{"version":"https://jsonfeed.org/version/1.1","title":"building and thinking","home_page_url":"https://journal.ehrlich.dev/","feed_url":"https://journal.ehrlich.dev/index.json","description":"Dev, philosophy, and research deep dives","authors":[{"name":"Bryan Ehrlich"}],"items":[{"id":"https://journal.ehrlich.dev/notes/2026-03-14-tooling/","url":"https://journal.ehrlich.dev/notes/2026-03-14-tooling/","title":"Claude Code planned and executed this entire redesign in one shot. The plan was …","summary":"Claude Code planned and executed this entire redesign in one shot. The plan was thorough enough that implementation was basically mechanical. Planning \u0026gt; coding.\n","content_html":"\u003cp\u003eClaude Code planned and executed this entire redesign in one shot. The plan was thorough enough that implementation was basically mechanical. Planning \u0026gt; coding.\u003c/p\u003e\n","date_published":"2026-03-14T14:30:00Z","date_modified":"2026-03-14T14:30:00Z"},{"id":"https://journal.ehrlich.dev/notes/2026-03-14-stream-launch/","url":"https://journal.ehrlich.dev/notes/2026-03-14-stream-launch/","title":"Rebuilt the journal as a stream. Notes and articles in one feed now. Temperature …","summary":"Rebuilt the journal as a stream. Notes and articles in one feed now. Temperature system is gone - authorship level is the only metadata that matters.\n","content_html":"\u003cp\u003eRebuilt the journal as a stream. Notes and articles in one feed now. Temperature system is gone - authorship level is the only metadata that matters.\u003c/p\u003e\n","date_published":"2026-03-14T14:00:00Z","date_modified":"2026-03-14T14:00:00Z","tags":["meta"]},{"id":"https://journal.ehrlich.dev/notes/2026-03-13-experiential-measure/","url":"https://journal.ehrlich.dev/notes/2026-03-13-experiential-measure/","title":"The experiential measure paper is getting close. The core argument - that …","summary":"The experiential measure paper is getting close. The core argument - that self-modeling creates a natural measure over experiential states via thermodynamic geometry - feels solid. The hard part is scoping the claims honestly.\n","content_html":"\u003cp\u003eThe experiential measure paper is getting close. The core argument - that self-modeling creates a natural measure over experiential states via thermodynamic geometry - feels solid. The hard part is scoping the claims honestly.\u003c/p\u003e\n","date_published":"2026-03-13T22:00:00Z","date_modified":"2026-03-13T22:00:00Z","tags":["research"]},{"id":"https://journal.ehrlich.dev/posts/options-primer-5-selling-options/","url":"https://journal.ehrlich.dev/posts/options-primer-5-selling-options/","title":"Options Primer Part 5: Selling Options","summary":"Part 5 of 5. Cash-secured puts, covered calls, and the volatility risk premium - why selling options has positive expected value.","content_html":"\u003cp\u003eEvery option has a buyer and a seller. The $100 call from \u003ca href=\"/posts/options-primer-3-volatility/\"\u003ePart 3\u003c/a\u003e? Someone sold it for $4.20 and kept the premium when WidgetCo crashed.\u003c/p\u003e\n\u003cp\u003eThis part covers the mechanics of selling and why, mathematically, selling options tends to be profitable.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"the-volatility-risk-premium\"\u003eThe volatility risk premium\u003c/h2\u003e\n\u003cp\u003eOptions are priced using implied volatility - the market\u0026rsquo;s estimate of future stock movement. Here\u0026rsquo;s the key fact: \u003cstrong\u003eIV is systematically higher than realized volatility.\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWhy? Options are insurance. Buyers pay a premium for protection against large moves. Sellers demand compensation for taking on tail risk. That compensation - the gap between implied and realized volatility - is the \u003cstrong\u003e\u003ca href=\"https://en.wikipedia.org/wiki/Volatility_risk_premium\"\u003evolatility risk premium\u003c/a\u003e\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003eEmpirically, the VIX (S\u0026amp;P 500 implied volatility) \u003ca href=\"https://caia.org/blog/2024/02/01/what-volatility-risk-premium\"\u003eexceeds subsequent realized volatility about 85% of the time\u003c/a\u003e. The average spread is about four volatility points.\u003c/p\u003e\n\u003cp\u003eThis means selling options has positive expected value. The risk premium compensates sellers for absorbing losses during market crashes, earnings disasters, and black swan events.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"pennies-in-front-of-a-steamroller\"\u003ePennies in front of a steamroller\u003c/h2\u003e\n\u003cp\u003eThe math: collect small premiums frequently, pay large claims rarely. On average, the premiums exceed the claims.\u003c/p\u003e\n\u003cp\u003eThis gets called \u0026ldquo;picking up pennies in front of a steamroller.\u0026rdquo; The metaphor overstates it - you\u0026rsquo;re not picking up pennies, you\u0026rsquo;re running an insurance business. Insurance companies are profitable. They just occasionally pay out large claims.\u003c/p\u003e\n\u003cp\u003eBut the metaphor captures something real: the P\u0026amp;L distribution is asymmetric. Months of steady gains, then one bad week erases them. You can be \u0026ldquo;right\u0026rdquo; 90% of the time and still lose money if the 10% hits hard enough.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"cash-secured-puts\"\u003eCash-secured puts\u003c/h2\u003e\n\u003cp\u003eA \u003cstrong\u003ecash-secured put (CSP)\u003c/strong\u003e: sell a put, hold cash to buy shares if assigned.\u003c/p\u003e\n\u003cp\u003eWidgetCo at $50. Sell a $45 put, 30 DTE, collect $1.50 ($150 per contract).\u003c/p\u003e\n\u003cp\u003eThe obligation: buy 100 shares at $45 if assigned. Need $4,500 cash to cover.\u003c/p\u003e\n\u003cdiv data-widget=\"csp-payoff\" data-strike=\"45\" data-premium=\"1.50\" data-price=\"50\"\u003e\u003c/div\u003e\n\u003cp\u003eDrag the stock price. Above $45: keep $150. At $43.50: breakeven. Below: losing money.\u003c/p\u003e\n\u003cp\u003eMaximum gain: $150. Maximum loss: $4,350 (stock to zero, minus premium).\u003c/p\u003e\n\u003cp\u003eStock drops to $44 at expiration. The put is exercised. You now own 100 shares at $45 cost basis. Time to sell calls.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"covered-calls\"\u003eCovered calls\u003c/h2\u003e\n\u003cp\u003eA \u003cstrong\u003ecovered call (CC)\u003c/strong\u003e: own shares, sell a call against them.\u003c/p\u003e\n\u003cp\u003eYou own 100 shares at $45. Stock at $44. Sell a $50 call, 30 DTE, collect $0.80 ($80).\u003c/p\u003e\n\u003cp\u003eThe obligation: sell shares at $50 if called away.\u003c/p\u003e\n\u003cdiv data-widget=\"cc-payoff\" data-strike=\"50\" data-premium=\"0.80\" data-cost-basis=\"45\" data-price=\"44\"\u003e\u003c/div\u003e\n\u003cp\u003eDrag the stock price. Below $50: keep shares and premium. Above $50: called away, miss upside beyond strike.\u003c/p\u003e\n\u003cp\u003eStock drifts to $47. Call expires worthless. Collect $80, still own shares.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"the-wheel\"\u003eThe wheel\u003c/h2\u003e\n\u003cp\u003eWhat if you just kept selling options no matter what happens?\u003c/p\u003e\n\u003cp\u003eSell a put. If it expires worthless, keep the premium, sell another. If you get assigned, you now own shares - so sell calls against them. If the call expires worthless, sell another. If you get called away, you\u0026rsquo;re back to cash - sell puts again.\u003c/p\u003e\n\u003cp\u003eThat\u0026rsquo;s the wheel. CSP → assigned → CC → called away → repeat.