In 1973, Fischer Black and Myron Scholes solved a problem that had defeated finance for seventy years: how to price an option without knowing where the stock is headed. Their formula didn't just settle an academic debate — it launched a derivatives market now worth trillions and quietly redefined what a share of stock actually is.

This is the finale of a four-episode arc. After Markowitz, Sharpe, and Modigliani-Miller, Black and Scholes bring together everything we've covered — hedging, CAPM, and capital structure — into a single closed-form result. Join Finance Papers for a section-by-section breakdown of the original text, stripped of textbook shortcuts.

Key Topics Covered in This Deep Dive:

The Seven Ideal Conditions: The frictionless, cost-free market the authors had to construct before the math could work — and why every subsequent advance in derivatives theory maps where one of these seven assumptions breaks.

The Hedge That Cancels Out: How a continuously adjusted long-stock, short-option position eliminates risk entirely, and why a riskless position must earn exactly the risk-free rate.

The Disappearing Expected Return: The most counterintuitive result in the paper — the stock's expected return drops out of the formula, meaning a bull and a bear must agree on the option's price.

Implied Volatility, Invented on the Spot: Why variance is the only unobservable input in the formula, and how the market built an entire concept — and later the VIX — just to estimate it.

Equity as an Option: The paper's most underread section. Why common stock in a leveraged firm is mathematically a call option on the firm's assets, and what that reveals about leverage, covenants, and default risk.

Paper Explored: Black, F., & Scholes, M. (1973), THE PRICING OF OPTIONS AND CORPORATE LIABILITIES. Journal of Political Economy, 81(3): 637–654.

🔗 Read the full paper here: https://doi.org/10.1086/260062

Finance Papers is conceptualized and curated by Luiz Gidrão, CFA, CAIA (founder of stock.cash and goa.capital). Hit play, open the paper PDF, and learn with us.

In 1958, two young economists published a paper that enraged corporate finance professors and quietly rewired the foundations of the field. Franco Modigliani and Merton Miller’s “The Cost of Capital, Corporation Finance and the Theory of Investment” claimed that, in a perfect market, a firm’s value is completely independent of how it mixes debt and equity—an idea so radical that many dismissed it as academic trickery.

In this third episode of Finance Papers, we strip away the folklore and walk through Modigliani & Miller’s original text section‑by‑section. Instead of treating “capital structure irrelevance” as a slogan, we focus on the arbitrage proof itself: the intellectual “magic trick” where investors reshuffle equity and debt between identical firms to expose any supposed free lunch.

Key Topics Covered in This Deep Dive:

• The Irrelevance Benchmark: Why, under no taxes, no distress costs, no agency problems and symmetric information, leverage alone cannot change firm value—only redistribute risk and cash flows across claimholders.

• Homemade Leverage: How investors can lever or de‑lever their own portfolios using borrowing and risk‑free assets, and why this undermines the idea that managers can create value purely by changing the debt‑equity mix.

• The Arbitrage Argument: Step‑by‑step through MM’s invariance proof, showing how any mispricing between levered and unlevered firms translates into a true free lunch until prices converge.

• From Irrelevance to Frictions: How the theorem doesn’t say “capital structure never matters”, but instead tells you exactly where to look when it does—tax shields, bankruptcy and distress costs, agency conflicts, and information asymmetry.

• Closing the Foundations: How Modigliani & Miller complete the structural base laid by Markowitz (portfolios) and Sharpe (CAPM), and set the stage for Fama’s formalization of “perfect markets” and the Efficient Market Hypothesis.

  
  

Paper Explored: Modigliani, F. & Miller, M. H. (1958), “The Cost of Capital, Corporation Finance and the Theory of Investment”, The American Economic Review, 48(3): 261–297.

🔗 Read the full paper: https://www.jstor.org/stable/1809766

Finance Papers is conceptualized and curated by Luiz Gidrão, CFA, CAIA (founder of ⁠stock.cash⁠ and ⁠goa.capital⁠). Hit play, open the PDF, and learn with us.

In 1952, a 25-year-old graduate student published a 14-page paper that changed Wall Street forever. Harry Markowitz’s "Portfolio Selection" laid the mathematical foundation for modern quantitative finance and earned him a Nobel Prize.

In this inaugural episode of Finance Papers, we strip away the fluff and do a section-by-section deep dive into the actual math and mechanics of Markowitz's original text.

Key Topics Covered:

• ​The Concentration Trap: Why trying to purely maximize returns mathematically forces you to put 100% of your money into a single stock.
• ​The Myth of Large Numbers: Why holding 60 railway stocks isn't true diversification when a systemic market crash hits.
• ​The Magic of Covariance: How the interaction between assets—not their individual volatility—defines true portfolio risk.
• ​The Computing Bottleneck: How the sheer impossibility of calculating 125,000 covariance pairs by hand in 1952 eventually led to the creation of Beta and CAPM.
• ​The Math of Gambling: Why rational humans willingly buy lottery tickets (the statistical concept of Skewness).

Paper Explored: Markowitz, H. (1952), PORTFOLIO SELECTION*. The Journal of Finance, 7: 77-91.

🔗 Read the full paper here: https://doi.org/10.1111/j.1540-6261.1952.tb01525.x

Finance Papers is conceptualized and curated by Luiz Gidrão, CFA, CAIA (founder of stock.cash and goa.capital). Hit play, open the paper PDF, and learn with us.

Following up on Harry Markowitz’s foundational concept of diversification, William Sharpe’s 1964 paper took Modern Portfolio Theory to the next level.

This seminal work is the practical key that unlocked MPT for the investing world, introducing the Capital Asset Pricing Model (CAPM) and fundamentally changing how we evaluate investment returns in relation to risk. Like Markowitz, Sharpe also received a Nobel Prize for this foundational contribution to finance.

Join Finance Papers for a focused breakdown of Sharpe's masterwork. We strip away the textbook simplifications to analyze how the model truly works from its original source text, section-by-section.

Key Topics Covered in This Deep Dive:

• The Risk-Free Addition: How introducing a risk-free borrowing and lending asset collapses the complex Efficient Frontier into a single, elegant visual: The Capital Market Line (CML).

• The Separation Theorem: Understanding why the "efficient" portfolio is identical for every investor, regardless of their personal risk tolerance.

• Defining Beta (β): The exact mathematical meaning of Sharpe’s original metric for systematic risk and its relationship to the total market portfolio.

• The Undiversifiable Problem: Why diversification (as taught by Markowitz) cannot eliminate market risk—only firm-specific risk.

• Market Equilibrium: How collective investor behavior forces asset prices to adjust to a state of balance based on risk premiums.

Paper Explored: Sharpe, W.F. (1964), CAPITAL ASSET PRICES: A THEORY OF MARKET EQUILIBRIUM UNDER CONDITIONS OF RISK*. The Journal of Finance, 19: 425-442.

🔗 Read the full paper: https://onlinelibrary.wiley.com/doi/10.1111/j.1540-6261.1964.tb02865.x

Finance Papers is conceptualized and curated by Luiz Gidrão, CFA, CAIA (founder of stock.cash and goa.capital). Hit play, open the PDF, and learn with us.