, February 10, 2026.

[8] White House crypto adviser statement, February 13, 2026. Yahoo Finance.

[9] K&L Gates legal analysis, January 29, 2026. "Crypto in 2026: The Democratization of Digital Assets."

Disclaimer

This report is provided for educational purposes only and does not constitute investment advice, a recommendation to buy or sell any security, or an offer of investment advisory services. The information contained herein is based on sources believed to be reliable but is not guaranteed as to accuracy or completeness.

Past performance is not indicative of future results. All investments involve risk, including possible loss of principal. The examples and case studies presented in this report, including references to MicroStrategy (MSTR), BitMine Immersion (BMNR), Bitcoin, and Ethereum, are used for educational illustration only and should not be interpreted as projections, predictions, or guarantees of future performance.

The mathematical frameworks and historical observations discussed herein describe potential payoff structures under specific conditions. There is no assurance that any security will exhibit convex behavior or that historical patterns will repeat. Cryptocurrency-related investments are subject to extreme volatility, regulatory uncertainty, and the potential for total loss of invested capital.

Institutional ownership data referenced in this report is derived from publicly available SEC filings (13F-HR). The presence of institutional investors in a security does not constitute an endorsement of that security, does not indicate institutional sentiment about future performance, and does not reduce investment risk. Institutions may increase, decrease, or eliminate positions at any time for reasons unrelated to their assessment of a security's value.

The views and strategies described may not be suitable for all investors and are subject to change without notice. Clients should consider their individual financial situation, risk tolerance, and investment objectives before making any investment decisions.

Patriot Advisory Group is a registered investment adviser. Advisory services are only offered to clients or prospective clients where Patriot Advisory Group and its representatives are properly licensed or exempt from licensure. This document does not constitute an offer to provide investment advisory services in any jurisdiction where prohibited. Patriot Advisory Group's standard advisory fee on assets under management is the sole basis of adviser compensation in connection with the positions discussed herein. No additional commissions, bonuses, or incentive-based compensation are received based on the purchase, sale, or holding of any specific security, including BMNR.

For additional information about Patriot Advisory Group, please refer to our Form ADV Part 2A brochure, which is available upon request.

February 17, 2026