\u003c/p\u003e\n\u003cp\u003eThe 45-day expiration comes from \u003ca href=\"/posts/options-primer-4-the-greeks/\"\u003ePart 4\u003c/a\u003e: theta accelerates near expiration, but so does gamma. \u003ca href=\"https://luckboxmagazine.com/techniques/the-magic-of-45-optimal-short-options-trade-duration/\"\u003eTastyTrade\u0026rsquo;s research\u003c/a\u003e found 45 DTE balances premium collection against the risk of sudden moves.\u003c/p\u003e\n\u003cp\u003eTry it. Sell puts until assigned, then sell calls until called away.\u003c/p\u003e\n\u003cdiv data-widget=\"wheel-sim\" data-starting-price=\"50\" data-iv=\"0.35\"\u003e\u003c/div\u003e\n\u003chr\u003e\n\u003ch2 id=\"the-missed-move\"\u003eThe missed move\u003c/h2\u003e\n\u003cp\u003eRun the wheel long enough and you\u0026rsquo;ll get called away right before a rally. Stock gaps up, your shares are gone at the strike price, and you watch the stock climb without you.\u003c/p\u003e\n\u003cp\u003eThat\u0026rsquo;s the trade-off. You collected premium along the way. But when the stock doubles, you don\u0026rsquo;t double with it.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"when-this-makes-sense\"\u003eWhen this makes sense\u003c/h2\u003e\n\u003cp\u003eThe wheel works when realized volatility stays below implied - when the market overestimates how much stocks will move. Historically, that\u0026rsquo;s most of the time. The strategy fits investors who want income over growth, can tolerate holding individual stocks through drawdowns, and have the capital to secure puts.\u003c/p\u003e\n\u003cp\u003eRetirees sometimes run wheels on blue chips for income. It\u0026rsquo;s active management with real risks, but the math favors sellers. If that sounds like work, JPMorgan\u0026rsquo;s \u003ca href=\"https://am.jpmorgan.com/us/en/asset-management/adv/products/jpmorgan-equity-premium-income-etf-etf-shares-46641q332\"\u003eJEPI\u003c/a\u003e and \u003ca href=\"https://am.jpmorgan.com/us/en/asset-management/adv/products/jpmorgan-nasdaq-equity-premium-income-etf-etf-shares-46654q203\"\u003eJEPQ\u003c/a\u003e run covered call strategies on the S\u0026amp;P 500 and Nasdaq respectively. You get the volatility risk premium without the position management.\u003c/p\u003e\n\u003cp\u003eThis series was about the math. Options are derivatives - their prices derive from stock prices, time, and volatility through functions we can write down. The Greeks measure sensitivity to each input. Buyers pay for movement. Sellers collect premium for absorbing risk.\u003c/p\u003e\n\u003cp\u003eWhether you trade options is a different question. But now you know what you\u0026rsquo;re looking at.\u003c/p\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cem\u003eThis is \u003ca href=\"/tags/options/\"\u003ePart 5 of a 5-part series\u003c/a\u003e on options. \u003ca href=\"/posts/options-primer-4-the-greeks/\"\u003ePart 4\u003c/a\u003e covers the Greeks. \u003ca href=\"/posts/options-primer-1-background/\"\u003ePart 1\u003c/a\u003e is where it started.\u003c/em\u003e\u003c/p\u003e\n","date_published":"2024-12-24T06:00:00Z","date_modified":"2024-12-24T06:00:00Z","tags":["options","finance","interactive"]},{"id":"https://journal.ehrlich.dev/posts/options-primer-4-the-greeks/","url":"https://journal.ehrlich.dev/posts/options-primer-4-the-greeks/","title":"Options Primer Part 4: The Greeks","summary":"Part 4 of 5. Delta, gamma, theta, vega - measuring your exposure to stock price, time, and volatility.","content_html":"\u003cp\u003eWidgetCo drops $5 tomorrow. How much do you lose?\u003c/p\u003e\n\u003cp\u003eIt depends. On where the stock started. On how much time is left. On whether volatility spiked with the drop. On how far away your strike is. The answer is a function of all these variables at once.\u003c/p\u003e\n\u003cp\u003eThe Greeks are tools for navigating this. But to understand what they\u0026rsquo;re actually measuring, we need to talk about the surface your option lives on.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"the-pricing-surface\"\u003eThe pricing surface\u003c/h2\u003e\n\u003cp\u003eAn option\u0026rsquo;s price depends on several inputs: stock price, time to expiration, implied volatility. Strike is fixed when you buy. That leaves three moving variables - which means the full picture is four-dimensional (three inputs plus price). There\u0026rsquo;s no way to draw that.\u003c/p\u003e\n\u003cp\u003eBut we can take lower dimensional slices. Hold IV constant and look at how price changes with stock and time. That\u0026rsquo;s a surface:\u003c/p\u003e\n\u003cdiv data-widget=\"pricing-surface\" data-strike=\"50\" data-iv=\"0.45\"\u003e\u003c/div\u003e\n\u003cp\u003eDrag to rotate. This is a $50 call with IV fixed at 45% - one slice of the larger space.\u003c/p\u003e\n\u003cp\u003eHold time constant too, and we get a 2d curve:\u003c/p\u003e\n\u003cdiv data-widget=\"delta-demo\" data-strike=\"50\" data-price=\"50\" data-iv=\"0.45\" data-dte=\"30\" data-minimal=\"true\"\u003e\u003c/div\u003e\n\u003cp\u003eSame option, but now we\u0026rsquo;re only watching how price changes with stock. We\u0026rsquo;ll use both views throughout.\u003c/p\u003e\n\u003cp\u003eBack to the surface: see those colored dots with arrows? Each dot is an option at a different position. The arrows show which way you\u0026rsquo;re exposed - how the option responds when things change, or when your option moves across the surface.\u003c/p\u003e\n\u003cp\u003eThose arrows are the Greeks. They work in the full space too, pointing in directions we can\u0026rsquo;t draw.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"delta---stock-price-sensitivity\"\u003eDelta - stock price sensitivity\u003c/h2\u003e\n\u003cp\u003eDelta measures how much your option moves when the stock moves $1. It\u0026rsquo;s the most immediate question: if WidgetCo goes up $1 tomorrow, how much do I make?\u003c/p\u003e\n\u003cp\u003eYou buy a $50 call, 30 days to expiration. It costs $2.85. Delta is 0.52.\u003c/p\u003e\n\u003cp\u003eIf WidgetCo goes from $50 to $51, your call gains roughly $0.52.\nIf WidgetCo drops from $50 to $49, your call loses roughly $0.52.\u003c/p\u003e\n\u003cdiv data-widget=\"delta-demo\" data-strike=\"50\" data-price=\"50\" data-iv=\"0.45\" data-dte=\"30\"\u003e\u003c/div\u003e\n\u003cp\u003eDrag the stock price. Watch how the \u003cspan style=\"color:#3b82f6; font-weight:bold;\"\u003eblue tangent line\u003c/span\u003e (delta) changes slope as you move.\u003c/p\u003e\n\u003cp\u003eDelta ranges from 0 to 1 for calls. Deep out of the money (stock way below strike), delta is near 0. At the money (stock at strike), delta is around 0.5. Deep in the money (stock way above strike), delta approaches 1 - moving almost 1:1 with the stock.\u003c/p\u003e\n\u003cp\u003eFor puts, delta is negative - ranging from -1 to 0. A put with delta -0.5 gains $0.50 when the stock drops $1.\u003c/p\u003e\n\u003cp\u003eAnother way to think about it: delta is how much stock you effectively own. A 0.52 delta call on 100 shares behaves like owning 52 shares. You paid $285, not $5,000 - the option amplifies your capital while exposing you to 52 shares worth of movement.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"theta---time-decay\"\u003eTheta - time decay\u003c/h2\u003e\n\u003cp\u003eEvery day, your option loses value. That\u0026rsquo;s theta.\u003c/p\u003e\n\u003cp\u003eYour $50 call has theta of -0.08. You\u0026rsquo;re losing $0.08 per share per day - $8 per contract - just by holding it.\u003c/p\u003e\n\u003cp\u003eWhy? An option is a bet that something will happen before a deadline. The value of that bet depends on how much time is left for it to play out. With 30 days, anything could happen. With 1 day, much less likely.