Strategy Lab

Black-Scholes: Price a European Option

An option’s price is built from five inputs, and only one of them — volatility — can’t simply be looked up. That’s the whole reason option traders disagree with each other: everyone sees the same spot price, strike, rate, and time to expiry. Volatility is the one number you have to estimate, and the model is unforgiving about how much that estimate matters.

How to read it: d₁ and d₂ are the intermediate terms that get fed into the cumulative normal distribution N(·) — think of N(d₂) loosely as “the model’s estimate of the probability this option finishes in the money.” Watch what happens to the price when you drag volatility versus when you drag time to expiry: volatility moves the price roughly linearly, but time enters through a square root, so halving the time to expiry removes noticeably less than half the value.

Watch out for: this prices a European call (exercisable only at expiry) with no dividends. American options, dividend-paying underlyings, and puts all need adjustments this formula doesn’t make.

Compound Growth: Project Your Savings

This is the forward-looking twin of CAGR: instead of asking “what rate did I actually earn,” it asks “if I keep contributing at this rate, what do I end up with.” The honest answer, for most people, is that the monthly contribution matters more than the starting capital over long horizons — the habit compounds harder than the lump sum.

How to read it: drag the return rate by just one percentage point and watch the final value move — over a 20-30 year horizon, that single point is usually worth far more than doubling the monthly contribution. That’s the entire case for caring about fees and asset allocation over trying to time contributions perfectly.

Watch out for: this assumes a constant annual return every year, which no real portfolio delivers. It’s a planning tool for the destination, not a forecast of the path.

CAGR: What Did You Actually Earn?

CAGR answers a narrower, more honest question than “total return”: not “how much did I make” but “what single, smooth annual rate would have produced that same result.” It’s the tool that turns a lumpy, real price history — the kind where a stock can be up 30x in a year, like SanDisk was in 2026 — into one comparable number.

How to read it: check the “total return” figure alongside the annualized one. A high CAGR over a short window (a year or two) can come from a single dramatic run rather than durable growth — the shorter the period, the less a smooth annual rate actually tells you about the ride it took to get there.

Watch out for: CAGR is blind to volatility along the way. Two investments with identical CAGR can have wildly different risk profiles if one arrived there smoothly and the other via a 50% drawdown and recovery.

WACC: Your Blended Cost of Capital

Every valuation model discounts future cash flows back to today at some rate, and WACC is how that rate gets built: a weighted blend of what equity holders require and what lenders charge, with debt’s cost reduced by the tax shield on interest. When the Fed moves rates, this is the actual mechanism by which “higher rates” becomes “lower stock valuations” — not vibes, arithmetic.

How to read it: notice that cost of equity, not cost of debt, usually does most of the work — equity is riskier, so it costs more, and most companies are financed more by equity than debt. Try dragging the equity share (E/V) rather than either cost input, and see how much the blended rate moves just from the financing mix.

Watch out for: cost of equity here is a raw input, but in practice it’s usually derived from CAPM (see that model) rather than guessed — the two tools are meant to be used together.

CAPM: Expected Return From Beta

Beta measures how much a stock amplifies the market’s moves — SanDisk carried a beta above 5 during its 2026 run, meaning it moved roughly five times whatever the S&P 500 did, in either direction. CAPM takes that number and turns it into a required return: the risk-free rate, plus beta times the extra return investors demand for holding the market at all.

How to read it: the “beta-weighted” figure in the readout is the actual risk premium this stock demands, over and above the risk-free rate. A beta of 2 doesn’t just mean “twice as volatile” — it means the market’s own risk premium gets counted twice when pricing this stock’s required return.

Watch out for: beta is estimated from historical price data and drifts over time — a stock’s beta during a speculative run (like a fresh spin-off with a thin trading history) can be a poor guide to its beta a year later.

Position Size: From Risk, Not Feel

Most position-sizing mistakes aren’t about picking the wrong stock — they’re about sizing a correct pick too large. This model flips the usual question: instead of “how many shares can I afford,” it asks “given how much I’m willing to lose if my stop gets hit, how many shares does that actually buy me.”

