{ "version": 1, "note": "Lesson content is data. Adding a lesson means adding an entry here, not changing engine code. All wording is original; concepts follow standard references (Hull) but no text is reproduced. 'reviewed_by' must be filled in by someone with real derivatives knowledge before release — null means not yet reviewed, and the app says so.", "lessons": [ { "id": "what-is-an-option", "order": 1, "title": "What is an option?", "summary": "Calls, puts, and why a right is not the same as an obligation.", "estimated_minutes": 5, "reviewed_by": null, "reviewed_on": null, "questions": [ { "type": "choice", "id": "q1-put-definition", "prompt": "What does a put option give its holder?", "choices": [ { "text": "The right to sell the underlying at the strike price", "correct": true, "explanation": "A put is the right to sell. It becomes worth exercising when the underlying is below the strike, because selling at the strike then beats selling at the market price." }, { "text": "An obligation to sell the underlying at the strike price", "correct": false, "explanation": "The holder of any option never has an obligation. The obligation sits with the seller of the put, who must buy if the holder decides to exercise." }, { "text": "The right to buy the underlying at the strike price", "correct": false, "explanation": "That is a call. Calls and puts point in opposite directions: a call is a right to buy, a put is a right to sell." } ], "teaching_note": "Holder of a put sells; holder of a call buys. The other side of each contract does whatever they are told." }, { "type": "choice", "id": "q2-right-vs-obligation", "setup": "A trader sells a call option and receives the premium. At expiry the buyer decides to exercise.", "prompt": "What can the seller do?", "choices": [ { "text": "Exercise their own side of the contract first", "correct": false, "explanation": "There is no second right to exercise. An option contract has exactly one right, and it belongs to the buyer." }, { "text": "Nothing — they must deliver at the strike price", "correct": true, "explanation": "The seller gave up the choice when they took the premium. Once the buyer exercises, the seller has to deliver at the strike regardless of what the market price is." }, { "text": "Decline, and keep the premium instead", "correct": false, "explanation": "The premium is payment for accepting the obligation, not a fee that can be forfeited to escape it. There is no opt-out on the seller side." } ], "teaching_note": "This asymmetry is what the premium pays for. It is also why a seller's downside is not capped the way a buyer's is." }, { "type": "numeric", "id": "q3-contract-size", "setup": "A listed call option is quoted at a premium of $2.30. One contract covers 100 shares.", "prompt": "What does one contract cost the buyer, ignoring fees?", "answer": 230, "tolerance": 0.01, "unit_prefix": "$", "hint": "Quoted premiums are per share. Multiply by the number of shares the contract covers.", "explanation": "$2.30 per share x 100 shares = $230 for one contract.", "teaching_note": "Premiums are quoted per share but traded per contract. A contract that looks like pocket change at $2.30 is a $230 commitment — and every cent of it can be lost." }, { "type": "choice", "id": "q4-short-put-view", "setup": "You have sold a put struck at $50 and received the premium.", "prompt": "What outcome suits your position best?", "choices": [ { "text": "The underlying falls well below $50", "correct": false, "explanation": "That is the outcome that hurts a short put. You would be obliged to buy at $50 something now worth far less, and the loss can be many times the premium received." }, { "text": "The underlying rises as far as possible", "correct": false, "explanation": "A big rise does not help beyond the strike. Your gain is capped at the premium whether the underlying finishes at $51 or $510 — the extra move is worth nothing to you." }, { "text": "The underlying stays above $50 through expiry", "correct": true, "explanation": "Above the strike the put is not worth exercising, so it expires worthless and you keep the premium. That is the whole of your upside." } ], "teaching_note": "A short option position has a capped gain and an uncapped, or barely capped, loss. Sellers are paid up front precisely because that shape is unattractive." }, { "type": "choice", "id": "adv-1", "min_depth": 3, "prompt": "For a share that pays no dividends, when is it optimal to exercise an American call before expiry?", "teaching_note": "Exercising early throws away the remaining time value and pays the strike sooner than necessary. Selling the option instead always realises at least as much.", "choices": [ { "text": "Whenever volatility falls", "correct": false, "explanation": "Lower volatility reduces the option's value but does not make exercising early preferable to selling." }, { "text": "On the last day before expiry", "correct": false, "explanation": "At expiry exercise is simply settlement — the question is about exercising strictly earlier." }, { "text": "As soon as it is in the money", "correct": false, "explanation": "Being in the money says nothing about time value, which early exercise discards." }, { "text": "Never — selling it is always at least as good", "correct": true, "explanation": "Right. With no dividend there is no compensating cash flow, so early exercise only surrenders time value." } ] } ], "cards": [ { "type": "title", "kicker": "Lesson 1", "icon": "contract", "title": "What is an option?", "subtitle": "A contract where one side buys a choice, and the other side takes on an obligation." }, { "type": "text", "heading": "The same idea, in plain numbers", "body": "You pay $5 today for the right to buy one share at $100 any time up to the expiry date. That $5 is gone either way — it is the price of the right, not a deposit.", "bullets": [ "Share ends at $130: you buy at $100, so the right was worth $30.", "Share ends at $101: the right is worth $1 — better than nothing, but you still paid $5.", "Share ends at $90: you simply do not use it. It expires worthless." ], "highlight": "A right you never have to use. That is the whole idea.", "max_depth": 2 }, { "type": "text", "heading": "The core idea", "icon": "handshake", "body": "An option is an agreement about a possible future trade. It fixes the price today — the strike — and gives the buyer until a set date to decide whether to go ahead. The seller does not get to decide. If the buyer exercises, the seller must deliver.", "bullets": [ "The buyer pays for the choice. That payment is the premium.", "The seller keeps the premium whatever happens, and carries the obligation." ], "highlight": "One side buys a choice. The other side is paid to give theirs up." }, { "type": "term", "term": "Call option", "icon": "call", "definition": "The right to buy the underlying at the strike price.", "example": "A call struck at $100 lets you buy at $100 even if the shares are trading at $130." }, { "type": "term", "term": "Put option", "icon": "put", "definition": "The right to sell the underlying at the strike price.", "example": "A put struck at $100 lets you sell at $100 even if the shares have fallen to $70." }, { "type": "compare", "heading": "Right versus obligation", "left": { "label": "Buyer (long)", "icon": "buyer", "tone": "gain", "tagline": "Pays the premium, holds the right.", "points": [ "Can always walk away.", "Worst case: the premium is lost in full.", "Exercising is choosing to use the right." ] }, "right": { "label": "Seller (short)", "icon": "seller", "tone": "loss", "tagline": "Receives the premium, carries the obligation.", "points": [ "No exit if the buyer exercises.", "Worst case: losses well beyond the premium.", "Assignment is being on the receiving end." ] }, "footnote": "This asymmetry is the whole product. The seller was paid up front precisely because they gave up the choice." }, { "type": "choice", "prompt": "Before moving on, commit to an answer.", "question": "You buy a call and the underlying finishes below the strike. What must you do at expiry?", "options": [ { "text": "Buy the shares at the strike anyway", "correct": false, "explanation": "That would be an obligation — and the buyer never has one. Buying at the strike when the market is cheaper would just lock in a worse price." }, { "text": "Nothing. You walk away and the premium is gone", "correct": true, "explanation": "The option is a right, not a duty. When exercising has no benefit, the rational move is to let it expire — losing exactly the premium, no more." }, { "text": "Pay the seller the difference", "correct": false, "explanation": "The buyer never owes anything beyond the premium already paid. Open-ended outcomes belong to the seller's side of the contract." } ] }, { "type": "text", "heading": "Four positions, not two", "icon": "split", "body": "Every option can be bought or sold, so there are four basic positions. Which one you hold decides what you want the underlying to do — and how badly it can go.", "bullets": [ "Long call — gains if the underlying rises well above the strike.", "Long put — gains if the underlying falls well below the strike.", "Short call — you were paid a premium, and want the underlying to stay below the strike.", "Short put — you were paid a premium, and want the underlying to stay above the strike." ] }, { "type": "payoff", "heading": "A first look — long call, strike $100, premium $5", "caption": "A preview: profit at expiry against where the underlying finishes. Lesson 2 teaches this diagram properly — for now, just notice the shape.", "spot_min": 60, "spot_max": 140, "legs": [ { "kind": "call", "side": "long", "strike": 100, "premium": 5 } ] }, { "type": "payoff", "heading": "The mirror image — long put, strike $100, premium $6", "caption": "Same idea, flipped: this position wants the underlying to fall, not rise.", "spot_min": 60, "spot_max": 140, "legs": [ { "kind": "put", "side": "long", "strike": 100, "premium": 6 } ] }, { "type": "text", "heading": "What defines a contract", "icon": "contract", "body": "Two options on the same company can behave completely differently. These are the fields that make them different.", "bullets": [ "Underlying — the asset the contract refers to.", "Strike — the fixed price written into the agreement.", "Expiry — the date the right runs out.", "Style — European options are exercisable only at expiry; American ones any time up to it.", "Premium — the market price of the contract itself.", "Contract size — one listed contract usually covers 100 shares, so a quoted premium is per share." ] }, { "type": "warning", "heading": "An option has a deadline", "body": "It is a wasting asset. Being right too late produces the same result as being wrong.", "points": [ "If the underlying never moves usefully past the strike, a long option expires worthless and 100% of the premium is gone.", "That is an ordinary outcome, not a rare failure — plenty of options end their life at zero.", "Selling inverts the shape: the premium arrives up front, but the loss is not capped by the amount received." ] }, { "type": "text", "heading": "The payoff as a function", "body": "A European call's value at expiry is max(S − K, 0), where S is the price at expiry and K the strike. The max is where the asymmetry lives: the downside is truncated at zero, and everything else about an option follows from that kink.", "bullets": [ "The writer holds the mirror image, −max(S − K, 0), which is bounded above by zero.", "The buyer's loss is bounded by the premium; the writer's is not.", "European exercise fixes the decision at one date, which is what makes the closed-form price tractable.", "American exercise adds an optimal-stopping problem, and for a non-dividend-paying share early exercise of a call is never optimal." ], "min_depth": 3 }, { "type": "summary", "heading": "What to remember", "takeaways": [ "An option gives the buyer a choice and the seller an obligation, at a price fixed today.", "A call is the right to buy. A put is the right to sell.", "A buyer's worst case is losing the whole premium. A seller's worst case is larger." ] } ], "advanced_order": 1 }, { "id": "payoff-at-expiry", "order": 2, "title": "Payoff at expiry", "summary": "Reading a payoff diagram, and where break-even actually sits.", "estimated_minutes": 7, "reviewed_by": null, "reviewed_on": null, "questions": [ { "type": "numeric", "id": "q1-call-break-even", "setup": "A call option is struck at $80. The premium paid was $7 per share.", "prompt": "At what finishing price does the position break even?", "answer": 87, "tolerance": 0.01, "unit_prefix": "$", "hint": "Exercising only recovers what the underlying is worth above the strike. You need to recover the premium as well.", "explanation": "Break-even for a long call is strike plus premium: $80 + $7 = $87. At $87 exercising is worth exactly $7, which cancels the premium and leaves nothing either way.", "teaching_note": "The bend in the chart sits at the strike, but break-even sits one premium further out. Between the two, exercising is worth doing and the trade still loses money." }, { "type": "numeric", "id": "q2-put-profit", "setup": "You bought a put struck at $100 for a premium of $6. At expiry the underlying finishes at $88.", "prompt": "What is your profit per share?", "answer": 6, "tolerance": 0.01, "unit_prefix": "$", "hint": "Work out what the put is worth at expiry first, then subtract what it cost.", "explanation": "The put is worth $100 - $88 = $12 at expiry. Subtracting the $6 premium leaves a profit of $6 per share.", "teaching_note": "Payoff and profit are different numbers. The payoff here is $12; the profit is $6. Every chart in this app plots the second one." }, { "type": "choice", "id": "q3-short-call-max-gain", "setup": "You sold a call struck at $100 and received $5.", "prompt": "What is the most this position can make, and the most it can lose?", "choices": [ { "text": "Make and lose the same amount either way", "correct": false, "explanation": "The position is deliberately lopsided. A capped gain paired with an uncapped loss is exactly the shape the seller is being paid to accept." }, { "text": "Make $5 at most; the loss has no fixed limit", "correct": true, "explanation": "The premium is the entire upside. On the downside the loss grows with the underlying, and since there is no ceiling on how far a price can rise, an uncovered short call has no fixed maximum loss." }, { "text": "Make $5 at most; lose $100 at most", "correct": false, "explanation": "The $100 strike is not a floor under the loss. If the underlying finishes at $250 the loss is $145 per share, and nothing in the contract stops it going further." } ], "teaching_note": "A payoff chart is cropped at the edge of the page. For a short call the line keeps falling past that edge — the chart ending is not the risk ending." }, { "type": "choice", "id": "q4-exercise-vs-profit", "setup": "A call struck at $100 cost $5. At expiry the underlying finishes at $101.", "prompt": "Should the holder exercise, and did the position work out?", "choices": [ { "text": "Do not exercise — the position lost money", "correct": false, "explanation": "Refusing to exercise throws away the $1 the contract is worth and turns a $4 loss into a $5 loss. Take the recovery even when the trade overall was a loser." }, { "text": "Exercise, for a $1 profit", "correct": false, "explanation": "$1 is the payoff, not the profit. The $5 premium was already spent, so the position is $4 down." }, { "text": "Exercise, but the position still lost $4", "correct": true, "explanation": "Exercising is worth $1, so it beats letting the option expire. But $1 recovered against $5 paid is a net loss of $4 — worth exercising and profitable are separate questions." } ], "teaching_note": "Finishing above the strike is not the same as finishing ahead. That gap is the whole reason break-even is a separate number to remember." }, { "type": "choice", "id": "adv-2", "min_depth": 3, "prompt": "Put-call parity relates a call and a put with the same strike and expiry. What enforces it?", "teaching_note": "Parity is not a modelling assumption — it follows from the two portfolios having identical payoffs in every state.", "choices": [ { "text": "An assumption of constant volatility", "correct": false, "explanation": "Parity does not depend on volatility at all." }, { "text": "Exchange rules on option listings", "correct": false, "explanation": "It is an economic relationship, not a listing convention." }, { "text": "Arbitrage: the two sides have identical payoffs, so any gap is a free profit", "correct": true, "explanation": "Right. It holds independently of any pricing model." }, { "text": "The Black-Scholes formula", "correct": false, "explanation": "Black-Scholes is consistent with parity, but parity holds without it." } ] } ], "cards": [ { "type": "title", "kicker": "Lesson 2", "icon": "chart", "title": "Payoff at expiry", "subtitle": "At the deadline there is no time left, so an option is worth exactly what it is worth right then." }, { "type": "text", "heading": "Finding break-even without a formula", "body": "Break-even is just the point where what the option pays back equals what you paid for it.", "bullets": [ "You paid $5 for a call struck at $100.", "At $103 the call pays $3. You are still $2 down.", "At $105 it pays $5, exactly what you paid. That is break-even.", "Above $105 you are ahead." ], "highlight": "Break-even for a call is the strike plus the premium. Nothing more clever than that.", "max_depth": 2 }, { "type": "text", "heading": "Intrinsic value is all that remains", "icon": "clock", "body": "Before expiry an option carries extra value because the underlying might still move. At expiry that possibility is gone. What is left is intrinsic value: the amount by which exercising beats the market, or nothing at all.", "bullets": [ "Call at expiry: underlying price minus strike, or zero — whichever is larger.", "Put at expiry: strike minus underlying price, or zero.", "Never negative. Nobody is forced to use a right." ] }, { "type": "payoff", "heading": "Long call — strike $100, premium $5", "caption": "Below $100 the call is not worth exercising and the loss stays flat at the $5 premium. Above $100 it tracks the underlying dollar for dollar.", "spot_min": 60, "spot_max": 140, "legs": [ { "kind": "call", "side": "long", "strike": 100, "premium": 5 } ] }, { "type": "text", "heading": "Reading the diagram", "icon": "compass", "body": "The horizontal axis is where the underlying finishes at expiry. The vertical axis is profit or loss on the position — the payoff, net of what you paid.", "bullets": [ "The flat section is the region where you simply don't exercise.", "The bend sits exactly at the strike.", "Break-even is strike plus premium: $105 here. Finishing at $102 still loses money, even though the call is worth exercising." ] }, { "type": "explore", "heading": "Drive the long call yourself", "prompt": "Drag the finishing price and watch where the position ends up. Find the exact price where the loss stops being the full premium — and the different price where the trade finally breaks even.", "spot_min": 60, "spot_max": 140, "spot_start": 95, "adjust_strike": true, "legs": [ { "kind": "call", "side": "long", "strike": 100, "premium": 5 } ] }, { "type": "payoff", "heading": "Long put — strike $100, premium $6", "caption": "The mirror image. Break-even is $94, the strike minus the premium. The gain is large but bounded, because the underlying cannot fall below zero.", "spot_min": 60, "spot_max": 140, "legs": [ { "kind": "put", "side": "long", "strike": 100, "premium": 6 } ] }, { "type": "text", "heading": "Payoff, or profit?", "icon": "scale", "body": "The two words get used loosely. Payoff usually means the value of the contract at expiry, before the premium. Profit subtracts what you paid, or adds what you received. Every diagram in this app plots profit, because that is the number that decides whether the position worked.", "highlight": "Every chart in this app plots profit — payoff net of the premium." }, { "type": "choice", "question": "A call struck at $100 cost $5. The underlying finishes at $103. How did the trade go?", "options": [ { "text": "A $3 profit — it finished above the strike", "correct": false, "explanation": "The $3 is the payoff, not the profit. The contract is worth $3 at expiry, but it cost $5 to own — so the position is still $2 underwater." }, { "text": "A $5 loss — the premium is always gone", "correct": false, "explanation": "The premium is only lost in full when the option expires worthless. Here exercising claws back $3, cutting the loss to $2." }, { "text": "A $2 loss, even though exercising was worth doing", "correct": true, "explanation": "Exercise recovers $3 of the $5 premium, so the net result is −$2. Worth exercising and profitable are not the same thing — that gap is exactly why break-even sits at $105, not $100." } ] }, { "type": "payoff", "heading": "Short call — strike $100, premium $5", "caption": "The long call flipped. The most you can make is the $5 premium. The line keeps falling to the right; this chart stops at $160 only because an axis has to end somewhere.", "spot_min": 60, "spot_max": 160, "legs": [ { "kind": "call", "side": "short", "strike": 100, "premium": 5 } ] }, { "type": "warning", "heading": "The line does not stop at the edge of the chart", "body": "A payoff diagram is cropped. For a short call, losses keep growing as the underlying rises, and there is no price at which they level off.", "points": [ "An uncovered short call has no fixed maximum loss.", "A short put's loss is bounded only by the underlying reaching zero, which can still be many times the premium received.", "Losing positions can require more collateral before expiry, so a position may become unsustainable well before the deadline." ] }, { "type": "text", "heading": "Payoff, profit, and the cost of carry", "body": "Payoff diagrams and profit diagrams are not the same picture, and conflating them is the most common error in reading one.", "bullets": [ "Payoff is the terminal value alone: max(S − K, 0). It is never negative for a holder.", "Profit subtracts the premium — and, strictly, the premium compounded to expiry, since it was paid up front.", "At typical rates and short maturities that compounding is small, which is why it is usually dropped.", "Put-call parity, C − P = S − K·e^(−rT), ties the two payoffs together and holds by arbitrage, not by assumption." ], "min_depth": 3 }, { "type": "summary", "heading": "What to remember", "takeaways": [ "At expiry an option is worth its intrinsic value and nothing more.", "Break-even is the strike adjusted by the premium, not the strike itself.", "The shape tells you the risk: flat lines are capped, sloping lines are not." ] } ], "advanced_order": 2 }, { "id": "why-use-options", "order": 3, "title": "Why use options — and what can go wrong", "summary": "Three real reasons people use options, and the honest downside of each.", "estimated_minutes": 8, "reviewed_by": null, "reviewed_on": null, "questions": [ { "type": "numeric", "id": "q1-protective-put-floor", "setup": "You hold shares bought at $100 and a put struck at $95 that cost $4. The underlying collapses to $30 by expiry.", "prompt": "What is your loss per share?", "answer": 9, "tolerance": 0.01, "unit_prefix": "$", "hint": "The shares fall all the way, but the put lets you sell at $95. Do not forget what the put itself cost.", "explanation": "The shares lose $70, and the put is worth $95 - $30 = $65, so the position is $5 down before the premium is counted. Subtracting the $4 premium gives a $9 loss per share — the same as it would be at $60, or at $1.", "teaching_note": "That $9 is the floor. Below the strike the loss stops moving, which is exactly what the premium bought." }, { "type": "choice", "id": "q2-protective-put-cost", "prompt": "What does that same protective put cost you when the underlying rises instead?", "choices": [ { "text": "The difference between the share price and the strike", "correct": false, "explanation": "That is what the put pays out when the underlying falls. On the way up the put is simply worth nothing, and the only cost is the premium." }, { "text": "The premium, so the upside is $4 per share worse than holding shares alone", "correct": true, "explanation": "The premium is paid whatever happens. Every profitable outcome is shifted down by $4 — that is the price of the floor, charged even when it turns out not to be needed." }, { "text": "Nothing, because the put expires worthless", "correct": false, "explanation": "The put expiring worthless is precisely the case where the $4 bought nothing. The money was still spent." } ], "teaching_note": "Protection is not free and is not a forecast. Most of the time insurance is money spent on an event that did not happen." }, { "type": "choice", "id": "q3-uncapped-side", "prompt": "Which position can lose more than it ever stood to gain?", "choices": [ { "text": "A long call", "correct": false, "explanation": "A buyer's downside is fixed at the premium on the day it is paid. Total loss is common, but it cannot exceed what was paid." }, { "text": "A long put", "correct": false, "explanation": "A long put can lose only the premium. Its gain is large but bounded, since the underlying cannot fall below zero — the opposite shape." }, { "text": "An uncovered short call", "correct": true, "explanation": "The gain stops at the premium received. The loss grows with the underlying and has no fixed limit, so the worst case is far larger than the best case." } ], "teaching_note": "Nothing in this app is a suggestion to sell options. Uncovered short positions can also demand more collateral before expiry, forcing a position closed at the worst possible moment." }, { "type": "choice", "id": "q4-real-market-frictions", "setup": "A payoff diagram puts break-even for your call at $105.", "prompt": "In a real market, where would break-even actually sit?", "choices": [ { "text": "Higher than $105, because trading costs work against you", "correct": true, "explanation": "You buy at the ask and sell at the bid, and commissions, exchange fees and taxes all come out of the result. Every one of them pushes the price you need to reach further away." }, { "text": "Exactly $105 — the diagram accounts for costs", "correct": false, "explanation": "The diagram is idealised. It assumes a single price, no spread, no fees and no early assignment, none of which hold in a real market." }, { "text": "Lower than $105, because option prices move before expiry", "correct": false, "explanation": "Value before expiry does move with volatility and time, which can let a position be closed early — but it does not lower the break-even at expiry, and costs still push in the other direction." } ], "teaching_note": "Every model in this app is a simplification. Treat the numbers as illustrations of how the pieces interact, not as prices you could actually trade at." }, { "type": "choice", "id": "adv-3", "min_depth": 3, "prompt": "An out-of-the-money call has an elasticity of about 12. What does that tell you?", "teaching_note": "High elasticity and low probability of payoff are two descriptions of the same position.", "choices": [ { "text": "The option costs 12 times the premium of an at-the-money call", "correct": false, "explanation": "Out-of-the-money options are cheaper, not dearer." }, { "text": "Delta is 12", "correct": false, "explanation": "Delta for a call lies between 0 and 1; elasticity scales it by share price over option price." }, { "text": "A 1% share move produces roughly a 12% option move — in both directions", "correct": true, "explanation": "Right, and the same leverage applies against you, from a position most likely to expire worthless." }, { "text": "The option is 12 times more likely to pay off", "correct": false, "explanation": "Elasticity is a sensitivity, not a probability — and high elasticity usually means the opposite." } ] } ], "cards": [ { "type": "title", "kicker": "Lesson 3", "icon": "shield", "title": "Why use options", "subtitle": "Three honest reasons — each with a cost attached." }, { "type": "text", "heading": "Why anyone bothers, in three lines", "body": "Options get used for three broad reasons. Each has a cost attached, and the cost is the part people forget.", "bullets": [ "Protection: a put limits how far a holding can fall — and you pay for that, every time, whether or not it falls.", "Leverage: a small premium controls a large position — which also means a small move against you wipes out a large share of it.", "Income: writing options collects premium — and takes on an obligation that can cost far more than the premium collected." ], "highlight": "Every one of the three is a trade, not a free lunch.", "max_depth": 2 }, { "type": "text", "heading": "Reason 1 — moving a risk you don't want", "icon": "umbrella", "body": "An investor holding shares can buy a put and set a floor under the position. The loss below the strike stops being theirs and becomes the problem of whoever sold the put.", "bullets": [ "The cost is the premium, paid whether or not the protection is ever needed.", "The trade-off is the same as insurance: a known, certain cost against an unknown, uncertain one." ] }, { "type": "payoff", "heading": "Protective put — shares at $100, put struck at $95 for $4", "caption": "Below $95 the line flattens: the loss stops at $9 per share. The premium shifts everything down by $4, so the upside is $4 worse than simply holding the shares.", "spot_min": 60, "spot_max": 140, "legs": [ { "kind": "underlying", "side": "long", "premium": 100 }, { "kind": "put", "side": "long", "strike": 95, "premium": 4 } ] }, { "type": "explore", "heading": "Test the insurance yourself", "prompt": "You hold shares bought at $100 and a put struck at $95 that cost $4. Drag the finishing price all the way down — the floor holds no matter how far you go. Then look at what the $4 premium does to the upside.", "spot_min": 60, "spot_max": 140, "spot_start": 100, "legs": [ { "kind": "underlying", "side": "long", "premium": 100 }, { "kind": "put", "side": "long", "strike": 95, "premium": 4 } ] }, { "type": "text", "heading": "Reason 2 — a known worst case", "icon": "lock", "body": "Buying an option costs a fixed amount, and that amount is the entire downside. Some people prefer a small, certain, capped loss to a large, uncertain one — even though the capped loss occurs far more often.", "bullets": [ "The premium buys exposure to a move without borrowing money.", "The trade-off is time: the move has to happen before expiry, not eventually." ] }, { "type": "text", "heading": "Reason 3 — being paid to take an obligation", "icon": "coins", "body": "Sellers receive the premium immediately. Someone has to be on the other side of every option, and that side is compensated for accepting an outcome they do not control.", "bullets": [ "The income is real and arrives up front.", "So does the obligation, and it is open-ended in a way the income is not." ] }, { "type": "choice", "prompt": "The honest version of the trade-off.", "question": "Which side of an option can lose more than it ever stood to gain?", "options": [ { "text": "The seller — the gain is capped at the premium, the loss is not", "correct": true, "explanation": "A seller's best case is keeping the premium. Their worst case is larger — for an uncovered short call there is no fixed limit at all. That imbalance is why selling options is not a beginner strategy." }, { "text": "The buyer — the premium compounds over time", "correct": false, "explanation": "A premium does not compound; once paid it is the buyer's entire downside. Losing it completely is common, but the amount is fixed on day one." }, { "text": "Neither — the two sides always net to zero", "correct": false, "explanation": "The two sides do net to zero against each other, but that is exactly the point: the seller's unlimited downside is the buyer's unlimited upside. Symmetry between the parties is not symmetry within one side's own risk." } ] }, { "type": "warning", "heading": "For a buyer, zero is a common outcome", "body": "Options are one of the few instruments where a total loss is unremarkable. If the underlying doesn't move far enough in the right direction before expiry, the contract simply ends at nothing.", "points": [ "Being directionally right but early still ends in a 100% loss of the premium.", "Repeated small losses add up fast, because each one is complete.", "A cheaper option is not a safer one. It is cheap because the market judges a payout less likely." ] }, { "type": "warning", "heading": "Selling is where the large losses live", "body": "A seller's profit is capped at the premium received. The loss is not capped in the same way.", "points": [ "An uncovered short call has no fixed maximum loss.", "A short put can lose far more than the premium if the underlying falls sharply.", "A broker can demand more collateral as the position moves against you, forcing it closed at the worst moment.", "Nothing in this app is a suggestion to sell options." ] }, { "type": "text", "heading": "What a payoff diagram leaves out", "icon": "question", "body": "Every chart here assumes a frictionless world. Real trading adds costs and complications, and they move break-even against you.", "bullets": [ "The bid/ask spread — you buy at the higher price and sell at the lower one.", "Commissions, exchange fees and taxes.", "Dividends, and for American-style options, early exercise or assignment.", "Liquidity — a thinly traded contract can be hard to leave at a fair price.", "Value before expiry, which depends on volatility as much as direction." ] }, { "type": "text", "heading": "Leverage, stated properly", "body": "Leverage in options is not a slogan — it is a ratio you can compute, and it is largest exactly where the option is most likely to expire worthless.", "bullets": [ "Elasticity, sometimes written lambda, is delta multiplied by S over the option price.", "A cheap out-of-the-money call can have an elasticity above 10: a 1% move in the share moves the option more than 10%.", "The same number applies on the way down, and the option has a floor at zero that the share does not.", "Higher elasticity is not a better trade. It is a smaller probability of any payoff at all." ], "min_depth": 3 }, { "type": "summary", "heading": "What to remember", "takeaways": [ "Options move risk between parties. The premium is the price of that transfer.", "A buyer's worst case is a total loss of the premium, and it happens often.", "A seller's worst case is larger than their best case, sometimes without a fixed limit.", "Diagrams are idealised. Real markets add costs the chart does not show." ] } ], "advanced_order": 3 }, { "id": "black-scholes-price", "order": 4, "title": "The Black-Scholes-Merton price", "summary": "A formula for what an option should be worth today, before it expires.", "estimated_minutes": 7, "reviewed_by": null, "reviewed_on": null, "questions": [ { "type": "numeric", "id": "q1-time-value-split", "setup": "A call's Black-Scholes-Merton price is $8.50. Its intrinsic value right now is $5.00.", "prompt": "How much of the price is time value?", "answer": 3.5, "tolerance": 0.01, "unit_prefix": "$", "hint": "Price splits cleanly into two pieces.", "explanation": "Price = intrinsic value + time value, so time value = $8.50 - $5.00 = $3.50.", "teaching_note": "Intrinsic value is what expiry arithmetic from lesson 2 already gives you. Time value is everything the model adds on top for the time still left on the clock.", "min_depth": 2 }, { "type": "choice", "id": "q2-five-inputs", "prompt": "Which of these is one of the five direct inputs to the Black-Scholes-Merton price?", "choices": [ { "text": "The company's most recent quarterly earnings", "correct": false, "explanation": "Earnings can move the spot price and the market's view of volatility, but the earnings number itself is not a direct input — only its effects are, through spot and volatility." }, { "text": "The option's trading volume today", "correct": false, "explanation": "Volume can affect how easily a position is traded, but it is not one of the five numbers the formula itself takes in." }, { "text": "Volatility of the underlying", "correct": true, "explanation": "One of the five: spot, strike, time to expiry, volatility, and the rate on a safe asset. Volatility is usually the hardest of the five to know in advance." } ], "teaching_note": "Only five numbers go in: spot, strike, time to expiry, volatility, and the rate on a safe asset. Everything else is either irrelevant to the model or already reflected in one of those five." }, { "type": "choice", "id": "q3-volatility-direction", "setup": "Two otherwise identical calls differ only in the volatility assumed for the underlying.", "prompt": "Which is worth more — the one priced with higher volatility, or lower?", "choices": [ { "text": "Higher volatility, because bigger swings raise the chance of a large payoff", "correct": true, "explanation": "A buyer's loss is capped at the premium no matter how bad the swing, but the gain is not capped. Bigger swings raise the value of that lopsided shape, which is exactly what higher volatility means here." }, { "text": "They are worth the same — volatility only changes the Greeks", "correct": false, "explanation": "Volatility is one of the five direct price inputs. It moves the price itself, not only its sensitivities." }, { "text": "Lower volatility, because the price is more predictable", "correct": false, "explanation": "Predictability does not raise value in this model. It is the size of the possible swings that raises it, not how confidently they can be forecast." } ], "teaching_note": "This is the single most important intuition in this lesson: volatility is not a risk being priced in, in the everyday sense. It is a source of value to a buyer, because a buyer's downside is fixed and the upside is not." }, { "type": "choice", "id": "q4-model-assumption", "setup": "A call is priced with Black-Scholes-Merton using today's volatility, rate and time to expiry.", "prompt": "Which assumption is baked into that price?", "choices": [ { "text": "The underlying will pay a large dividend right before expiry", "correct": false, "explanation": "The plain formula in this lesson assumes no dividends at all. A real dividend would need an adjustment this version does not include." }, { "text": "Volatility stays constant for the rest of the option's life", "correct": true, "explanation": "One of the model's core simplifications. Real volatility moves around, but the formula treats it as fixed and known for the whole remaining life of the option." }, { "text": "The option can be exercised at any time before expiry", "correct": false, "explanation": "That describes an American option. Black-Scholes-Merton prices the European kind, exercisable only at expiry itself." } ], "teaching_note": "Every model number in this app rests on assumptions like these. Treat the price as an illustration of how the five inputs interact, not as a number you could actually trade at." }, { "type": "choice", "id": "adv-4", "min_depth": 3, "prompt": "The implied-volatility surface is not flat across strikes. Which Black-Scholes assumption does that most directly contradict?", "teaching_note": "A flat surface is what a single constant volatility would produce. The observed shape is the market disagreeing with that assumption.", "choices": [ { "text": "That interest rates are known", "correct": false, "explanation": "Rate uncertainty is a minor effect for short-dated equity options." }, { "text": "That trading is frictionless", "correct": false, "explanation": "Frictions widen quoted spreads but do not produce the systematic shape across strikes." }, { "text": "That volatility is a single constant over the life of the option", "correct": true, "explanation": "Right — a smile or skew is the market pricing strikes as if volatility differed between them." }, { "text": "That the option is European", "correct": false, "explanation": "Exercise style affects early-exercise premia, not the shape of the implied surface across strikes." } ] } ], "cards": [ { "type": "title", "kicker": "Lesson 4", "icon": "formula", "title": "The Black-Scholes-Merton price", "subtitle": "A formula for what an option should be worth today, before it ever gets to expiry." }, { "type": "text", "heading": "What the formula is actually doing", "body": "Strip away the notation and Black-Scholes asks one question: what is a fair price today for a payoff that has not happened yet?", "bullets": [ "It weighs every price the share might end at.", "It weighs them by how likely the model thinks they are.", "It discounts the answer back to today, because money now is not money later." ], "highlight": "An average of what the option might pay, discounted to today. The symbols are bookkeeping on top of that idea.", "max_depth": 2 }, { "type": "text", "heading": "Why intrinsic value isn't enough", "icon": "clock", "body": "Lesson 2 priced an option at expiry, where there is no time left and the value is whatever exercising is worth right then. Before expiry, the underlying might still move, and that possibility is worth something. Pricing it needs a model.", "bullets": [ "At expiry, value is just intrinsic value — lesson 2.", "Before expiry, there is still time for the underlying to move, and that chance has to count for something.", "Black-Scholes-Merton (1973) was the first widely used formula for pricing that chance." ], "highlight": "Before expiry, an option is worth its intrinsic value plus time value — the price of the time still left on the clock." }, { "type": "text", "heading": "The five inputs", "icon": "chart", "body": "The formula takes exactly five numbers and turns them into a price.", "bullets": [ "Spot — the underlying's price right now.", "Strike — fixed by the contract.", "Time to expiry — how long the option has left, in years.", "Volatility — how much the underlying's price tends to swing, annualised.", "The rate on a safe asset — the return available from a safe, short-term investment over that time." ], "highlight": "The same five inputs feed both the price and the Greeks — lesson 5." }, { "type": "term", "term": "N(x)", "icon": "formula", "definition": "The cumulative normal distribution. It turns a distance, measured in standard deviations, into a probability between 0 and 1 — the same bell-curve function from any introductory statistics course.", "example": "N(0) = 0.50 — zero standard deviations from average is a coin flip. N(1) is about 0.84, and N(-1) is about 0.16 — its mirror image." }, { "type": "text", "heading": "d1 and d2 — moneyness, measured in standard deviations", "icon": "target", "body": "d1 and d2 both answer the same question: how far is the spot price from the strike, not in dollars, but in standard deviations of the underlying's expected movement? The bigger the number, the further in the money the option is expected to finish.", "bullets": [ "ln(S/K) compares spot to strike on a percentage basis — zero exactly at the money.", "Dividing by volatility times the square root of time rescales that gap by how much movement is expected over what is left.", "d2 uses this distance directly — N(d2) is the plain probability of finishing in the money.", "d1 adds a small correction (+ volatility squared over 2) from the mathematics of compounding growth, and turns out to equal the option's delta from lesson 5." ] }, { "type": "equation", "heading": "The call price, piece by piece", "terms": [ { "text": "C", "caption": "Call price" }, { "text": "=" }, { "text": "S", "caption": "Spot price", "tone": "gain" }, { "text": "x" }, { "text": "N(d1)", "caption": "Weight: how likely, and how much, you would receive", "tone": "gain" }, { "text": "-" }, { "text": "K.e^(-rT)", "caption": "Strike, discounted to today", "tone": "loss" }, { "text": "x" }, { "text": "N(d2)", "caption": "Probability of finishing in the money", "tone": "loss" } ], "footnote": "The value of what you would receive, weighted by likelihood and size — minus the discounted cost of what you would pay, weighted by how likely you will actually pay it." }, { "type": "text", "heading": "Reading it piece by piece", "icon": "compass", "body": "In words: the price is the value of receiving the stock, if you end up exercising, minus the discounted