\u003c/p\u003e\n\u003cdiv data-widget=\"theta-demo\" data-strike=\"50\" data-price=\"50\" data-iv=\"0.45\" data-dte=\"30\"\u003e\u003c/div\u003e\n\u003cp\u003eDrag the time slider. Watch how the \u003cspan style=\"color:#ef4444; font-weight:bold;\"\u003ered tangent line\u003c/span\u003e (theta) steepens as expiration approaches.\u003c/p\u003e\n\u003cp\u003eTheta accelerates near expiration. With 30 days left, you lose $8/day. With 7 days left, you might lose $15/day. At 30 days, losing one day costs you 1/30th of your remaining time. At 7 days, losing one day costs you 1/7th. Same absolute time, bigger percentage hit.\u003c/p\u003e\n\u003cp\u003eHere\u0026rsquo;s what theta feels like in practice: You buy a call. The stock does nothing for a week. You check your position. It\u0026rsquo;s down 5%.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"gamma---deltas-rate-of-change\"\u003eGamma - delta\u0026rsquo;s rate of change\u003c/h2\u003e\n\u003cp\u003eDelta isn\u0026rsquo;t constant. It changes as the stock moves.\u003c/p\u003e\n\u003cp\u003eThat\u0026rsquo;s gamma - how fast delta changes per $1 stock move.\u003c/p\u003e\n\u003cp\u003eYour $50 call has delta 0.52 and gamma 0.04.\u003c/p\u003e\n\u003cp\u003eIf WidgetCo goes from $50 to $51:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eYour call gains $0.52 (from delta)\u003c/li\u003e\n\u003cli\u003eYour new delta is 0.56 (old delta + gamma)\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eIf it continues to $52:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eYou gain roughly $0.56 on that dollar (using your new delta)\u003c/li\u003e\n\u003cli\u003eNew delta is 0.60\u003c/li\u003e\n\u003c/ul\u003e\n\u003cdiv data-widget=\"gamma-demo\" data-strike=\"50\" data-price=\"50\" data-iv=\"0.45\" data-dte=\"30\"\u003e\u003c/div\u003e\n\u003cp\u003eMove the stock price. Watch how the \u003cspan style=\"color:#8b5cf6; font-weight:bold;\"\u003epurple tangent line\u003c/span\u003e (gamma) shows whether delta is accelerating or decelerating.\u003c/p\u003e\n\u003cp\u003eWhy care? Gamma tells you how unstable your exposure is. High gamma means your delta can flip quickly. At $50, your call has delta 0.52 - you\u0026rsquo;re moderately exposed. But gamma is high. If the stock moves $5, your delta might be 0.75 or 0.25 depending on direction.\u003c/p\u003e\n\u003cp\u003eLow gamma means stability. Deep ITM, delta is 0.95 and gamma is tiny. The stock can move $5 and delta barely changes - you\u0026rsquo;re locked in at near-1:1 exposure.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"vega---volatility-sensitivity\"\u003eVega - volatility sensitivity\u003c/h2\u003e\n\u003cp\u003eVega measures how much your option moves when IV changes by 1 percentage point.\u003c/p\u003e\n\u003cp\u003eWhy does IV matter? Remember: IV is the market\u0026rsquo;s estimate of how much the stock might move. Higher IV means bigger expected swings. Options are bets on movement - if bigger moves are expected, your option is worth more. When IV drops, the market is saying \u0026ldquo;actually, calmer than we thought\u0026rdquo; - and your option loses value because the big move you were betting on now seems less likely.\u003c/p\u003e\n\u003cp\u003eYour $50 call has vega of 0.08. If IV goes from 45% to 46%, your call gains $0.08 per share - $8 per contract.\u003c/p\u003e\n\u003cdiv data-widget=\"vega-demo\" data-strike=\"50\" data-price=\"50\" data-iv=\"0.45\" data-dte=\"30\"\u003e\u003c/div\u003e\n\u003cp\u003eDrag the IV slider. Watch how the \u003cspan style=\"color:#22c55e; font-weight:bold;\"\u003egreen tangent line\u003c/span\u003e (vega) shows sensitivity to volatility changes.\u003c/p\u003e\n\u003cp\u003eThis is what makes earnings plays tricky. Before earnings, IV is high - uncertainty is priced in. After earnings, uncertainty resolves, IV collapses. You can be right about direction and still lose because IV dropped faster than the stock moved.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"back-to-the-surface\"\u003eBack to the surface\u003c/h2\u003e\n\u003cp\u003eThose arrows from the first surface:\u003c/p\u003e\n\u003cdiv data-widget=\"pricing-surface\" data-strike=\"50\" data-iv=\"0.45\"\u003e\u003c/div\u003e\n\u003cp\u003eNow you can read them.\u003c/p\u003e\n\u003cp\u003eEach point has two arrows: \u003cspan style=\"color:#3b82f6; font-weight:bold;\"\u003eblue (delta)\u003c/span\u003e points along stock price, \u003cspan style=\"color:#ef4444; font-weight:bold;\"\u003ered dashed (theta)\u003c/span\u003e points toward expiration.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCalm waters\u003c/strong\u003e (top left) - ATM, 45 days out. Both arrows moderate length. You have time, you have exposure.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eThe cliff\u003c/strong\u003e (bottom left) - ATM, 5 days out. The red theta arrow is longer - time decay is accelerating. The blue delta arrow is similar - stock sensitivity hasn\u0026rsquo;t changed much.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eDeep ITM\u003c/strong\u003e (right side) - $12 above strike, 30 days out. The blue delta arrow is long - this option moves nearly 1:1 with stock.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"how-they-interact\"\u003eHow they interact\u003c/h2\u003e\n\u003cp\u003eThe Greeks don\u0026rsquo;t operate in isolation. On any given day, all of them are working simultaneously.\u003c/p\u003e\n\u003cdiv data-widget=\"interactive-surface\" data-strike=\"50\" data-iv=\"0.45\"\u003e\u003c/div\u003e\n\u003cp\u003eDrag the point around. Watch delta and theta change as you move across the surface. Near expiration (the cliff), theta accelerates. Deep in the money, delta approaches 1. At the money with time left, you\u0026rsquo;re in the sweet spot - moderate exposure to everything.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"greek-summary\"\u003eGreek summary\u003c/h2\u003e\n\u003ctable\u003e\n  \u003cthead\u003e\n      \u003ctr\u003e\n          \u003cth\u003eGreek\u003c/th\u003e\n          \u003cth\u003eMeasures\u003c/th\u003e\n          \u003cth\u003eRange\u003c/th\u003e\n      \u003c/tr\u003e\n  \u003c/thead\u003e\n  \u003ctbody\u003e\n      \u003ctr\u003e\n          \u003ctd\u003eDelta\u003c/td\u003e\n          \u003ctd\u003eStock price sensitivity\u003c/td\u003e\n          \u003ctd\u003e0 to 1 (calls), -1 to 0 (puts)\u003c/td\u003e\n      \u003c/tr\u003e\n      \u003ctr\u003e\n          \u003ctd\u003eGamma\u003c/td\u003e\n          \u003ctd\u003eDelta\u0026rsquo;s rate of change\u003c/td\u003e\n          \u003ctd\u003eAlways positive\u003c/td\u003e\n      \u003c/tr\u003e\n      \u003ctr\u003e\n          \u003ctd\u003eTheta\u003c/td\u003e\n          \u003ctd\u003eTime decay per day\u003c/td\u003e\n          \u003ctd\u003eNegative for long options\u003c/td\u003e\n      \u003c/tr\u003e\n      \u003ctr\u003e\n          \u003ctd\u003eVega\u003c/td\u003e\n          \u003ctd\u003eIV sensitivity\u003c/td\u003e\n          \u003ctd\u003eAlways positive\u003c/td\u003e\n      \u003c/tr\u003e\n  \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003eWhen you\u0026rsquo;re long options: delta is your directional bet, gamma means big moves help you, theta works against you, vega helps on IV spikes.\u003c/p\u003e\n\u003cp\u003eWhen you\u0026rsquo;re short options, the signs flip. Theta works for you. Vega works against you. That\u0026rsquo;s \u003ca href=\"/posts/options-primer-5-selling-options/\"\u003ePart 5\u003c/a\u003e.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"what-the-greeks-dont-tell-you\"\u003eWhat the Greeks don\u0026rsquo;t tell you\u003c/h2\u003e\n\u003cp\u003eThe Greeks are local. They tell you what happens for small moves right now. They don\u0026rsquo;t tell you which direction the stock moves, what IV will do, or how big the move will be.