How to read it: the risk percentage (R) is the number that actually decides whether a string of losing trades wipes out an account or just dents it. Move the stop closer to the entry and watch the position size grow — same dollar risk, more shares, because each share now risks less.

Watch out for: this sizes a single position in isolation. It says nothing about correlation across a whole portfolio — five “independent” 1%-risk positions in the same sector can behave like one 5%-risk position in a sell-off.

Kelly Criterion: Optimal Bet Size

Kelly answers a genuinely different question than simple position sizing: given a real, honestly-estimated edge, what fraction of capital maximizes long-run growth rate rather than just limiting any single loss? It’s the framework behind “how much should I actually risk on a name with a beta this extreme” — the exact question a stock like SanDisk or PayPal’s post-collapse repricing raises.

How to read it: if the result goes negative, that’s not a bug — it’s Kelly telling you the edge as estimated doesn’t justify betting at all. In practice, most disciplined users size at “half-Kelly” (shown in the readout) rather than full Kelly, since full Kelly is mathematically optimal but brutally volatile.

Watch out for: the entire result hinges on your win-probability estimate being honest. Kelly amplifies a bad edge estimate just as readily as a good one — garbage in, confidently-sized garbage out.

Want it with real data instead of assumed inputs? The tool below computes f* from each ticker’s actual two-year daily return history, not a guess.

Value at Risk: Maximum Expected Loss

VaR answers “how much could I plausibly lose” at a chosen confidence level and time horizon — a portfolio-level companion to position sizing rather than a single-trade tool. The z-score is doing the real work here: it’s the statistical multiplier that converts a confidence level (95%, 99%) into a number of standard deviations.

How to read it: notice volatility scales with the square root of time, not time itself — a 10-day VaR is roughly 3.16× the 1-day figure, not 10×. That square-root relationship is one of the more counter-intuitive facts in risk management and shows up constantly.

Watch out for: VaR assumes returns are roughly normally distributed. Real markets have fatter tails than that — the “1-in-20” loss VaR describes at 95% confidence tends to happen more often than the normal distribution predicts, especially during crises.

Sharpe Ratio: Was the Return Worth It?

A 15% return sounds identical whether it came from a boring, low-volatility strategy or a wild one that also could have lost 40%. Sharpe ratio is the tool that separates those two cases: excess return over the risk-free rate, divided by the volatility paid to get it.

How to read it: above 1 is generally considered good, above 2 very good, above 3 excellent — but the number is only as honest as the volatility input. Compare two portfolios with the same return: the one with the higher Sharpe ratio got there more smoothly, which usually means it survives the next drawdown better too.

Watch out for: Sharpe penalizes upside volatility exactly as much as downside volatility, even though investors only actually mind the downside. Sortino ratio (a close cousin, not included here) fixes that by only counting downside deviation.

Bollinger Bands: Distance From the Mean

Bollinger Bands are nothing more exotic than a moving average plus and minus a multiple of the standard deviation — but that simplicity is the point. The bands widen and narrow automatically as volatility changes, so a “breakout” beyond the band means something very different in a calm market than in an already-turbulent one.

How to read it: at k = 2 (the default and most common setting), roughly 95% of a normal distribution falls inside the bands — so a price sitting outside them is, loosely, a “one-in-twenty” event under that assumption, not automatically a signal to act.

Watch out for: “the price touched the upper band” is not, by itself, a sell signal — in a strong trend, price can walk along a band for weeks. The bands describe dispersion, not direction.

RSI: Momentum, Quantified

RSI compresses a stretch of up-moves and down-moves into a single 0-100 reading by comparing their average sizes. Wilder’s original convention — above 70 overbought, below 30 oversold — is a starting heuristic, not a law; a stock in a strong uptrend can sit above 70 for a long stretch without reversing.

How to read it: the RS ratio in the readout (average gain divided by average loss) is the number RSI is actually built from — RSI itself is just that ratio rescaled onto a 0-100 range so it’s comparable across different stocks and time periods.

Watch out for: this widget takes average gain/loss directly as inputs rather than computing them from a raw price series — in practice you’d derive those from the last N periods (classically 14) of actual price data first.