cost of paying the strike, if you end up exercising. Both pieces are weighted by how likely that outcome is.", "bullets": [ "K.e^(-rT).N(d2) is the expected, discounted cost of paying the strike.", "S.N(d1) is the expected, discounted value of receiving the stock — no extra discounting needed, because a stock's own expected growth already accounts for it.", "N(d1) is not just a probability-like weight — it also equals the call's delta from lesson 5. That is not a coincidence: the formula was originally derived from a hedging argument, not a probability one." ] }, { "type": "equation", "heading": "The put price mirrors the call", "terms": [ { "text": "P", "caption": "Put price" }, { "text": "=" }, { "text": "K.e^(-rT)", "caption": "Strike, discounted to today", "tone": "gain" }, { "text": "x" }, { "text": "N(-d2)", "caption": "Probability of finishing in the money", "tone": "gain" }, { "text": "-" }, { "text": "S", "caption": "Spot price", "tone": "loss" }, { "text": "x" }, { "text": "N(-d1)", "caption": "Weight: how likely, and how much, you would give up", "tone": "loss" } ], "footnote": "Same shape, flipped: the value of receiving the strike, if you end up exercising, minus the discounted value of giving up the stock." }, { "type": "pricer_explore", "heading": "Watch the price form", "prompt": "Drag the underlying price. Notice the call is worth something even before it is in the money — that is time value, priced in because the underlying could still get there.", "option_type": "call", "strike": 100, "volatility": 0.25, "time_to_expiry": 0.5, "rate": 0.04, "spot_min": 60, "spot_max": 140, "spot_start": 90, "focus": "price" }, { "type": "text", "heading": "Time value peaks at the money", "icon": "target", "body": "Time value is largest when the outcome is least certain — when the underlying sits right at the strike, nobody yet knows whether the option will finish in the money. Deep in or deep out of the money, the outcome is close to settled, and time value shrinks toward zero.", "bullets": [ "Deep in the money: price is close to intrinsic value plus a small cushion.", "Deep out of the money: price is close to zero — a long shot, priced as one.", "At the money: time value is at its largest." ] }, { "type": "compare", "heading": "Intrinsic value versus time value", "left": { "label": "Intrinsic value", "icon": "target", "tone": "neutral", "tagline": "What exercising is worth right now.", "points": [ "Zero for an out-of-the-money option.", "Grows one-for-one once in the money.", "The only thing left once expiry actually arrives." ] }, "right": { "label": "Time value", "icon": "clock", "tone": "neutral", "tagline": "What the remaining chance is worth.", "points": [ "Largest when the outcome is least certain.", "Shrinks as expiry approaches.", "Reaches exactly zero at expiry." ] }, "footnote": "Price = intrinsic value + time value, always. Lesson 5's Theta is the rate at which the second piece bleeds away." }, { "type": "warning", "heading": "A model, not a market", "body": "Black-Scholes-Merton is an idealisation, not a forecast and not a live quote.", "points": [ "It assumes a constant, known volatility and a constant, known rate — neither holds in a real market.", "It assumes European exercise, no dividends, and frictionless trading with no fees or bid/ask spread.", "A real quoted price can, and routinely does, differ from the model price." ] }, { "type": "choice", "prompt": "Before moving on, commit to an answer.", "question": "A deep out-of-the-money call is very unlikely to pay off. What does the Black-Scholes-Merton price say it should be worth?", "options": [ { "text": "Exactly zero, since it is very unlikely to be exercised", "correct": false, "explanation": "Very unlikely is not the same as impossible. As long as there is any chance at all of finishing in the money, the price stays above zero — just barely." }, { "text": "Small but positive — there is still some chance it finishes in the money", "correct": true, "explanation": "The whole point of a probability-based model: a low-probability payoff still has a price, just a small one. Zero is reserved for the moment of expiry, once the chance is actually gone." }, { "text": "The same as an at-the-money call, since both could still move", "correct": false, "explanation": "At the money the outcome is closer to a coin flip; deep out of the money it needs a much larger move. The price reflects that gap in likelihood directly." } ] }, { "type": "text", "heading": "The assumptions you are buying into", "body": "The formula is exact given its assumptions. Every one of them is an approximation of a real market, and knowing which one is failing is most of what using the model well consists of.", "bullets": [ "The share follows geometric Brownian motion: returns are lognormal, with no jumps.", "Volatility and the discount rate are constant and known over the life of the option.", "Trading is continuous and frictionless — no spread, no fees, no limit on shorting.", "The hedge can be rebalanced continuously, which is what makes the risk-neutral argument work at all.", "Real markets break all five. Jumps and stochastic volatility are why the implied-volatility surface is not flat." ], "min_depth": 3 }, { "type": "summary", "heading": "What to remember", "takeaways": [ "At expiry, value is intrinsic value. Before expiry, it is intrinsic value plus time value.", "Black-Scholes-Merton turns five inputs — spot, strike, time, volatility, the safe-asset rate — into a price.", "It is a model built on simplifying assumptions, not a live market quote." ] } ], "advanced_order": 4 }, { "id": "the-greeks", "order": 5, "title": "The Greeks", "summary": "Five numbers that say how an option's price reacts when one input moves.", "estimated_minutes": 9, "reviewed_by": null, "reviewed_on": null, "questions": [ { "type": "numeric", "id": "q1-delta-step", "setup": "An option has a gamma of 0.02 and a delta of 0.58 when the underlying is at $100.", "prompt": "If the underlying rises by $1, what is the option's new delta, approximately?", "answer": 0.6, "tolerance": 0.01, "hint": "Gamma is delta's own rate of change per $1 move.", "explanation": "New delta is approximately old delta plus gamma times the move: 0.58 + 0.02 x 1 = 0.60.", "teaching_note": "This is the local, one-step version of what gamma measures — a reasonable approximation for a $1 move, much less reliable for a $20 one. That breakdown at larger moves is exactly why it is an approximation, not a guarantee." }, { "type": "choice", "id": "q2-gamma-definition", "prompt": "Which Greek directly measures how fast delta itself is changing?", "choices": [ { "text": "Gamma", "correct": true, "explanation": "Gamma is defined exactly this way: delta's own delta, or how much delta shifts for a $1 move in the underlying." }, { "text": "Theta", "correct": false, "explanation": "Theta measures the effect of time passing on the price, not how fast delta is changing." }, { "text": "Vega", "correct": false, "explanation": "Vega measures the effect of volatility on the price. Delta's own rate of change has its own name." } ], "teaching_note": "Gamma is largest when the underlying sits right at the strike — exactly where the outcome is least settled and delta has the most room to swing." }, { "type": "choice", "id": "q3-theta-meaning", "prompt": "A trader holds a long call. As one day passes with nothing else changing, what does a negative theta mean for the position?", "choices": [ { "text": "Theta has no effect unless the underlying also moves", "correct": false, "explanation": "Theta is defined as the effect of time passing alone, holding everything else fixed — including the underlying. It does not need the underlying to move to matter." }, { "text": "The position loses a little value purely from time passing", "correct": true, "explanation": "Exactly what a negative theta says: with the underlying, volatility and rate all held still, one day passing costs the position a small amount of value." }, { "text": "The position gains value purely from time passing", "correct": false, "explanation": "That would describe a positive theta. A negative theta means value is lost as each day passes, all else equal — the definition of time decay." } ], "teaching_note": "Every long option position is, in this sense, in a race against its own theta — it needs to be right before time value finishes decaying away." }, { "type": "choice", "id": "q4-delta-sign-flip", "setup": "A long call and a long put share the same strike, expiry and underlying.", "prompt": "Which of these Greeks has the OPPOSITE sign for the put compared with the call?", "choices": [ { "text": "Gamma", "correct": false, "explanation": "Gamma is positive for both a long call and a long put — being long either one means becoming more exposed to the underlying as it moves toward the strike, in the same sense." }, { "text": "Vega", "correct": false, "explanation": "Vega is positive for both — more uncertainty raises the value of either right, regardless of which direction it pays off in." }, { "text": "Delta", "correct": true, "explanation": "Delta captures direction: positive for a long call, which gains as the underlying rises, and negative for a long put, which gains as it falls. It is the one Greek that flips sign between the two." } ], "teaching_note": "Delta is the direction-taking Greek. Everything else on this page (gamma, vega) cares about the SIZE of an uncertain outcome, not which way it breaks — which is why they agree in sign between a call and a put." }, { "type": "numeric", "id": "adv-5", "min_depth": 3, "setup": "An option is priced at $8.00 with delta 0.55 and gamma 0.04. The share rises by $2.00.", "prompt": "Estimate the new option price using the second-order approximation.", "answer": 9.18, "tolerance": 0.03, "unit_prefix": "$", "explanation": "delta·ΔS + ½·gamma·ΔS² = 0.55×2 + 0.5×0.04×4 = 1.10 + 0.08 = $1.18, so about $9.18.", "teaching_note": "The gamma term is what a delta-only estimate misses. It grows with the square of the move, so it matters most exactly when the move is large.", "hint": "Add the delta term and the gamma term, then add both to the starting price." } ], "cards": [ { "type": "title", "kicker": "Lesson 5", "icon": "insights", "title": "The Greeks", "subtitle": "Five numbers that say how an option's price reacts when one input moves." }, { "type": "text", "heading": "Delta, with actual numbers", "body": "Delta says how much the option price moves when the share moves by $1. A call with delta 0.50 moves about 50 cents.", "bullets": [ "Share up $1, delta 0.50: the option gains roughly $0.50.", "Share down $1: it loses roughly $0.50.", "Deep in the money, delta approaches 1 and the option tracks the share almost one for one.", "Far out of the money, delta approaches 0 and the share can move without the option caring much." ], "highlight": "Roughly. Delta itself changes as the share moves — that is what gamma measures.", "max_depth": 2 }, { "type": "text", "heading": "Why the price alone isn't enough", "icon": "compass", "body": "A price is a single snapshot. The Greeks say what happens to that price next — if the underlying ticks up, if a day passes, if volatility jumps. They come straight out of the same Black-Scholes-Merton formula from the last lesson.", "highlight": "Each Greek answers one 'what if' about the price." }, { "type": "term", "term": "Delta", "icon": "target", "definition": "How much the option's price moves for a $1 move in the underlying. A call's delta runs from 0 to 1; a put's runs from -1 to 0.", "example": "At $100 spot and a $100 strike, a 6-month call here has a delta around 0.58 — its price moves about $0.58 for the next $1 the underlying moves, for a moment, before delta itself moves too." }, { "type": "pricer_explore", "heading": "Watch delta move", "prompt": "Drag the underlying from deep out of the money to deep in the money. Delta starts near 0, and finishes near 1 for a call.", "option_type": "call", "strike": 100, "volatility": 0.25, "time_to_expiry": 0.5, "rate": 0.04, "spot_min": 60, "spot_max": 140, "spot_start": 100, "focus": "delta" }, { "type": "term", "term": "Gamma", "icon": "scale", "definition": "How fast delta itself changes as the underlying moves — delta's own delta. It is largest when the underlying sits right at the strike.", "example": "High gamma means delta can shift quickly, so a position's exposure to the underlying changes fast even without a large price move." }, { "type": "pricer_explore", "heading": "Where gamma peaks", "prompt": "Drag the underlying toward the $100 strike and watch gamma rise, then fall away again on the far side.", "option_type": "call", "strike": 100, "volatility": 0.25, "time_to_expiry": 0.5, "rate": 0.04, "spot_min": 60, "spot_max": 140, "spot_start": 70, "focus": "gamma" }, { "type": "term", "term": "Vega", "icon": "waves", "definition": "How much the price moves for a one-percentage-point rise in volatility. Vega is positive for both long calls and long puts — more uncertainty raises the price of the choice either way.", "example": "At $100 spot and 25% volatility, this call has a vega around 0.28 — a jump to 26% volatility should add roughly $0.28 to the price, on its own." }, { "type": "pricer_explore", "heading": "Volatility, priced", "prompt": "Drag volatility up and down and watch the price move even though the underlying has not.", "option_type": "call", "strike": 100, "volatility": 0.25, "time_to_expiry": 0.5, "rate": 0.04, "spot_min": 60, "spot_max": 140, "spot_start": 100, "adjust_volatility": true, "focus": "vega" }, { "type": "term", "term": "Theta", "icon": "clock", "definition": "How much the price falls as one day passes, with nothing else changing. Usually negative for a long option — this is time decay, the cost of the clock running down.", "example": "At $100 spot with 6 months left, this call's theta is around -$0.02 per day: a small, steady bleed while everything else sits still." }, { "type": "pricer_explore", "heading": "Time decay in motion", "prompt": "Drag time to expiry down toward zero. Watch the price fall toward the intrinsic value from lesson 2, even with the underlying held still.", "option_type": "call", "strike": 100, "volatility": 0.25, "time_to_expiry": 0.5, "rate": 0.04, "spot_min": 60, "spot_max": 140, "spot_start": 100, "adjust_time": true, "focus": "theta" }, { "type": "text", "heading": "Rho — the quiet one", "icon": "coins", "body": "Rho measures the price's sensitivity to the rate on a safe asset. It matters most for long-dated options, and tends to be the smallest of the five effects for the shorter-dated contracts most learners start with.", "bullets": [ "Positive for calls, negative for puts.", "Usually the last of the five Greeks anyone checks." ] }, { "type": "warning", "heading": "The