\u003c/p\u003e\n\u003cp\u003eYou can have perfect Greek management and still lose money because the stock went the wrong way.\u003c/p\u003e\n\u003chr\u003e\n\u003cp\u003e\u003ca href=\"/posts/options-primer-5-selling-options/\"\u003ePart 5\u003c/a\u003e: you stop buying options and start selling them.\u003c/p\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cem\u003eThis is \u003ca href=\"/tags/options/\"\u003ePart 4 of a 5-part series\u003c/a\u003e on options. \u003ca href=\"/posts/options-primer-3-volatility/\"\u003ePart 3\u003c/a\u003e covers volatility. \u003ca href=\"/posts/options-primer-5-selling-options/\"\u003ePart 5\u003c/a\u003e covers selling options.\u003c/em\u003e\u003c/p\u003e\n","date_published":"2024-12-24T05:00:00Z","date_modified":"2024-12-24T05:00:00Z","tags":["options","finance","interactive"]},{"id":"https://journal.ehrlich.dev/posts/options-primer-3-volatility/","url":"https://journal.ehrlich.dev/posts/options-primer-3-volatility/","title":"Options Primer Part 3: Volatility","summary":"Part 3 of 5. Historical vs implied volatility, IV crush, and why the stock can go up while your call goes down.","content_html":"\u003cp\u003eThe trading simulator in \u003ca href=\"/posts/options-primer-2-what-options-are/\"\u003ePart 2\u003c/a\u003e was rigged. The stock drifted whichever way made your positions profitable.\u003c/p\u003e\n\u003cp\u003eReal markets are more complicated. WidgetCo is about to have a bad quarter.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"two-kinds-of-volatility\"\u003eTwo kinds of volatility\u003c/h2\u003e\n\u003cp\u003eVolatility measures how much a stock moves. But there are two different measurements, and understanding the gap between them is half of options trading.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eHistorical volatility (HV)\u003c/strong\u003e looks backward. Take the last 30 days of prices. Calculate how much the stock moved day-to-day. Annualize it. That\u0026rsquo;s HV.\u003c/p\u003e\n\u003cp\u003eWidgetCo\u0026rsquo;s HV is about 22%. Over the past month, the stock has been moving at a 22% annualized rate. Not wild, not dead. Normal industrial company stuff.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eImplied volatility (IV)\u003c/strong\u003e looks forward. Or rather, it\u0026rsquo;s what the market \u003cem\u003eimplies\u003c/em\u003e about the future by how it prices options.\u003c/p\u003e\n\u003cp\u003eTo understand IV, we need to talk about pricing models.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"the-spherical-cow\"\u003eThe spherical cow\u003c/h2\u003e\n\u003cp\u003ePhysicists have a joke about simplifying a cow to a sphere to make the math tractable. The \u003ca href=\"https://en.wikipedia.org/wiki/Spherical_cow\"\u003espherical cow\u003c/a\u003e isn\u0026rsquo;t accurate, but it\u0026rsquo;s useful.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model\"\u003eBlack-Scholes\u003c/a\u003e is the spherical cow of options pricing. It assumes stock prices move smoothly in a random walk, volatility stays constant, you can trade continuously with no fees, and a bunch of other things that aren\u0026rsquo;t true. The model won a Nobel Prize anyway, because it\u0026rsquo;s useful.\u003c/p\u003e\n\u003cp\u003eMore sophisticated models exist: \u003ca href=\"https://en.wikipedia.org/wiki/Binomial_options_pricing_model\"\u003ebinomial trees\u003c/a\u003e that let prices jump discretely, \u003ca href=\"https://en.wikipedia.org/wiki/Monte_Carlo_methods_for_option_pricing\"\u003eMonte Carlo simulations\u003c/a\u003e that handle path-dependent options, \u003ca href=\"https://en.wikipedia.org/wiki/Heston_model\"\u003estochastic volatility models\u003c/a\u003e that let volatility itself fluctuate. Those are beyond scope here.\u003c/p\u003e\n\u003cp\u003eFor this series, we\u0026rsquo;re trying to learn to drive, not build the engine. I implemented Black-Scholes for the interactive widgets on this page. It\u0026rsquo;s a black box: stock price, strike, time to expiration, interest rate, volatility in. Option price out. Good enough to illustrate the concepts.\u003c/p\u003e\n\u003cp\u003eThe trick: we already know the option\u0026rsquo;s price - it\u0026rsquo;s whatever the market is trading at. Stock price, strike, expiration, interest rate - all known. Only one unknown: volatility.\u003c/p\u003e\n\u003cp\u003eSo, given all these caveats, if we run my flawed black box in reverse, we get a value for \u003ccode\u003evolatility\u003c/code\u003e. That\u0026rsquo;s the (according-to-my-flawed-algorithm) \u003cem\u003eimplied volatility\u003c/em\u003e - what the market implies about future movement by how it prices options today.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"log-returns\"\u003eLog returns\u003c/h2\u003e\n\u003cp\u003eLog returns are how HV and IV are actually expressed mathematically. Understanding them helped the concept click for me.\u003c/p\u003e\n\u003cp\u003eWhy \u003cem\u003elog\u003c/em\u003e returns instead of regular percent changes?\u003c/p\u003e\n\u003cp\u003eSimple return: stock goes from $100 to $110, that\u0026rsquo;s +10%. Goes from $110 to $100, that\u0026rsquo;s -9.1%. You\u0026rsquo;re back where you started, but +10% and -9.1% don\u0026rsquo;t cancel.\u003c/p\u003e\n\u003cp\u003eLog return: \\(\\ln(110/100) = +9.53%\\). \\(\\ln(100/110) = -9.53%\\). They cancel exactly. Log returns are symmetric and additive - they compound correctly over time. This is why volatility is measured in log space and why pricing models assume \u003ca href=\"https://en.wikipedia.org/wiki/Log-normal_distribution\"\u003elog-normal\u003c/a\u003e price distributions.\u003c/p\u003e\n\u003cp\u003eThe formula for historical volatility:\u003c/p\u003e\n\u003cp\u003e$$\\sigma = \\sqrt{\\frac{252}{n} \\sum_{i=1}^{n} \\left( \\ln \\frac{P_i}{P_{i-1}} \\right)^2}$$\u003c/p\u003e\n\u003cp\u003eTake each day\u0026rsquo;s log return, square it, average them, take the square root. The 252 is trading days per year - it annualizes the daily volatility.\u003c/p\u003e\n\u003cp\u003eWhat does this number actually mean? An IV of 20% says the market expects the stock\u0026rsquo;s annual log returns to have a standard deviation of 20%. If returns are roughly normal:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e~68% of the time, the stock ends the year within ±20% of where it started\u003c/li\u003e\n\u003cli\u003e~95% of the time, within ±40%\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eFor daily movement, divide by \\(\\sqrt{252} \\approx 16\\). A 20% annual volatility means roughly 1.25% expected daily movement. WidgetCo at 22% HV moves about 1.4% on a typical day.\u003c/p\u003e\n\u003cp\u003eWhen you see \u0026ldquo;IV: 25%\u0026rdquo; on Yahoo Finance or Robinhood, that\u0026rsquo;s this number - annualized standard deviation of log returns.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"back-to-widgetco\"\u003eBack to WidgetCo\u003c/h2\u003e\n\u003cp\u003eWidgetCo\u0026rsquo;s IV is 25%. The market expects slightly more movement than has been happening. Maybe the new PE ownership adds uncertainty.\u003c/p\u003e\n\u003cp\u003eWhen IV \u0026gt; HV, options are \u0026ldquo;expensive\u0026rdquo; - you\u0026rsquo;re paying for more movement than has occurred.\u003c/p\u003e\n\u003cp\u003eWhen IV \u0026lt; HV, options are \u0026ldquo;cheap\u0026rdquo; - you\u0026rsquo;re paying for less movement than has occurred.\u003c/p\u003e\n\u003cp\u003eNeither tells you which way the stock moves.\u003c/p\u003e\n\u003cp\u003eLet\u0026rsquo;s see what happens when volatility changes.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"the-recall\"\u003eThe recall\u003c/h2\u003e\n\u003cp\u003eIt\u0026rsquo;s Q3 2020. WidgetCo\u0026rsquo;s new manufacturing line has been running for six months. The PE firm pushed an aggressive timeline - 18 months to double capacity. Quality control got squeezed.\u003c/p\u003e\n\u003cp\u003eNews breaks on a Tuesday morning: WidgetCo recalling the Widget Pro line. Stress fractures. Twelve injuries reported, no deaths. Lawsuits pending.\u003c/p\u003e\n\u003cp\u003eStock opens at $102. By lunch, $88. By close, $79. Over the next week, it bottoms at $65.\u003c/p\u003e\n\u003cp\u003eBefore the recall: IV at 25%.\nDuring the crash: spikes to 65%.\nA week later: settles around 55%.\u003c/p\u003e\n\u003cp\u003eThe stock dropped 36%. IV nearly tripled.\u003c/p\u003e\n\u003cp\u003eUncertainty. Will there be more recalls? How big are the lawsuits? Could the company go bankrupt? Nobody knows.\u003c/p\u003e\n\u003cp\u003eOptions are insurance. That uncertainty gets priced into them the same way hurricane season gets priced into Florida homeowners policies.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"your-call-during-the-crash\"\u003eYour call during the crash\u003c/h2\u003e\n\u003cp\u003eSay you held a $100 call from Part 2. You bought it for $4.20 when the stock was at $102.\u003c/p\u003e\n\u003cp\u003eBy close, the stock is at $79. Your $100 call is now out of the money - if expiration were today, it would be worthless. But expiration is 30 days away.\u003c/p\u003e\n\u003cdiv data-widget=\"crash-call\" data-strike=\"100\" data-entry-price=\"4.20\" data-constant-iv=\"25\"\u003e\u003c/div\u003e\n\u003cp\u003eDrag the slider to watch IV rise as the stock falls. Two lines: orange shows actual value (IV spiking), gray shows hypothetical value (IV constant at 25%).\u003c/p\u003e\n\u003cp\u003eAt $79, the stock needs to rally 27% to reach your $100 strike. With 25% IV, the market expects about 25% movement \u003cem\u003eper year\u003c/em\u003e. A 27% move in 30 days? The math says nearly impossible. Your call is worthless.\u003c/p\u003e\n\u003cp\u003eBut IV isn\u0026rsquo;t 25% anymore. It\u0026rsquo;s 65%. The market is saying: we have no idea what happens next. Lawsuits, more recalls, bankruptcy, or maybe a surprise recovery. With that much uncertainty, a 27% swing isn\u0026rsquo;t crazy. Your call is worth $0.84.\u003c/p\u003e\n\u003cp\u003eThat\u0026rsquo;s what IV \u003cem\u003eis\u003c/em\u003e. It\u0026rsquo;s the market\u0026rsquo;s estimate of how much the stock might move. Higher IV means bigger expected swings.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"volatility-collapse\"\u003eVolatility collapse\u003c/h2\u003e\n\u003cp\u003eQ1 2021. WidgetCo stabilizes around $52. The lawsuits are pending. Management is \u0026ldquo;restructuring.\u0026rdquo; Everyone\u0026rsquo;s waiting for the next earnings call.\u003c/p\u003e\n\u003cp\u003eIV spikes to 85% in the days before earnings. Why? Because nobody knows what will be announced. Revenue up or down? More recalls? Bankruptcy? A surprise turnaround? The stock could go anywhere. That uncertainty is what IV measures.\u003c/p\u003e\n\u003cp\u003eYou buy a $55 call for $3.20. The option is expensive because the market expects big moves.\u003c/p\u003e\n\u003cp\u003eEarnings drop. Stock goes from $52 to $56. You were right about direction.\u003c/p\u003e\n\u003cp\u003ePost-earnings: Your $55 call is worth $2.10. You lost $1.10.\u003c/p\u003e\n\u003cdiv data-widget=\"iv-crush\" data-pre-iv=\"85\" data-post-iv=\"40\" data-strike=\"55\" data-pre-price=\"52\" data-post-price=\"56\"\u003e\u003c/div\u003e\n\u003cp\u003eIV collapsed from 85% to 40%. The earnings announcement resolved the uncertainty. Now everyone knows: revenue down, restructuring plan in place, lawsuits ongoing. The range of possible futures just narrowed. The market no longer expects 85% annualized movement - more like 40%.\u003c/p\u003e\n\u003cp\u003eYour option was priced for a world of 85% volatility. Now it\u0026rsquo;s repriced for 40%. The stock moved $4 in your favor, but the expected-movement component of your option\u0026rsquo;s value shrank faster than the stock-moved-toward-strike component grew.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"the-volatility-smile\"\u003eThe volatility smile\u003c/h2\u003e\n\u003cp\u003eIV isn\u0026rsquo;t one number. Every option has its own IV, depending on the strike price.\u003c/p\u003e\n\u003cdiv data-widget=\"vol-smile\" data-atm-iv=\"45\" data-skew=\"0.15\"\u003e\u003c/div\u003e\n\u003cp\u003eDrag the slider to see IV at different strikes.\u003c/p\u003e\n\u003cp\u003eThe curve isn\u0026rsquo;t flat - it\u0026rsquo;s a \u003cstrong\u003esmile\u003c/strong\u003e. Out-of-the-money puts (left side) have higher IV than at-the-money options (center). That\u0026rsquo;s \u003cstrong\u003eskew\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003eThink about what a $35 put actually insures against. WidgetCo at $52 falling to $35 - that\u0026rsquo;s a 33% crash. In a world where that happens, it\u0026rsquo;s making headlines. The company is in crisis. Volatility spikes. We watched exactly this during the recall.\u003c/p\u003e\n\u003cp\u003eIf you\u0026rsquo;re selling that insurance, you price it accordingly. Extreme scenarios cost more. When we run Black-Scholes backward on these pricier contracts, we get higher implied volatility.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"the-dark-years\"\u003eThe dark years\u003c/h2\u003e\n\u003cp\u003e2021-2023 in the simulator. WidgetCo stuck between $45-60. Three CEOs. Constant uncertainty.\u003c/p\u003e\n\u003cdiv data-widget=\"trading-sim\" data-cooperative=\"false\" data-price=\"52\" data-iv=\"0.45\" data-chaos=\"true\"\u003e\u003c/div\u003e\n\u003cp\u003eNo cooperative mode.\u003c/p\u003e\n\u003cp\u003eA few things to notice as you trade:\u003c/p\u003e\n\u003cp\u003eIV stays elevated. Options are perpetually expensive. The premium you pay reflects real uncertainty about this company\u0026rsquo;s future.\u003c/p\u003e\n\u003cp\u003eEvery earnings is an IV collapse. Before earnings: IV spikes. After earnings: IV collapses. Regardless of which way the stock moves.\u003c/p\u003e\n\u003cp\u003eNews gaps aren\u0026rsquo;t tradeable after the fact. New CEO announced: stock gaps 15% overnight. If you weren\u0026rsquo;t positioned before, you missed it. After the gap, IV collapses - the news is priced in.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"what-this-means\"\u003eWhat this means\u003c/h2\u003e\n\u003cp\u003eWhen you buy an option, you\u0026rsquo;re not just betting on direction.\u003c/p\u003e\n\u003cp\u003eThe option\u0026rsquo;s value depends on IV. If you buy when IV is high and it drops, you lose money even if the stock moves your way. If you buy when IV is low and it spikes, you make money even if the stock barely moves.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"/posts/options-primer-2-what-options-are/\"\u003ePart 2\u003c/a\u003e mentioned that delta isn\u0026rsquo;t the whole story. Now you see why. The stock can go up and your call can go down - if IV falls faster than the stock rises.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"/posts/options-primer-4-the-greeks/\"\u003ePart 4\u003c/a\u003e formalizes all of this. Delta measures your exposure to stock movement. Vega measures your exposure to IV. Theta measures what time costs you. They\u0026rsquo;re all partial derivatives of the option price. But you can\u0026rsquo;t understand what they\u0026rsquo;re measuring until you understand volatility.