Greeks describe a moment, not a guarantee", "body": "Every Greek is a snapshot from the current model, not a promise about what happens next.", "points": [ "They assume a small move. A large jump can behave very differently from what delta alone predicts — that gap is what gamma is for, and even gamma is only an approximation.", "All five recalculate constantly as the underlying, volatility and time all change together.", "None of this changes the basic shape from lessons 1 and 2: a long option can still expire worthless, in full." ] }, { "type": "choice", "prompt": "Before moving on, commit to an answer.", "question": "A long call and a long put share the same strike and expiry. One Greek has the SAME sign for both. Which one?", "options": [ { "text": "Gamma", "correct": true, "explanation": "Gamma (and vega) do not care about direction — both a call and a put get more sensitive near the strike, and both get more valuable as volatility rises. Delta and rho are the ones that split by direction." }, { "text": "Delta", "correct": false, "explanation": "Delta is exactly where they differ — positive for the call, negative for the put. That is the whole point of delta: it captures direction." }, { "text": "Rho", "correct": false, "explanation": "Rho splits by direction too — positive for a call, negative for a put, mirroring how a higher rate helps a deferred purchase and hurts a deferred sale." } ] }, { "type": "text", "heading": "Second order: gamma, theta and the cost of hedging", "body": "Delta alone is a straight line drawn against a curve. Gamma measures how fast that line goes wrong, and theta is roughly what you pay for the privilege.", "bullets": [ "A second-order estimate of the price change is delta·ΔS + ½·gamma·ΔS², which is why a large move surprises a delta-only view.", "Gamma is greatest near the strike and close to expiry, exactly where a hedger has to rebalance most often.", "For a long option, gamma is positive and theta is negative: convexity is not free, it is rented daily.", "A delta-hedged long option therefore profits when realised volatility exceeds the implied volatility it was priced at, and loses when it does not." ], "min_depth": 3 }, { "type": "summary", "heading": "What to remember", "takeaways": [ "Delta: exposure to the underlying. Gamma: how fast that exposure changes.", "Vega: exposure to volatility. Theta: the daily cost of time passing.", "All five come from the same model as the price, and describe a moment — not a forecast." ] } ], "advanced_order": 5 }, { "id": "options-strategies", "order": 6, "title": "Options strategies — and when to use them", "summary": "Combining legs to fit a view of the market, and what each combination costs you.", "estimated_minutes": 9, "reviewed_by": null, "reviewed_on": null, "questions": [ { "type": "numeric", "id": "q1-spread-net-cost", "setup": "A bull call spread costs $8 for the long $100 call and returns $4 from selling the $110 call.", "prompt": "What is the net cost to open this spread, per share?", "answer": 4, "tolerance": 0.01, "unit_prefix": "$", "hint": "One leg costs money, the other brings money in.", "explanation": "$8 paid minus $4 received = $4 net cost per share.", "teaching_note": "A spread's whole appeal is right here: a smaller cash outlay than a lone call, in exchange for giving up the gains above the second strike." }, { "type": "choice", "id": "q2-straddle-at-strike", "setup": "A trader opens a straddle: long a $100 call for $8 and long a $100 put for $6.", "prompt": "The underlying finishes at exactly $100. What happens to the position?", "choices": [ { "text": "Both legs expire worthless; the $14 in premium is a total loss", "correct": true, "explanation": "At exactly the strike, neither the call nor the put is worth exercising, so both expire worthless. The full $14 paid for the two premiums is gone — the worst outcome for a straddle held to expiry." }, { "text": "The call and put cancel out, so the position breaks even", "correct": false, "explanation": "Breaking even needs the underlying to move away from $100 by the size of the combined premium, not sit still at the strike. Sitting still is the straddle's worst outcome, not a neutral one." }, { "text": "One leg is exercised automatically to offset the loss on the other", "correct": false, "explanation": "Nothing is exercised automatically just because it would help. An option is exercised only when doing so is worth something, and at the strike neither one is." } ], "teaching_note": "A straddle is a bet on the SIZE of a move, not its direction — and the underlying sitting still is exactly the scenario it loses to." }, { "type": "choice", "id": "q3-spread-vs-naked", "prompt": "What makes a bull call spread's short leg different from the naked short call in lesson 3?", "choices": [ { "text": "Spreads are never assigned early, unlike a naked short", "correct": false, "explanation": "Assignment risk on the short leg of a spread is the same as any other short American-style option. What actually differs here is that the position's overall loss is bounded, not that assignment cannot happen." }, { "text": "The short leg is covered by the long call underneath it, capping the combined loss", "correct": true, "explanation": "The long call bought first means that if the underlying keeps rising, the gain on the long leg offsets the loss on the short leg above the second strike — the combination caps out instead of running open-ended." }, { "text": "It isn't different — the risk is identical", "correct": false, "explanation": "It is different. This short call sits above a long call already owned, which caps the combined position. A naked short call in lesson 3 has no such cap." } ], "teaching_note": "This is the general pattern behind every covered short in this lesson: a short leg is only as safe as whatever sits next to it." }, { "type": "choice", "id": "q4-match-strategy-to-goal", "setup": "An investor already owns shares and wants some income now, while accepting a cap on the upside if the shares rally hard.", "prompt": "Which strategy fits that description?", "choices": [ { "text": "Straddle", "correct": false, "explanation": "A straddle needs no existing share position and is a bet on a big move, not a way to generate income from shares already held." }, { "text": "Protective put", "correct": false, "explanation": "A protective put costs money and caps the DOWNSIDE, not the upside. It is insurance, not income, and does not fit 'wants income now'." }, { "text": "Covered call", "correct": true, "explanation": "Selling a call against shares already owned brings in premium immediately and caps the upside above the strike — exactly the trade-off described." } ], "teaching_note": "Matching a strategy to a goal comes down to two questions: what view of the market does it need, and what is it willing to give up? Income now, in exchange for capped upside, is the covered call's whole trade." }, { "type": "numeric", "id": "adv-6", "min_depth": 3, "setup": "You sell a $100 put for $4.00 and buy a $95 put for $1.50, both expiring on the same date.", "prompt": "What is the most this position can lose?", "answer": 2.5, "tolerance": 0.01, "unit_prefix": "$", "explanation": "The strike width is $5.00 and the net credit is $4.00 − $1.50 = $2.50. Maximum loss is width minus credit: $5.00 − $2.50 = $2.50.", "teaching_note": "The long put is what turns an open-ended obligation into a known worst case. Selling the $100 put alone would risk far more.", "hint": "Work out the strike width, then subtract the credit you took in." } ], "cards": [ { "type": "title", "kicker": "Lesson 6", "icon": "strategy", "title": "Options strategies", "subtitle": "Combining legs to fit a view of the market — and what each combination costs you." }, { "type": "text", "heading": "A spread, counted out", "body": "Buy the $100 call for $6 and sell the $110 call for $2. Net cost $4.", "bullets": [ "Share ends below $100: both expire worthless. You lose the $4.", "Share ends at $106: your call pays $6, the one you sold pays nothing. You are $2 up.", "Share ends at $130: yours pays $30, the one you sold costs you $20. Net $10, less the $4 — capped at $6." ], "highlight": "You gave up the far upside to pay less at the start. That is the whole trade.", "max_depth": 2 }, { "type": "text", "heading": "A strategy is just several legs added together", "icon": "split", "body": "Every diagram so far has shown one leg. A strategy opens more than one at the same time, and the combined position is simply the sum of what each leg is worth — the same arithmetic from lesson 2, run once per leg and added up.", "bullets": [ "The shape of the combined line is new; the maths behind it is not.", "Every strategy below trades away some upside, some downside, or both, in exchange for a different cost or a different risk shape." ], "highlight": "Nothing here is a suggestion to use any particular strategy. Which one fits, if any, depends on a view of the market and a tolerance for risk that only the trader can judge." }, { "type": "payoff", "heading": "Bull call spread — long $100 call for $8, short $110 call for $4", "caption": "Net cost $4. The gain caps out at $6 once the underlying clears $110; the loss caps out at the $4 paid. A cheaper, lower-conviction version of a lone long call.", "spot_min": 70, "spot_max": 140, "legs": [ { "kind": "call", "side": "long", "strike": 100, "premium": 8 }, { "kind": "call", "side": "short", "strike": 110, "premium": 4 } ] }, { "type": "text", "heading": "When to reach for a bull call spread", "icon": "compass", "body": "Fits a view that the underlying rises, but not without limit, alongside a preference for a known, smaller cost over a lone call's larger premium.", "bullets": [ "Modestly bullish — not aiming to catch an unlimited move.", "Willing to give up gains above $110 in exchange for paying less up front.", "Both the best and worst case are fixed before the trade opens." ] }, { "type": "payoff", "heading": "Bear put spread — long $100 put for $6, short $90 put for $2", "caption": "Net cost $4. The mirror image of the call spread: profits as the underlying falls, capped at $6 once it clears $90 on the way down, with the loss capped at the $4 paid.", "spot_min": 60, "spot_max": 130, "legs": [ { "kind": "put", "side": "long", "strike": 100, "premium": 6 }, { "kind": "put", "side": "short", "strike": 90, "premium": 2 } ] }, { "type": "text", "heading": "When to reach for a bear put spread", "icon": "compass", "body": "The bearish counterpart to the call spread, with the same trade-off in the opposite direction.", "bullets": [ "Modestly bearish — not aiming to catch an unlimited fall.", "Cheaper than a lone put, at the cost of a capped payout below $90.", "Defined risk in both directions before the trade opens." ] }, { "type": "payoff", "heading": "Straddle — long $100 call for $8, long $100 put for $6", "caption": "Net cost $14. Profits on a big move either way; loses if the underlying sits near $100 through expiry, since both premiums decay together.", "spot_min": 60, "spot_max": 140, "legs": [ { "kind": "call", "side": "long", "strike": 100, "premium": 8 }, { "kind": "put", "side": "long", "strike": 100, "premium": 6 } ] }, { "type": "explore", "heading": "Feel the straddle's shape", "prompt": "Drag the finishing price away from $100 in either direction and watch the position turn around. Then bring it back to $100 and see both legs lose at once.", "spot_min": 60, "spot_max": 140, "spot_start": 100, "legs": [ { "kind": "call", "side": "long", "strike": 100, "premium": 8 }, { "kind": "put", "side": "long", "strike": 100, "premium": 6 } ] }, { "type": "warning", "heading": "A straddle needs a big move, not just any move", "body": "Two premiums are paid up front, and both decay if the underlying does not move enough.", "points": [ "Break-even sits exactly two premiums away from the strike on either side — here, $86 and $114.", "The most common outcome for a straddle held to expiry is a loss, because on most days the underlying does not move that far.", "This is a bet on the SIZE of a move, not its direction — the opposite of the spreads above." ] }, { "type": "payoff", "heading": "Covered call — own shares at $100, short a $110 call for $4", "caption": "The $4 premium arrives immediately. Upside is capped at $14 per share (the $10 gain to $110, plus the $4 premium); the downside is the same shape as owning the shares outright, just cushioned by the $4 already collected.", "spot_min": 60, "spot_max": 140, "legs": [ { "kind": "underlying", "side": "long", "premium": 100 }, { "kind": "call", "side": "short", "strike": 110, "premium": 4 } ] }, { "type": "text", "heading": "When to reach for a covered call", "icon": "coins", "body": "For someone who already owns the shares and has a neutral-to-mildly-bullish view — willing to trade away upside beyond $110 for income collected today.", "bullets": [ "Only makes sense if you are comfortable owning the shares anyway.", "Reduces, but does not remove, the loss if the shares fall.", "The most upside given away happens exactly when the shares rally hard — the scenario the call caps." ] }, { "type": "payoff", "heading": "Protective put — own shares at $100, long a $90 put for $2", "caption": "A cheaper, lower floor than lesson 3's example: the loss stops at $12 per share no matter how far the shares fall, for a $2 premium.", "spot_min": 60, "spot_max": 140, "legs": [ { "kind": "underlying", "side": "long", "premium": 100 }, { "kind": "put", "side": "long", "strike": 90, "premium": 2 } ] }, { "type": "compare", "heading": "Selling income versus buying protection", "left": { "label": "Covered call", "icon": "coins", "tone": "neutral", "tagline": "Own the shares, sell a call, collect the premium now.", "points": [ "Upside capped at the strike plus the premium.", "Premium is income, received immediately.", "Downside unchanged in shape, just cushioned by the premium." ] }, "right": { "label": "Protective put", "icon": "umbrella", "tone": "neutral", "tagline": "Own the shares, buy a put, pay the premium now.", "points": [ "Downside capped at the strike, minus the premium.", "Premium is a cost, paid immediately.", "Upside unchanged in shape, just reduced by the premium." ] }, "footnote": "One collects income and caps the upside; the other pays for a floor and caps nothing on the way up. Neither is free, and neither suits every position." }, { "type": "warning", "heading": "Adding legs changes the shape, not the honesty required", "body": "Every strategy above still needs the same plain statement of risk as a single option.", "points": [ "A spread's short leg is only safe because the long leg beside it covers it — remove that cover and the short leg is naked again, with the risk from lesson 3.", "All of these diagrams are at expiry, in