\u003c/p\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cem\u003eThis is \u003ca href=\"/tags/options/\"\u003ePart 3 of a 5-part series\u003c/a\u003e on options. \u003ca href=\"/posts/options-primer-2-what-options-are/\"\u003ePart 2\u003c/a\u003e covers what options are. \u003ca href=\"/posts/options-primer-4-the-greeks/\"\u003ePart 4\u003c/a\u003e covers the Greeks.\u003c/em\u003e\u003c/p\u003e\n","date_published":"2024-12-24T04:00:00Z","date_modified":"2024-12-24T04:00:00Z","tags":["options","finance","interactive"]},{"id":"https://journal.ehrlich.dev/posts/options-primer-2-what-options-are/","url":"https://journal.ehrlich.dev/posts/options-primer-2-what-options-are/","title":"Options Primer Part 2: What Options Are","summary":"Part 2 of 5. Calls, puts, strikes, and premiums. Interactive trading simulator included.","content_html":"\u003ch2 id=\"whats-an-option\"\u003eWhat\u0026rsquo;s an option?\u003c/h2\u003e\n\u003cp\u003eHere\u0026rsquo;s how \u003ca href=\"https://en.wikipedia.org/wiki/Option_(finance)\"\u003eWikipedia defines it\u003c/a\u003e:\u003c/p\u003e\n\u003cblockquote\u003e\n\u003cp\u003eIn finance, an option is a contract which conveys to its owner, the holder, the right, but not the obligation, to buy or sell a specific quantity of an underlying asset or instrument at a specified strike price on or before a specified date.\u003c/p\u003e\n\u003c/blockquote\u003e\n\u003cp\u003eOne phrase at a time:\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u0026ldquo;The right, but not the obligation\u0026rdquo;\u003c/strong\u003e - You \u003cem\u003ecan\u003c/em\u003e do something, but you don\u0026rsquo;t \u003cem\u003ehave\u003c/em\u003e to. If exercising the option would lose you money, you just\u0026hellip; don\u0026rsquo;t. You walk away.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u0026ldquo;To buy or sell\u0026rdquo;\u003c/strong\u003e - There are two flavors. The right to \u003cem\u003ebuy\u003c/em\u003e is called a \u003cstrong\u003ecall\u003c/strong\u003e. The right to \u003cem\u003esell\u003c/em\u003e is called a \u003cstrong\u003eput\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u0026ldquo;At a specified strike price\u0026rdquo;\u003c/strong\u003e - The price is locked in when you buy the option. Doesn\u0026rsquo;t matter what happens to the market afterward.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u0026ldquo;On or before a specified date\u0026rdquo;\u003c/strong\u003e - Options expire. After that date, they\u0026rsquo;re worthless. (We\u0026rsquo;ll use \u0026ldquo;American-style\u0026rdquo; options, which can be exercised anytime before expiration.)\u003c/p\u003e\n\u003cp\u003eThe catch? You pay for this right upfront. That payment is called the \u003cstrong\u003epremium\u003c/strong\u003e. If you never use the option, you lose the premium.\u003c/p\u003e\n\u003ch2 id=\"meet-widgetco\"\u003eMeet WidgetCo\u003c/h2\u003e\n\u003cp\u003eWe need a stock to trade. Enter WidgetCo (ticker: WGT).\u003c/p\u003e\n\u003cp\u003e\u003cem\u003e(\u0026ldquo;Widget\u0026rdquo; is a \u003ca href=\"https://en.wikipedia.org/wiki/Widget\"\u003eplaceholder name\u003c/a\u003e economists use for a generic product. WidgetCo is our fictional company - a bland, stable industrial manufacturer. Think of it as \u0026ldquo;Generic Corp.\u0026rdquo;)\u003c/em\u003e\u003c/p\u003e\n\u003cp\u003eWidgetCo was founded in 1952 by Earl Widget in Akron, Ohio. Three generations of the Widget family ran it until 2019, when the last heir sold to a private equity consortium. Revenue: $2.3 billion. Employees: 12,000.\u003c/p\u003e\n\u003cp\u003eStock\u0026rsquo;s at $100.\u003c/p\u003e\n\u003ch2 id=\"whats-a-call-option\"\u003eWhat\u0026rsquo;s a call option?\u003c/h2\u003e\n\u003cp\u003eA call is a contract. It says: I have the right to buy this stock at a fixed price, no matter what happens.\u003c/p\u003e\n\u003cp\u003eWidgetCo is at $100. You buy a \u003cstrong\u003e$100 call\u003c/strong\u003e expiring in 30 days. It costs you $4.\u003c/p\u003e\n\u003cp\u003eThat $100 is the \u003cstrong\u003estrike price\u003c/strong\u003e. It\u0026rsquo;s locked in. Doesn\u0026rsquo;t matter what happens next.\u003c/p\u003e\n\u003cp\u003eSay WidgetCo announces a breakthrough widget. Stock jumps to $150. Your call says: \u0026ldquo;I can buy at $100.\u0026rdquo; So you do. You buy at $100, sell at $150, pocket $50 per share. Minus the $4 premium, that\u0026rsquo;s $46 profit per share.\u003c/p\u003e\n\u003cp\u003eOr say WidgetCo\u0026rsquo;s factory burns down. Stock drops to $60. Your call still says \u0026ldquo;I can buy at $100\u0026rdquo; - but why would you? You wouldn\u0026rsquo;t. The contract expires unused. You\u0026rsquo;re out the $4 you paid, nothing more.\u003c/p\u003e\n\u003cp\u003eThat $4 is the \u003cstrong\u003epremium\u003c/strong\u003e. It\u0026rsquo;s what you pay for the right. If you never use the right, you lose the premium. If you use it, your profit is whatever you made minus the premium.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eA note on pricing:\u003c/strong\u003e Options are quoted per share, but each contract controls 100 shares. So a \u0026ldquo;$4 option\u0026rdquo; actually costs $400 to buy. When you exercised at $100 and sold at $150, you made $50 per share - but that\u0026rsquo;s $5,000 total (100 × $50), minus the $400 you paid, for $4,600 profit.\u003c/p\u003e\n\u003cp\u003eThis series quotes prices per share because that\u0026rsquo;s how you\u0026rsquo;ll see them displayed. Just remember: multiply by 100 for the actual cash changing hands.\u003c/p\u003e\n\u003cp\u003eDrag the slider to see how your call\u0026rsquo;s value at expiration depends on the stock price:\u003c/p\u003e\n\u003cdiv data-widget=\"payoff\" data-type=\"call\" data-strike=\"100\" data-premium=\"4\" data-stock-price=\"100\"\u003e\u003c/div\u003e\n\u003cp\u003eBelow $100: flat at -$4. You lost the premium, nothing more. Above $100: slopes upward. You\u0026rsquo;re making money.\u003c/p\u003e\n\u003ch2 id=\"whats-a-put-option\"\u003eWhat\u0026rsquo;s a put option?\u003c/h2\u003e\n\u003cp\u003eA put is the mirror image. It says: I have the right to sell this stock at a fixed price, no matter what happens.\u003c/p\u003e\n\u003cp\u003eWidgetCo is at $100. You buy a \u003cstrong\u003e$100 put\u003c/strong\u003e expiring in 30 days. Costs $3.50.\u003c/p\u003e\n\u003cp\u003eSay WidgetCo\u0026rsquo;s CEO gets arrested for fraud. Stock crashes to $50. Your put says: \u0026ldquo;I can sell at $100.\u0026rdquo; So you buy shares at $50 on the open market, then immediately sell them at $100 using your put. That\u0026rsquo;s $50 profit, minus the $3.50 premium = $46.50.\u003c/p\u003e\n\u003cp\u003eOr say WidgetCo announces record earnings. Stock jumps to $130. Your put says \u0026ldquo;I can sell at $100\u0026rdquo; - but the market will pay $130. Why sell at $100? You wouldn\u0026rsquo;t. Contract expires unused. You\u0026rsquo;re out $3.50.\u003c/p\u003e\n\u003cdiv data-widget=\"payoff\" data-type=\"put\" data-strike=\"100\" data-premium=\"3.50\" data-stock-price=\"100\"\u003e\u003c/div\u003e\n\u003cp\u003eAbove $100: flat at -$3.50. Below $100: slopes upward.\u003c/p\u003e\n\u003ch2 id=\"try-it---make-some-money\"\u003eTry it - make some money\u003c/h2\u003e\n\u003cp\u003eTime to play. The simulator below lets you buy calls and puts on WidgetCo, then fast-forward time to see what happens.\u003c/p\u003e\n\u003cdiv data-widget=\"trading-sim\" data-cooperative=\"true\" data-price=\"100\" data-iv=\"0.25\"\u003e\u003c/div\u003e\n\u003cp\u003eThe stock is being nice to you right now. Cooperative mode: it tends to drift in whatever direction makes your positions profitable. This is practice.\u003c/p\u003e\n\u003cp\u003eTry this:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eBuy a $100 call for around $4\u003c/li\u003e\n\u003cli\u003eHit \u0026ldquo;Fast Forward\u0026rdquo; to advance a week\u003c/li\u003e\n\u003cli\u003eWatch the stock drift up. Your call gains value.\u003c/li\u003e\n\u003cli\u003eSell it. You just made money.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003eNow try a put:\u003c/p\u003e\n\u003col\u003e\n\u003cli\u003eReset the simulator\u003c/li\u003e\n\u003cli\u003eBuy a $100 put for around $3.50\u003c/li\u003e\n\u003cli\u003eFast forward. Stock drifts down.\u003c/li\u003e\n\u003cli\u003eYour put is now worth more. Sell it. Profit.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003ePlay around. Get a feel for it.\u003c/p\u003e\n\u003ch2 id=\"the-relationship-isnt-11\"\u003eThe relationship isn\u0026rsquo;t 1:1\u003c/h2\u003e\n\u003cp\u003eAs you played, you might have noticed: the stock moved $8, but your option didn\u0026rsquo;t move $8. It moved less.\u003c/p\u003e\n\u003cp\u003eThat\u0026rsquo;s \u003cstrong\u003edelta\u003c/strong\u003e - the ratio of option movement to stock movement.\u003c/p\u003e\n\u003cp\u003eYour call has delta of about +0.5. That means for every $1 the stock moves up, your call gains about $0.50. Your put has delta of about -0.5 - it gains $0.50 for every $1 the stock moves \u003cem\u003edown\u003c/em\u003e.\u003c/p\u003e\n\u003cp\u003eWe\u0026rsquo;ll formalize delta in \u003ca href=\"/posts/options-primer-4-the-greeks/\"\u003ePart 4\u003c/a\u003e. For now, just notice: options amplify your exposure (you paid $4 to control $100 of stock) but they don\u0026rsquo;t move 1:1.\u003c/p\u003e\n\u003ch2 id=\"why-delta-isnt-constant\"\u003eWhy delta isn\u0026rsquo;t constant\u003c/h2\u003e\n\u003cp\u003eWhen the stock is at $100 and your strike is $100:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eCall delta is around +0.5\u003c/li\u003e\n\u003cli\u003ePut delta is around -0.5\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eBut watch what happens as the stock moves.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eStock rises to $110:\u003c/strong\u003e\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eCall delta approaches +1.0 (deep \u0026ldquo;in the money\u0026rdquo; - moves almost 1:1 with stock)\u003c/li\u003e\n\u003cli\u003ePut delta approaches 0 (deep \u0026ldquo;out of the money\u0026rdquo; - barely moves)\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003e\u003cstrong\u003eStock falls to $90:\u003c/strong\u003e\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003eCall delta approaches 0 (deep OTM, barely moves)\u003c/li\u003e\n\u003cli\u003ePut delta approaches -1.0 (deep ITM, moves almost 1:1 inverse with stock)\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThis is the second derivative from \u003ca href=\"/posts/options-primer-1-background/\"\u003ePart 1\u003c/a\u003e. Delta itself has a rate of change. That\u0026rsquo;s called \u003cstrong\u003egamma\u003c/strong\u003e. We\u0026rsquo;ll cover it in \u003ca href=\"/posts/options-primer-4-the-greeks/\"\u003ePart 4\u003c/a\u003e.\u003c/p\u003e\n\u003ch2 id=\"what-youre-actually-buying\"\u003eWhat you\u0026rsquo;re actually buying\u003c/h2\u003e\n\u003cp\u003eIf options were perfectly priced, directional betting would break even. The $4 premium already reflects the market\u0026rsquo;s probability estimate. Over enough trades, wins and losses balance out.\u003c/p\u003e\n\u003cp\u003eSo what are you paying for?\u003c/p\u003e\n\u003cp\u003eExposure to volatility - whether the stock moves more or less than the market expects.\u003c/p\u003e\n\u003cp\u003eThe option expires eventually. Every day that passes, it loses value. The right to buy at $100 is worth something today. Worth less tomorrow. Worth nothing after expiration.\u003c/p\u003e\n\u003cp\u003eAnd if the stock might move a lot, options are worth more. Dead calm? Worth less.\u003c/p\u003e\n\u003cp\u003eWhen you buy options, you\u0026rsquo;re betting the stock moves more than expected. When you sell them (\u003ca href=\"/posts/options-primer-5-selling-options/\"\u003ePart 5\u003c/a\u003e), you\u0026rsquo;re betting it moves less.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"/posts/options-primer-3-volatility/\"\u003ePart 3\u003c/a\u003e covers volatility - the key input to option pricing.\u003c/p\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cem\u003eThis is \u003ca href=\"/tags/options/\"\u003ePart 2 of a 5-part series\u003c/a\u003e on options. \u003ca href=\"/posts/options-primer-1-background/\"\u003ePart 1\u003c/a\u003e covers background. \u003ca href=\"/posts/options-primer-3-volatility/\"\u003ePart 3\u003c/a\u003e covers volatility.\u003c/em\u003e\u003c/p\u003e\n","date_published":"2024-12-24T02:00:00Z","date_modified":"2024-12-24T02:00:00Z","tags":["options","finance","interactive"]},{"id":"https://journal.ehrlich.dev/posts/options-primer-1-background/","url":"https://journal.ehrlich.dev/posts/options-primer-1-background/","title":"Options Primer Part 1: Background","summary":"Part 1 of 5. What stocks are, what derivatives measure, and why options pricing borrows from calculus.","content_html":"\u003cp\u003eThis is part one of a five-part series on options. We\u0026rsquo;re building toward understanding how options are priced and how to trade them. But first, two pieces of background: what stocks are, and what derivatives are.\u003c/p\u003e\n\u003cp\u003eNot financial derivatives - calculus derivatives. Rates of change. The options world borrowed the word \u0026ldquo;derivative\u0026rdquo; from math because options prices are derived from stock prices. To understand how that derivation works, you need to understand what a derivative measures.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eSkip ahead:\u003c/strong\u003e If you already know what stocks are and you\u0026rsquo;re comfortable with calculus (slopes of curves, rates of change), \u003ca href=\"/posts/options-primer-2-what-options-are/\"\u003eskip to Part 2\u003c/a\u003e.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"whats-a-stock\"\u003eWhat\u0026rsquo;s a stock?\u003c/h2\u003e\n\u003cp\u003eA stock is a piece of ownership in a company.\u003c/p\u003e\n\u003cp\u003eWidgetCo has 1 million shares outstanding. If you own 1,000 shares, you own 0.1% of WidgetCo. If WidgetCo is worth $100 million total, your 0.1% stake is worth $100,000. That\u0026rsquo;s $100 per share.\u003c/p\u003e\n\u003cp\u003eThe stock price is just: (what the market thinks the company is worth) / (number of shares).\u003c/p\u003e\n\u003cp\u003eWhen people say \u0026ldquo;WidgetCo is at $100,\u0026rdquo; they mean one share costs $100. If the company does well - makes money, grows, announces good news - people want to own it. Demand goes up, price goes up. If the company does poorly, people sell, price goes down.\u003c/p\u003e\n\u003cp\u003eA stock is a slice of a company. The price moves based on what people think that slice is worth.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"the-math-we-need\"\u003eThe math we need\u003c/h2\u003e\n\u003cp\u003eOptions are priced using rates of change. How fast does the option price change when the stock price moves? How fast does it decay as time passes? These are the \u0026ldquo;Greeks\u0026rdquo; you might have heard of - delta, gamma, theta, vega. They\u0026rsquo;re all measuring rates of change.\u003c/p\u003e\n\u003cp\u003eTo understand them, we need to understand one concept from calculus: the derivative. Not the financial kind - the mathematical kind.\u003c/p\u003e\n\u003ch2 id=\"slope-of-a-line\"\u003eSlope of a line\u003c/h2\u003e\n\u003cp\u003eStart simple. A straight line on a graph.