an idealised market. Real trading adds the bid/ask spread, commissions, and the cost of trading more than one contract at once.", "None of this is a recommendation to use any particular strategy. Which one, if any, fits depends on a market view and a tolerance for risk that belong to the trader, not to a diagram." ] }, { "type": "choice", "prompt": "Before moving on, commit to an answer.", "question": "An investor expects a company's earnings announcement to move the stock a lot, but has no view on which direction. Which strategy fits that view best?", "options": [ { "text": "Bull call spread", "correct": false, "explanation": "A bull call spread needs the underlying to rise — it has a direction built in. An investor with no directional view is not expressing the view this strategy needs." }, { "text": "Covered call", "correct": false, "explanation": "A covered call needs shares already owned and a neutral-to-bullish view with capped upside — it is built for calm markets, the opposite of an expected big move." }, { "text": "Straddle", "correct": true, "explanation": "A straddle profits from a big move in EITHER direction and does not need to guess which way — exactly the shape for an expected big move with no directional view. It still loses if the move does not happen." } ] }, { "type": "text", "heading": "Spreads as building blocks", "body": "Every strategy in this lesson is a sum of the same two payoffs. Once that is clear, the shape of any of them can be reasoned out rather than memorised.", "bullets": [ "A vertical spread is long one strike, short another, same expiry — bounded on both ends because the two kinks cancel beyond the far strike.", "A collar is a protective put funded by a written call; put-call parity says a collar on a holding is close to a spread on its own.", "A straddle is long volatility with no directional view: it profits on a move either way, and bleeds if nothing happens.", "Credit strategies collect premium up front and carry the obligation — max loss on a vertical credit spread is the strike width less the credit." ], "min_depth": 3 }, { "type": "summary", "heading": "What to remember", "takeaways": [ "A strategy's combined payoff is just its legs added together.", "Spreads trade a smaller cost for a capped payout in both directions.", "A straddle is a bet on the size of a move, not its direction.", "Covered calls and protective puts start from an existing share position and cap one side of it." ] } ], "advanced_order": 8 }, { "id": "path-dependent-options", "order": 7, "title": "Options that watch the path", "summary": "Barriers, averages and baskets — when the route matters, not just the destination.", "estimated_minutes": 7, "reviewed_by": null, "reviewed_on": null, "questions": [ { "type": "choice", "id": "q1-knockout-direction", "setup": "You buy a down-and-out call struck at 100, with a barrier at 85. Over the next six months the price dips to 84, recovers, and finishes at 120.", "prompt": "What is the option worth at expiry?", "choices": [ { "text": "Nothing", "correct": true, "explanation": "It touched 85 on the way, which killed it. Being right about the direction did not help: the contract had already ceased to exist before the price recovered. This is the risk a knock-out adds that a plain call does not have." }, { "text": "20 — it finished 20 above the strike", "correct": false, "explanation": "That is what the plain call would pay. The knock-out died when the price touched 85, and a dead option does not come back to life however well things go afterwards." }, { "text": "15 — the distance from the barrier to the finish", "correct": false, "explanation": "The barrier is a trigger, not a strike. It decides whether the option exists; it never enters the payoff calculation." } ], "teaching_note": "A knock-out can be perfectly right about direction and still pay nothing. That is precisely why it costs less than the plain option." }, { "type": "numeric", "id": "q2-in-out-parity", "setup": "A plain put on the same underlying, same strike and same expiry is worth 7.50. The down-and-in version of it is worth 3.20.", "prompt": "What is the down-and-OUT put worth?", "answer": 4.3, "tolerance": 0.02, "unit_prefix": "$", "explanation": "In-out parity: 7.50 - 3.20 = 4.30. Holding both the knock-in and the knock-out leaves you holding the plain option, because exactly one of the two is alive at expiry — so the two must add up to it.", "hint": "If you owned BOTH the knock-in and the knock-out, what would you have?", "teaching_note": "This identity is exact, and it is the quickest sanity check on any barrier price.", "min_depth": 2 }, { "type": "choice", "id": "q3-asian-cheaper", "prompt": "Why is an average-price (Asian) call usually cheaper than the otherwise identical European call?", "choices": [ { "text": "It expires earlier, so there is less time value", "correct": false, "explanation": "Both options run to the same expiry date. The difference is what the payoff looks at, not when it is settled." }, { "text": "Averaging smooths out the extremes, so the number the payoff depends on is less volatile", "correct": true, "explanation": "Exactly. An average wanders less than the endpoint it is drawn from, and an option on something less volatile is worth less. Nothing is being taken away unfairly — you are being paid less because you are buying exposure to a steadier quantity." }, { "text": "The seller keeps the dividends", "correct": false, "explanation": "Dividends affect both contracts the same way. The averaging is what makes the difference." } ], "teaching_note": "Averaging also makes the payoff much harder to manipulate on a single date, which is why Asians are common in commodity and currency markets." }, { "type": "choice", "id": "q4-basket-correlation", "setup": "You are comparing two baskets of three shares each. Everything is the same except that basket A's members move almost in lockstep, while basket B's move nearly independently.", "prompt": "Which basket call costs more, and why?", "choices": [ { "text": "Basket B, because independent assets create more possible outcomes", "correct": false, "explanation": "More combinations of individual outcomes, but a narrower distribution for the BLEND — the independent moves partly cancel. It is the blend the option pays on." }, { "text": "They cost the same, since the members have the same volatility", "correct": false, "explanation": "Individual volatilities are only half the story. How the members move relative to each other decides the volatility of the blend, and that is what the option is written on." }, { "text": "Basket A, because assets moving together make the blend more volatile", "correct": true, "explanation": "Right. When members move together the blend swings almost as much as a single share, so the option on it is dear. In basket B one member rising while another falls leaves the blend nearly unchanged — a steadier blend, and a cheaper option." } ], "teaching_note": "Correlation is also the least stable input in the whole calculation, and it has a habit of rising towards 1 in a crash — exactly when a diversified basket is being relied on." }, { "type": "numeric", "id": "adv-7", "min_depth": 3, "setup": "A Monte Carlo price using 10,000 paths has a standard error of $0.40.", "prompt": "By what factor must the number of paths increase to bring the standard error down to $0.10?", "answer": 16, "tolerance": 0.5, "explanation": "Error falls with 1/√N. Quartering the error needs 4² = 16 times the paths — 160,000 in total.", "teaching_note": "This is why variance reduction matters. Buying accuracy with raw path count gets expensive very quickly.", "hint": "The error falls with the square root of the path count, so work backwards from a factor of four." } ], "cards": [ { "type": "title", "icon": "compass", "kicker": "EXOTIC OPTIONS", "title": "Options that watch the path", "subtitle": "So far the only thing that mattered was where the price finished. Some contracts also care how it got there." }, { "type": "text", "heading": "Why the journey can matter", "body": "For a plain option only the final price counts. For a path-dependent option, how it got there counts too.", "bullets": [ "A barrier option can be knocked out by one touch of a level, then recover — and still pay nothing.", "An Asian option pays on the average price, so a single closing spike barely moves it.", "Two shares can end at exactly the same price and these options pay completely different amounts." ], "highlight": "Same ending, different journey, different payout.", "max_depth": 2 }, { "type": "text", "icon": "chart", "heading": "Everything so far looked at one number", "body": "A plain call or put settles on the price at expiry. What happened in between — a crash and a recovery, a slow drift, a wild month — leaves no trace. Two very different journeys that end at the same price pay exactly the same.", "highlight": "A vanilla option has no memory." }, { "type": "term", "icon": "waves", "term": "Path-dependent option", "definition": "An option whose payoff depends on the route the price took, not only on where it ended up.", "example": "Two shares both finish the year at 120. One climbed steadily; the other fell to 70 first. A vanilla call pays the same on both. A path-dependent option need not." }, { "type": "text", "icon": "flag", "heading": "Barrier options", "body": "A barrier option adds a trigger level. A knock-out dies the moment the price touches it and pays nothing ever after. A knock-in does not exist until the price touches it, and pays nothing if it never does.", "bullets": [ "The barrier is a switch, not a strike — it decides whether the option exists, and never enters the payoff sum.", "Down barriers sit below today's price; up barriers sit above it.", "Both kinds are CHEAPER than the plain option they are carved from." ] }, { "type": "compare", "heading": "Two ways to attach a trigger", "left": { "label": "Knock-out", "icon": "lock", "tone": "neutral", "tagline": "Alive now. Dead if the level is touched.", "points": [ "Starts as an ordinary option.", "One touch of the barrier ends it permanently.", "Recovery afterwards does not revive it." ] }, "right": { "label": "Knock-in", "icon": "target", "tone": "neutral", "tagline": "Dormant now. Alive once the level is touched.", "points": [ "Pays nothing unless the barrier is reached.", "From the moment it is, behaves like a plain option.", "If the level is never touched, it expires worthless." ] }, "footnote": "Own both halves and you own the plain option: exactly one of them is alive at expiry. So their prices must add up to it — a check worth remembering." }, { "type": "warning", "heading": "Cheaper is not better value", "body": "A barrier option costs less because it pays in fewer situations, not because it is a bargain. The discount is the market charging you for a smaller set of outcomes.", "points": [ "A knock-out can be completely right about direction and still pay nothing, because the price dipped through the barrier on the way. That is a route to losing 100% of the premium that a plain option does not have.", "The nearer the barrier sits to today's price, the cheaper the option and the more likely it is to be triggered. Cheap barrier options are cheap for a reason.", "Some combinations can never pay at all. A down-and-out put struck below its own barrier is worth exactly nothing: to finish in the money the price must first pass through the level that kills it." ] }, { "type": "term", "icon": "scale", "term": "Asian (average-price) option", "definition": "An option that settles on the AVERAGE price over its life instead of the final price.", "example": "A one-year Asian call might average the closing price on every trading day, then pay the amount that average finished above the strike." }, { "type": "text", "icon": "insights", "heading": "Why an average costs less", "body": "An average wanders less than the endpoint it is drawn from — a spike on one day is diluted by every other day. The option pays on a steadier quantity, so it is worth less than the vanilla.", "bullets": [ "It is also far harder to manipulate. Moving one day's closing price barely shifts a year's average.", "That is why Asians are common in commodity and currency markets, where a business is hedging an average exposure anyway." ], "highlight": "You are not being short-changed. You are buying exposure to something calmer." }, { "type": "text", "icon": "split", "heading": "Basket options: one contract, several underlyings", "body": "A basket option pays on a weighted blend of several shares rather than one. Its price is NOT the sum of the prices of the individual options, and the gap between the two is the whole point of the instrument.", "bullets": [ "If the members move together, the blend is nearly as volatile as one share and the option is dear.", "If they move independently, one rising while another falls leaves the blend steadier — so the option is cheap.", "Diversification shows up in an option price as lower volatility, and therefore as a smaller premium." ], "highlight": "Correlation, not volatility alone, sets the price of a basket option." }, { "type": "warning", "heading": "The input most likely to be wrong", "body": "Correlation is estimated from history, it is not constant, and it has a well-documented habit of rising towards 1 in a crash — precisely when a diversified basket is being relied on to hold up.", "points": [ "A basket option that looks cheap because of a low correlation estimate is only cheap if that estimate holds.", "Diversification is real, but it is weakest exactly when it is needed most." ] }, { "type": "text", "icon": "formula", "heading": "Why these need simulation", "body": "Black-Scholes gives an exact formula because a European option depends on one number. Once the payoff depends on the whole route, or on several assets at once, there is usually no formula at all. So the price is estimated instead: generate many possible futures, work out what the contract would have paid on each, and average them.", "bullets": [ "That average is an ESTIMATE, not a value. Run it again with different random numbers and you get a slightly different answer.", "A simulated price should always arrive with an error bar. A price quoted to the cent with no stated uncertainty is claiming a precision it does not have.", "Four times the work halves the error, and no more — the error falls with the square root of the effort." ], "highlight": "You can try all of this yourself on the Advanced tab of the Sandbox." }, { "type": "text", "heading": "What Monte Carlo costs to get right", "body": "Path-dependent payoffs generally have no closed form, so they are simulated. Simulation converges slowly, and knowing how slowly is the difference between a usable number and a confident wrong one.", "bullets": [ "The standard error of a Monte Carlo estimate falls with 1/√N: to halve the error you need four times the paths.", "Discrete time steps add bias on top of that noise — a barrier can be crossed between two steps and never observed.", "Continuity corrections and finer steps reduce that bias at a direct cost in compute.", "Variance-reduction techniques — antithetic variates, control variates — buy accuracy far more cheaply than raw path count." ], "min_depth": 3 }, { "type": "summary", "heading": "What to take away", "takeaways": [ "A vanilla option looks only at the finishing price. Barrier and Asian options look at the route.", "A knock-in plus a knock-out equals the plain option, because exactly one of them survives to expiry.", "Barrier options are cheaper because they pay in fewer situations — you can be right about direction and still get nothing.", "Averaging damps the extremes, so Asian options cost less than their vanilla twins.", "For a basket, correlation sets the price — and it is the input that moves most when it matters most.", "These contracts have no formula, so their prices are simulated, and every simulated price comes with an uncertainty." ] } ], "advanced_order": 7, "advanced": true }, { "id": "volatility-is-not-constant", "order": 8, "title": "Volatility is not a constant", "summary": "The volatility smile, what it tells you, and the Heston model that reproduces it.", "estimated_minutes": 7, "reviewed_by": null, "reviewed_on": null, "questions": [ { "type": "choice", "id": "q1-what-implied-vol-is", "prompt": "What does \"implied volatility\" actually measure?", "choices": [ { "text": "The volatility number that makes Black-Scholes return the price the option is trading at", "correct": true, "explanation": "It is the formula run backwards. You know the market price, you solve for the one input you cannot observe. It is a way of quoting a price in more comparable units — not a measurement of anything in the world." }, { "text": "A forecast of how much the underlying will move, published by the exchange", "correct": false, "explanation": "Nobody publishes it as a forecast. It is derived from prices, and it embeds supply, demand and risk appetite as well as any view about future movement." }, { "text": "How much the underlying has actually moved over the past year", "correct": false, "explanation": "That is realised (historical) volatility, computed from past prices. Implied volatility is backed out of today's option prices and can differ from it substantially." } ], "teaching_note": "Implied volatility is a price expressed in different units. Treating it as a prediction is one of the most common mistakes in the subject." }, { "type": "choice", "id": "q2-smile-contradiction", "setup": "On the same underlying, on the same day, options at different strikes imply different volatilities.", "prompt": "Why is that a problem for Black-Scholes?", "choices": [ { "text": "It means the options are mispriced and there is an arbitrage", "correct": false, "explanation": "Tempting, but no. The pattern is stable, persistent and present in every liquid market. It is a sign the model is incomplete, not that traders have made an error." }, { "text": "The model assumes one constant volatility for the underlying, so it cannot have several at once", "correct": true, "explanation": "Exactly. Volatility in Black-Scholes is a property of the SHARE, not of the contract, so every option on it should imply the same number. That they do not is the market telling us the model's picture of how prices move is too simple." }, { "text": "It means the rate on a safe asset is wrong", "correct": false, "explanation": "The rate affects all strikes in a predictable way and cannot produce a curved pattern across them." } ], "teaching_note": "A model failing in a stable, repeatable way is far more informative than one failing randomly — it tells you what to fix." }, { "type": "numeric", "id": "q3-variance-vs-volatility", "setup": "The Heston model is written in terms of VARIANCE rather than volatility. A set of parameters starts with a variance of 0.09.", "prompt": "What is that as a volatility, in percent?", "answer": 30, "tolerance": 0.5, "unit_suffix": "%", "explanation": "Volatility is the square root of variance: sqrt(0.09) = 0.30, or 30%. Mixing the two up is the single easiest mistake to make with this model, and it produces answers that are wrong by a wide margin rather than a little.", "hint": "Volatility and variance are related by a square root.", "teaching_note": "Check which one a formula wants before putting a number into it.", "min_depth": 2 }, { "type": "choice", "id": "q4-negative-correlation", "prompt": "In Heston, what does a NEGATIVE correlation between the price and its volatility produce?", "choices": [ { "text": "Lower option prices at every strike", "correct": false, "explanation": "It redistributes value between strikes rather than lowering everything. Low strikes get dearer; high strikes get cheaper." }, { "text": "A perfectly flat implied volatility across strikes", "correct": false, "explanation": "A flat line is what Black-Scholes predicts. Correlation is what TILTS the curve away from flat." }, { "text": "A fatter left tail, making downside protection more expensive", "correct": true, "explanation": "Prices falling while volatility spikes is exactly what equity markets do in a sell-off. Modelling it makes large falls more likely than Black-Scholes assumes, so puts at low strikes cost more — which is the downward-sloping skew real markets show." } ], "teaching_note": "This is called the leverage effect, and reproducing it is the main reason to reach for Heston over Black-Scholes." }, { "type": "choice", "id": "adv-8", "min_depth": 3, "prompt": "In the Heston model, what does a negative correlation between the share's shocks and the variance's shocks produce?", "teaching_note": "It is the parameter that lets the model reproduce the shape equity markets actually show.", "choices": [ { "text": "Higher implied volatility at high strikes only", "correct": false, "explanation": "That describes a positive skew, seen in some commodities, not equity index options." }, { "text": "Nothing — correlation only affects the price level", "correct": false, "explanation": "Correlation is precisely what controls the asymmetry of the distribution." }, { "text": "A flat surface across strikes", "correct": false, "explanation": "Zero correlation with no vol-of-vol gets closer to flat; negative correlation tilts it." }, { "text": "A downward skew — low strikes priced at higher implied volatility", "correct": true, "explanation": "Right. Falling prices coinciding with rising variance fattens the left tail, which is what equity skew looks like." } ] } ], "cards": [ { "type": "title", "icon": "waves", "kicker": "STOCHASTIC VOLATILITY", "title": "Volatility is not a constant", "subtitle": "Black-Scholes asks for one volatility. The market answers with a different one at every strike." }, { "type": "text", "heading": "The smile, without the mathematics", "body": "Black-Scholes assumes one volatility for a share. Ask the market what volatility its option prices imply and you get a different answer for every strike.", "bullets": [ "Plot implied volatility against strike and the line curves rather than lying flat.", "For equities it is usually a skew: low strikes imply higher volatility.", "That is the market pricing crashes as more likely than a single fixed volatility would allow." ], "highlight": "The smile is not the market being wrong. It is the model being simple.", "max_depth": 2 }, { "type": "text", "icon": "formula", "heading": "The assumption we quietly accepted", "body": "Black-Scholes needs a single number for volatility, held fixed for the life of the option. That is a property of the SHARE, not of any particular contract — so every option on the same underlying, expiring on the same day, should need the same number.", "highlight": "One underlying, one volatility. That is what the model claims." }, { "type": "term", "icon": "question", "term": "Implied volatility", "definition": "The volatility you have to put into Black-Scholes to get the price the option is actually trading at.", "example": "A call is trading at 6.20. Feed in 24% and the formula says 6.20. The implied volatility is 24% — the formula run backwards." }, { "type": "text", "icon": "lightbulb", "heading": "It is a price in different units", "body": "Implied volatility is not a measurement of the world and not a forecast anyone published. It is a way of restating a price so that options at different strikes and expiries can be compared. Treating it as a prediction of future movement is one of the commonest mistakes in the subject.", "highlight": "Quoting an option in implied volatility is like quoting a bond in yield." }, { "type": "compare", "heading": "What the model says, and what markets do", "left": { "label": "Black-Scholes predicts", "icon": "chart", "tone": "neutral", "tagline": "A flat line across strikes.", "points": [ "Volatility belongs to the share.", "Every strike must imply the same figure.", "Anything else would be a contradiction." ] }, "right": { "label": "Markets actually show", "icon": "insights", "tone": "neutral", "tagline": "A curve, usually tilted.", "points": [ "Low strikes imply HIGHER volatility.", "The pattern is stable and shows up everywhere.", "It has been there in equities since 1987." ] }, "footnote": "The curve is called the volatility smile, and its tilt is called the skew. A model failing in a stable, repeatable way tells you exactly what to fix." }, { "type": "text", "icon": "shield", "heading": "Why the tilt points that way", "body": "In equity markets, prices and volatility move in opposite directions. Calm markets drift upward; frightened markets fall hard and fast. So a large drop is more likely, and arrives with more turbulence, than a model with fixed volatility would ever expect.", "bullets": [ "That makes low-strike puts — crash protection — dearer than Black-Scholes says.", "It also means the flat-volatility model systematically understates the chance of a severe fall." ], "highlight": "The skew is the market pricing in the fact that crashes are faster than rallies." }, { "type": "text", "icon": "waves", "heading": "The Heston model", "body": "Heston (1993) fixes the assumption directly: instead of a constant, volatility becomes a second random process with its own motion, pulled back over time towards a long-run level, and correlated with the price itself.", "bullets": [ "Variance is dragged towards a long-run level — turbulent periods calm down, quiet ones eventually stir.", "It has a volatility of its own, so it wanders rather than following a fixed path.", "Its shocks are correlated with the price's, which is what creates the tilt." ] }, { "type": "equation", "heading": "How variance moves under Heston", "terms": [ { "text": "change in v", "caption": "how variance moves over a small step", "tone": "neutral" }, { "text": "=" }, { "text": "k (0 - v) dt", "caption": "pulled back towards its long-run level, at speed k", "tone": "neutral" }, { "text": "+" }, { "text": "x sqrt(v) dW", "caption": "plus a random shock, sized by the volatility of volatility x", "tone": "neutral" } ], "footnote": "In words: variance drifts back towards where it usually sits, while being knocked about along the way. The share price gets its own equation, and the two random shocks are correlated — that correlation is what tilts the smile." }, { "type": "term", "icon": "scale", "term": "Variance, not volatility", "definition": "Heston is written in terms of variance — the SQUARE of volatility. A 20% volatility is a variance of 0.04.", "example": "Set a long-run variance of 0.09 and you are saying volatility settles around 30%, because sqrt(0.09) = 0.30." }, { "type": "text", "icon": "target", "heading": "What each dial does", "body": "Five numbers define a Heston world, and each one changes the shape of the smile in its own way.", "bullets": [ "Volatility today: where the whole curve sits.", "Long-run volatility: where it drifts towards over longer expiries.", "Speed of return: how quickly today's level stops mattering.", "Volatility of volatility: how far the wings lift — more of it means fatter tails on both sides.", "Correlation: the tilt. Negative pushes value into low strikes; positive does the reverse." ], "highlight": "The Heston tab in the Sandbox puts all five on sliders." }, { "type": "warning", "heading": "A more elaborate model is not a truer one", "body": "Heston reproduces the smile. That is a real achievement and it is not the same as being right about the future.", "points": [ "It has five parameters where Black-Scholes has one, which means five things to estimate wrongly. Several quite different combinations often fit the same market prices about equally well.", "It assumes volatility moves smoothly, with no jumps. Real markets gap — and the days that matter most are usually the days they gap.", "It assumes those five parameters are constants. In practice they are re-fitted constantly, which is an admission that they are not.", "It keeps every frictionless-market assumption Black-Scholes made: no spreads, no fees, no limits on trading.", "Fitting today's smile is not predicting tomorrow's." ] }, { "type": "text", "icon": "formula", "heading": "Two ways to get a price out of it", "body": "Heston has no simple formula for an option price, but it does have one for the distribution of the future price. Ordinary calls and puts can be recovered from that by an integral — fast and essentially exact. Anything path-dependent has to be simulated instead.", "bullets": [ "The two methods share almost no machinery, so when they agree on a vanilla option, that is real evidence both are working.", "The simulation carries an extra error the formula does not: stepping through time in chunks is itself an approximation, and no error bar describes it. More steps are the fix, not more paths." ] }, { "type": "text", "heading": "Heston: variance with a life of its own", "body": "Heston's answer to the smile is to stop treating variance as a constant and give it its own stochastic process, mean-reverting around a long-run level.", "bullets": [ "Variance reverts at rate kappa toward a long-run level theta, driven by its own volatility-of-volatility.", "The correlation between the share's shocks and the variance's shocks is what produces skew — negative correlation tilts it the way equity markets actually trade.", "Vanilla options still have a semi-analytic price through the characteristic function, so calibration is tractable.", "The Feller condition, 2·kappa·theta > vol-of-vol², keeps variance from reaching zero; calibrated equity parameters often violate it, which matters for simulation