\u003c/p\u003e\n\u003cp\u003eThe \u003cstrong\u003eslope\u003c/strong\u003e measures steepness: rise over run. How much the line goes up (or down) for each unit it moves across.\u003c/p\u003e\n\u003cp\u003eDrag the two points below. Watch the slope update.\u003c/p\u003e\n\u003cdiv data-widget=\"slope\"\u003e\u003c/div\u003e\n\u003cp\u003eA steep line has a large slope. A flat line has slope near zero. A line going downward has negative slope.\u003c/p\u003e\n\u003cp\u003eFor a straight line, the slope is the same everywhere. Pick any two points, calculate rise/run, you get the same number.\u003c/p\u003e\n\u003ch2 id=\"curves-dont-have-constant-slope\"\u003eCurves don\u0026rsquo;t have constant slope\u003c/h2\u003e\n\u003cp\u003eNow consider a curve. The parabola y = x², for instance.\u003c/p\u003e\n\u003cp\u003eThis curve bends. It\u0026rsquo;s steep in some places, flat in others. At x = -2, it\u0026rsquo;s diving down. At x = 0, it\u0026rsquo;s flat - the bottom of the U. At x = 2, it\u0026rsquo;s climbing up.\u003c/p\u003e\n\u003cp\u003eThere isn\u0026rsquo;t one slope. The steepness changes depending on where you are.\u003c/p\u003e\n\u003cp\u003eSo we ask a different question: what\u0026rsquo;s the slope \u003cem\u003eat a specific point\u003c/em\u003e?\u003c/p\u003e\n\u003ch2 id=\"zoom-in-until-it-looks-straight\"\u003eZoom in until it looks straight\u003c/h2\u003e\n\u003cp\u003eHere\u0026rsquo;s the key insight. If you zoom in far enough on any smooth curve, it starts to look like a straight line.\u003c/p\u003e\n\u003cp\u003eThat local straightness is what we measure. The line that just barely touches the curve at one point - the \u003cstrong\u003etangent line\u003c/strong\u003e - has a slope. That slope is the derivative at that point.\u003c/p\u003e\n\u003cp\u003eDrag the point along the curve. Watch the tangent line rotate. The slope display shows the derivative at each position.\u003c/p\u003e\n\u003cdiv data-widget=\"tangent\" data-start-x=\"1\"\u003e\u003c/div\u003e\n\u003cp\u003eAt x = 0, the tangent is flat. Slope is 0.\nAt x = 1, the tangent tilts up. Slope is 2.\nAt x = -1, the tangent tilts down. Slope is -2.\u003c/p\u003e\n\u003cp\u003eThe derivative of y = x² is 2x. Plug in any x value, get the slope at that point.\u003c/p\u003e\n\u003ch2 id=\"thats-a-derivative\"\u003eThat\u0026rsquo;s a derivative\u003c/h2\u003e\n\u003cp\u003eThe derivative at a point = the slope of the curve at that point.\u003c/p\u003e\n\u003cp\u003eDifferent notation, same idea:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003ef\u0026rsquo;(x) - \u0026ldquo;f prime of x\u0026rdquo;\u003c/li\u003e\n\u003cli\u003edy/dx - \u0026ldquo;the derivative of y with respect to x\u0026rdquo;\u003c/li\u003e\n\u003cli\u003eThe slope of the tangent line\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eThe derivative tells you: how fast is y changing when x changes? If the derivative is large and positive, y is climbing steeply. If it\u0026rsquo;s small and negative, y is drifting slowly downward. If it\u0026rsquo;s zero, y is momentarily flat.\u003c/p\u003e\n\u003ch2 id=\"the-derivative-as-its-own-curve\"\u003eThe derivative as its own curve\u003c/h2\u003e\n\u003cp\u003eThe derivative at each point is just a number. If we plot all those numbers, we get a new curve: the \u003cstrong\u003ederivative function\u003c/strong\u003e.\u003c/p\u003e\n\u003cp\u003eBelow, the top graph shows y = x³ with its tangent line (red). The bottom graph shows f\u0026rsquo;(x) = 3x² - the derivative curve. The orange line is the tangent to \u003cem\u003ethat\u003c/em\u003e curve - the \u003cstrong\u003esecond derivative\u003c/strong\u003e.\u003c/p\u003e\n\u003cdiv data-widget=\"tangent\" data-start-x=\"-2\" data-show-second-derivative=\"true\"\u003e\u003c/div\u003e\n\u003cp\u003eDrag either point. Three things update together:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\u003cstrong\u003eRed line\u003c/strong\u003e: slope of the cubic (first derivative, f\u0026rsquo;(x) = 3x²)\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eGreen dot\u003c/strong\u003e: your position on the derivative curve\u003c/li\u003e\n\u003cli\u003e\u003cstrong\u003eOrange line\u003c/strong\u003e: slope of the derivative curve (second derivative, f\u0026rsquo;\u0026rsquo;(x) = 6x)\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eAt x = 0, the red tangent is flat (slope = 0) and the orange line is also flat - the derivative curve has a minimum there. Move right: the orange line tilts up, meaning the slope is \u003cem\u003eincreasing\u003c/em\u003e. Move left: the orange line tilts down, meaning the slope is \u003cem\u003eincreasing in the other direction\u003c/em\u003e (becoming less negative).\u003c/p\u003e\n\u003cp\u003eThe second derivative tells you how the slope is changing. In options, delta (first derivative) measures price sensitivity. Gamma (second derivative) measures how fast that sensitivity changes - crucial for understanding risk.\u003c/p\u003e\n\u003chr\u003e\n\u003ch2 id=\"coming-up\"\u003eComing up\u003c/h2\u003e\n\u003cp\u003e\u003ca href=\"/posts/options-primer-2-what-options-are/\"\u003ePart 2\u003c/a\u003e introduces options through a company called WidgetCo. You\u0026rsquo;ll buy calls and puts in a trading simulator and watch your positions gain and lose value.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"/posts/options-primer-3-volatility/\"\u003ePart 3\u003c/a\u003e covers volatility - the uncertainty that determines what options are worth.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"/posts/options-primer-4-the-greeks/\"\u003ePart 4\u003c/a\u003e formalizes the Greeks. They\u0026rsquo;re derivatives applied to option prices. Delta measures sensitivity to stock price. Gamma measures how delta changes. Theta measures sensitivity to time. Vega measures sensitivity to volatility.\u003c/p\u003e\n\u003cp\u003e\u003ca href=\"/posts/options-primer-5-selling-options/\"\u003ePart 5\u003c/a\u003e: you stop buying options and start selling them. You become the insurance company.\u003c/p\u003e\n\u003cp\u003eThe math you just learned - slopes, rates of change, derivatives - that\u0026rsquo;s the foundation. Everything else builds on it.\u003c/p\u003e\n\u003chr\u003e\n\u003cp\u003e\u003cem\u003e\u003ca href=\"/posts/options-primer-2-what-options-are/\"\u003eContinue to Part 2: What Options Are\u003c/a\u003e\u003c/em\u003e\u003c/p\u003e\n","date_published":"2024-12-24T01:00:00Z","date_modified":"2024-12-24T01:00:00Z","tags":["options","finance","math"]},{"id":"https://journal.ehrlich.dev/posts/hello/","url":"https://journal.ehrlich.dev/posts/hello/","title":"Hello","summary":"Starting a journal. First up: a five-part series on options trading.","content_html":"\u003cp\u003eStarting a journal. \u0026ldquo;Blog\u0026rdquo; feels a bit dated but it\u0026rsquo;s the same thing. You can find it at \u003ca href=\"https://blog.ehrlich.dev\"\u003eblog.ehrlich.dev\u003c/a\u003e or \u003ca href=\"https://journal.ehrlich.dev\"\u003ejournal.ehrlich.dev\u003c/a\u003e - same place.\u003c/p\u003e\n\u003cp\u003eI use LLMs to help write this. If something reads like an AI wrote it, that\u0026rsquo;s a failure. Let me know: \u003ca href=\"mailto:journal@ehrlich.dev\"\u003ejournal@ehrlich.dev\u003c/a\u003e.\u003c/p\u003e\n\u003cp\u003eFirst up: people sometimes ask about options trading when it comes up that I do it for fun. The answer takes a while, so I wrote it down. Five parts, starting with \u003ca href=\"/posts/options-primer-1-background/\"\u003ewhat a derivative is\u003c/a\u003e.\u003c/p\u003e\n","date_published":"2024-12-24T00:00:00Z","date_modified":"2024-12-24T00:00:00Z","tags":["meta"]}]}