schemes." ], "min_depth": 3 }, { "type": "summary", "heading": "What to take away", "takeaways": [ "Implied volatility is a price restated in different units, not a forecast and not a measurement.", "Options at different strikes imply different volatilities — a stable, worldwide pattern Black-Scholes cannot account for.", "In equities the curve tilts: crash protection is expensive because falls are faster and more violent than rises.", "Heston makes volatility a second random process, correlated with the price, which reproduces that tilt.", "It is written in variance; volatility is the square root of it.", "More parameters mean more ways to be wrong. Reproducing today's prices is not the same as knowing what happens next." ] } ], "advanced_order": 6, "advanced": true }, { "id": "structured-products", "order": 9, "title": "Structured products: taking the wrapper off", "summary": "A bond plus some options, sold as one certificate — and how to see what is really inside.", "estimated_minutes": 8, "reviewed_by": null, "reviewed_on": null, "questions": [ { "type": "numeric", "id": "q1-bond-leg", "setup": "A five-year note promises to return your 100 in full at maturity. The rate on a safe asset is 4%, continuously compounded.", "prompt": "What does that promise cost to buy today?", "answer": 81.87, "tolerance": 0.6, "unit_prefix": "$", "explanation": "100 x e^(-0.04 x 5) = 81.87. So about 82 of your 100 goes on the bond that provides the protection, and only the remaining 18 or so is available to buy the upside. This is the whole economics of a capital-protected note in one sum.", "hint": "Discount 100 back five years at 4% continuous compounding.", "teaching_note": "When rates fall, the bond costs more and less is left for options — which is why these products all but vanished in the near-zero-rate years.", "min_depth": 2 }, { "type": "choice", "id": "q2-reverse-convertible-shape", "setup": "A one-year reverse convertible pays a 12% coupon whatever happens. If the underlying finishes below its starting level, you are repaid in shares instead of cash.", "prompt": "What is the shape of that position?", "choices": [ { "text": "Both gain and loss capped, like a spread", "correct": false, "explanation": "The gain is capped, but nothing caps the loss. If the underlying goes to zero you are left with worthless shares and the coupon." }, { "text": "Gain capped at the coupon; loss uncapped all the way to zero", "correct": true, "explanation": "You have bought a bond and SOLD a put. However far the underlying rises you get the coupon and nothing more; however far it falls, you absorb it. A high coupon is not generosity — it is the premium for insurance you have written." }, { "text": "Gain uncapped; loss limited to the coupon", "correct": false, "explanation": "That is the shape of owning an option, not selling one. Here the upside stops dead at the coupon." } ], "teaching_note": "Any time a product offers a coupon far above the going rate, look for the option that has been sold to pay for it." }, { "type": "choice", "id": "q3-barrier-not-a-floor", "setup": "A barrier reverse convertible has a strike at 100 and a barrier at 65. During the year the underlying touches 60, then recovers to finish at 64.", "prompt": "What do you get back, ignoring the coupon?", "choices": [ { "text": "The full 100 — the price recovered above the barrier", "correct": false, "explanation": "Recovering above the BARRIER does not help; only recovering above the STRIKE would. It finished at 64, below the strike of 100." }, { "text": "Shares worth 65 — the barrier acts as a floor", "correct": false, "explanation": "This is the most common and most expensive misreading of this product. The barrier is a trigger, not a floor. It never limits the loss." }, { "text": "Shares worth 64 — you absorb the fall from 100, not from 65", "correct": true, "explanation": "The barrier decides WHETHER the sold put wakes up, not how much it can cost you. Once breached it is an ordinary put struck at 100, so the loss is measured from 100 down. Breaching a 65% barrier means losing at least 35%, not at most 35%." } ], "teaching_note": "These products pay well most of the time and occasionally lose a lot. That shape flatters a short track record." }, { "type": "choice", "id": "q4-issue-margin", "setup": "You add up the parts of a note sold at 100 and find the bond and options are worth 96.", "prompt": "What is that 4 difference?", "choices": [ { "text": "The issuer's costs and profit, paid by the buyer on day one", "correct": true, "explanation": "It is not a scandal — an issuer has real costs and is entitled to a margin. But a buyer who cannot see it cannot judge whether the deal is fair, and the brochure will not print it. Being able to add up the parts yourself is the whole skill." }, { "text": "A safety buffer that gets returned at maturity", "correct": false, "explanation": "Nothing is returned. It is spent when you buy." }, { "text": "A pricing error — the parts must add up to the price", "correct": false, "explanation": "They add up to the VALUE. The price is what you pay, and the two are not the same thing for anything sold with a margin." } ], "teaching_note": "Sell the note back the next day and you would get roughly the value, not the price. That gap is why these are held to maturity." }, { "type": "choice", "id": "adv-9", "min_depth": 3, "prompt": "A note is advertised as capital protected. What risk does that phrase leave out?", "teaching_note": "Protection is a promise from a company, and companies can fail. The payoff diagram cannot show that.", "choices": [ { "text": "That interest rates might fall", "correct": false, "explanation": "Rates affect the bond component's cost, not whether protection is honoured." }, { "text": "That the index might rise", "correct": false, "explanation": "A rising index is the favourable case for the holder." }, { "text": "None — protection means the money always comes back", "correct": false, "explanation": "It is a promise from the issuer, which is a very different statement." }, { "text": "Issuer credit risk — the protection is only as good as the issuer", "correct": true, "explanation": "Right. If the issuer defaults, the protection defaults with it." } ] } ], "cards": [ { "type": "title", "icon": "wallet", "kicker": "STRUCTURED PRODUCTS", "title": "Taking the wrapper off", "subtitle": "A single certificate with a headline number. Inside: a bond and some options you already understand." }, { "type": "text", "heading": "Taking one apart on the table", "body": "A capital-protected note promising your money back plus some of the index return is two ordinary things in one wrapper.", "bullets": [ "Most of your money buys a bond that grows back to the full amount by maturity.", "What is left buys call options for the upside.", "That is why the upside is only partial — the leftover only stretches so far." ], "highlight": "Nothing exotic inside. A bond and some options, sold as one product.", "max_depth": 2 }, { "type": "text", "icon": "contract", "heading": "What is being sold", "body": "Structured products are the derivatives most often sold to ordinary savers. They arrive with a headline — \"100% capital protection with 70% of any rise\", \"12% coupon\" — and a brochure that rarely takes the wrapper off. Underneath, every one of them is a bundle of things covered in earlier lessons.", "highlight": "Nothing in this lesson is new mathematics. It is all recognition." }, { "type": "term", "icon": "shield", "term": "Capital-protected note", "definition": "A bond that grows back to your money by maturity, plus call options bought with whatever is left over.", "example": "Of 100 invested for five years at 4%, about 82 buys the bond that guarantees the 100 back. The other 18 buys the upside." }, { "type": "equation", "heading": "A capital-protected note, decomposed", "terms": [ { "text": "the note", "caption": "what you bought", "tone": "neutral" }, { "text": "=" }, { "text": "a bond", "caption": "grows back to your money by maturity", "tone": "gain" }, { "text": "+" }, { "text": "calls", "caption": "bought with whatever the bond did not use", "tone": "gain" } ], "footnote": "The participation rate is not a policy decision by the issuer. It is simply how many calls the leftover money could buy." }, { "type": "text", "icon": "coins", "heading": "Why participation collapsed when rates did", "body": "The bond has to reach the full amount by maturity, so the lower the interest rate, the more it costs today and the less remains for options.", "bullets": [ "At 4% over five years the bond costs about 82, leaving 18 for upside.", "At 0.5% it costs about 98, leaving 2 — barely enough to buy anything.", "That is why these products were everywhere when rates were high and almost extinct when rates were near zero." ], "highlight": "The protection is real. It is paid for with the interest you would otherwise have earned." }, { "type": "warning", "heading": "What \"protected\" leaves out", "body": "The word does a lot of work in a brochure. Four things it does not cover:", "points": [ "It applies AT MATURITY only. Sell early and you get the market price, which can be well below par.", "You get your money back in nominal terms. Over five years of inflation, with no dividends and no interest, that can still be a real loss.", "Participation is on the PRICE of the underlying, not its total return. Dividends stay with the issuer, and over five years those are substantial.", "It is an unsecured promise by a bank. If the issuer fails, the protection fails with it — Lehman Brothers issued capital-protected notes." ] }, { "type": "term", "icon": "coins", "term": "Reverse convertible", "definition": "A bond paying a high fixed coupon, funded by SELLING a put on the underlying.", "example": "A 12% coupon on a one-year note. If the underlying finishes below its starting level, you are repaid in shares rather than cash." }, { "type": "payoff", "heading": "The shape you have taken on", "caption": "A reverse convertible without its bond leg: you receive a premium (the coupon) and have sold someone the right to hand you the shares. Gains stop at the coupon; losses do not stop.", "spot_min": 40, "spot_max": 160, "legs": [ { "kind": "put", "side": "short", "strike": 100, "premium": 12 } ] }, { "type": "compare", "heading": "What you got, and what you gave", "left": { "label": "What you receive", "icon": "coins", "tone": "gain", "tagline": "A coupon well above the going rate.", "points": [ "Paid whatever the underlying does.", "Known in advance, to the penny.", "And that is the entire upside." ] }, "right": { "label": "What you sold", "icon": "warning", "tone": "loss", "tagline": "The right to be handed the shares.", "points": [ "If the price falls, they are yours at the strike.", "Your loss grows all the way down to zero.", "The underlying doubling adds nothing." ] }, "footnote": "A high coupon is not generosity. It is the premium for insurance you have written — which is why it is high." }, { "type": "text", "icon": "flag", "heading": "Adding a barrier", "body": "The most widely sold version softens the sold put into a DOWN-AND-IN put: it only wakes up if the underlying ever falls through a level well below today's price. Most of the time it never does, the coupon is paid, and the product looks like an unusually good bond.", "bullets": [ "The sold option is cheaper, so the coupon is lower than a plain reverse convertible would pay.", "The result is a run of good years punctuated by an occasional large loss." ], "highlight": "That shape looks safest for longest, and flatters a short track record." }, { "type": "warning", "heading": "The barrier is a trigger, not a floor", "body": "This is the most expensive misreading in the whole subject, and the product invites it.", "points": [ "The barrier decides WHETHER the sold put comes alive. It never limits what the put can cost you.", "Once breached, it is an ordinary sold put struck at the original level — so the loss is measured from the STRIKE, not from the barrier.", "A 65% barrier that breaks means losing at least 35% of the underlying, not at most 35%.", "And the gain is still capped at the coupon, however far the underlying recovers." ] }, { "type": "text", "icon": "scale", "heading": "The number the brochure does not print", "body": "Add up what the bond and the options are actually worth, and a note sold at 100 will usually come to less than 100. The difference is the issuer's costs and profit, and the buyer pays it the moment they buy.", "bullets": [ "This is not a scandal. Issuers have real costs and are entitled to a margin.", "But it is why selling the note back the next day returns roughly the value, not the price.", "Being able to add up the parts yourself is the entire skill this lesson is teaching." ], "highlight": "The Structured tab in the Sandbox does the addition for you, and shows the gap." }, { "type": "text", "icon": "lightbulb", "heading": "How to read any of them", "body": "Four questions will take the wrapper off almost any structured product you meet.", "bullets": [ "What is the bond leg, and what is it worth today?", "Which options have been BOUGHT, and which have been SOLD? The sold ones are where the risk lives.", "What is the worst case, stated as a number rather than a mood?", "What do the parts add up to, and how does that compare with the price?" ] }, { "type": "text", "heading": "Where the cost hides, and who you are lending to", "body": "Decomposing a note tells you what it is worth. Two things it will not tell you sit outside the payoff diagram entirely.", "bullets": [ "The fee is the gap between what you pay and what the components cost to assemble — it is embedded, not invoiced.", "Capital protection is the issuer's promise, so the note carries their credit risk. If the issuer fails, protection fails with it.", "Notes are typically illiquid: exiting early means the issuer's own bid, not a market price.", "Price the pieces separately and the answer to 'is this good value' usually stops being a matter of opinion." ], "min_depth": 3 }, { "type": "summary", "heading": "What to take away", "takeaways": [ "Every structured product is a bond plus options. Recognising the parts is the whole skill.", "A capital-protected note spends most of your money on the bond; low rates leave almost nothing for upside.", "\"Protected\" means at maturity, in nominal terms, without dividends, and only if the issuer survives.", "A reverse convertible caps your gain at the coupon and leaves your loss uncapped — because you sold a put.", "A barrier is a trigger, not a floor. Breached, the loss is measured from the strike.", "A note sold at 100 is usually worth less than 100 on day one, and the gap is paid by the buyer." ] } ], "advanced_order": 9, "advanced": true } ] }