The 56
Options Strategies

This is a curriculum, not a reference. A reference can be entered at any point; a curriculum has an order, and the order is the argument.

The order here is: arithmetic, then price, then sensitivity, then volatility, then risk, then mechanics — and only then strategy. Six foundation units precede the first structure. That sequencing is deliberate and it is where most options education fails. Courses that open with the covered call produce students who can execute a trade and cannot say what they are short.

Sixteen units. Sixty-two chapters. One engine.

Hours assume study rather than viewing: reading, working the labs, and completing the assessments. Passive consumption of the same material takes about a third of the time and produces about a tenth of the retention.

The through-line. Every module builds one more component of a payoff and pricing engine. By Module 10 the student has watched a complete system assemble itself — array summation, plateau detection, edge-slope diagnostics, parity assertions, and Black–Scholes valuation for the time-dependent structures. This is not a programming course. It is a claim about what understanding options actually means: not memorising fifty-six shapes, but holding the one procedure that generates them.

One chain, one book. Every premium, breakeven and maximum in this manual is computed from a single option chain (Appendix A) at fixed assumptions: spot 100, volatility 25 percent, interest and dividends zero, expiries at 30 and 90 days, strikes 85 to 115. Any two examples can therefore be compared directly, and any figure can be recomputed by hand from the chain. No number in this book was typed in.

Three routes through the same material. All three assume Part I is completed first, without exception.

The Practitioner Path — 20 hours. For someone who will trade a small number of structures and needs to trade them properly.

F1–F6 → Module 1 → Module 3 → Module 4 (Chapters 17–20) → Module 6 (Chapter 33) →

Module 2 → Capstone A

This covers the primitives, verticals, volatility, the iron condor and the stock-plus-option family. It deliberately omits ratios, calendars and arbitrage. A practitioner who has mastered these and can compute a required win rate before every entry is better equipped than one who has skimmed all fifty-six.

The Full Curriculum — 66 hours. All units in order. This is the intended route and the one the assessments are built against.

The Instructor Path — 66 hours plus preparation. The full curriculum, plus the module labs run as facilitated sessions, plus the assessment bank. See the instructor notes below.

Prerequisite gates. These are enforced rather than advisory:

• No strategy module may begin until F1–F6 assessments are passed at 80 percent.
• Module 7 (Ratios) requires Module 3 (Verticals) — ratios are verticals with the counts broken.
• Module 8 (Calendars) requires F2 (Pricing) and F4 (Volatility) — the curve is model-generated.
• Module 9 (Synthetics) requires F2 — parity is the entire content.
• Capstone A requires Modules 1–6. Capstone B requires all ten.

On completing this curriculum a student can do the following, without reference material.

Derive. Construct the payoff of any of the fifty-six structures from its leg definitions, locate its kinks, plateaus and breakevens, and state its risk class.

Compute. Produce maximum profit, maximum loss and every breakeven for any structure, and compute the win rate a credit structure requires to break even.

Diagnose. Read the slope at both extremes of a payoff chart and state the net contract exposure there — the single discipline that separates defined risk from disguised risk.

Translate. Convert freely between calls, puts and shares using put–call parity, and name the synthetic equivalent of any stock-plus-option position.

Price. Decompose any premium into intrinsic and extrinsic value, and state which of the six Black–Scholes inputs dominates a given position.

Anticipate. Predict the sign of net delta, gamma, theta and vega for any structure before computing them, and identify which greek is driving its day-to-day result.

Size. Determine position size from maximum loss or from a stress scenario, never from premium or margin, and explain why a book of correlated credit structures is one position.

Recognise. Identify early assignment exposure, pin risk and the full transaction cost of a multi-leg structure, and state what each does to a payoff diagram's promises.

Judge. Given a market view, select an appropriate structure and defend the choice on grounds other than payoff shape — because for most views, several structures share one shape.

The last outcome is the whole curriculum. If a student can only choose by diagram, they have learned pictures.

Assessment is designed to be resistant to pattern-matching. A student who has memorised fifty-six diagrams should fail; a student who holds the generating procedure should pass comfortably.

Four instruments.

1 — Unit checks (formative, ungraded). Two questions at the end of every chapter and foundation unit, with worked answers. 124 in total across the manual. These are for immediate self-correction, not for scoring.

2 — Module examinations (graded, 80 percent to pass). One per module. Structure:

Section D is marked on the defence, not the choice. Several answers are acceptable; only some justifications are.

3 — Capstone A: The Position Audit. The student is given six live positions with legs, entry prices and account size, and must for each: classify risk, compute all critical values, identify the dominant greek, state the required win rate where applicable, and recommend hold, reduce or close with reasoning. Three of the six are deliberately mis-sized. One is described by its owner as defined risk and is not.

4 — Capstone B: The Structure Brief. The student receives a market view, a capital constraint, a liquidity constraint and an assignment constraint, and must produce a two-page brief: the structure chosen, two rejected alternatives with reasons, the full risk arithmetic, the sizing calculation, the exit rule set before entry, and the three scenarios in which the position fails. Marked against the rubric below.

What is deliberately not assessed. Win rates on paper trades, backtest results, and anything resembling a simulated account balance. These measure luck over short horizons and teach students to attribute outcomes to skill.

Capstone B is marked against five criteria, each scored 0–4. A pass requires 15 of 20 with no criterion below 2.

On the top band. The 4-level descriptors are not decoration. Each one names a specific misconception this manual sets out to remove: that risk class is static, that gross numbers are the numbers, that margin is a risk budget, that the diagram distinguishes trades, and that losses come only from being wrong.

Notes for anyone teaching this material.

Do not open with the covered call. It is the conventional first lesson and it is the worst one. It teaches a structure whose true nature — a short put — cannot be explained until parity is in place, and it establishes early that options are an income technique rather than a risk transformation. Open with F1 and the payoff of a single call.

Teach the tool, not the taxonomy. The fastest route to fifty-six structures is one procedure. Build the payoff engine live in front of the class in Module 1, extend it in every subsequent module, and let the strategies arrive as data. Students who see the engine stop asking which strategy to memorise next.

Insist on the edge-slope check. Make it a ritual. Before any discussion of any structure, ask the room: what is the slope at the far right, and at the far left? It takes ten seconds and it is the single habit most likely to prevent a catastrophic position.

Run Section D disagreements openly. In the judgement sections, several answers are defensible. Where two students choose differently and both defend well, mark both fully and say so. The objective is a defensible process, not a house view.

Handle the credit-structure question directly. Students arrive having seen premium selling marketed as reliable income. Do not dismiss it and do not endorse it. Put the arithmetic on the board: 1.52 collected, 3.48 at risk, 69.6 percent required. Then ask them to find their own measured hit rate. The arithmetic does the teaching.

Do not simulate profits. Paper-trading competitions teach outcome attribution over horizons far too short for the tails to appear. Assess process instead. A student whose reasoning is sound and whose quarter was negative has passed.

Pace. The natural pace is one module per week with the lab run mid-week. Modules 8 and 9 routinely need two weeks; Modules 1 and 2 routinely need less than one. Resist compressing Part I, which is where the temptation is greatest and the cost highest.

What this manual does not do.

It does not recommend trades. Every structure in it can lose money, and several can lose more than the amount invested. Nothing here constitutes advice, a forecast or a price target.

It does not present returns. There are no performance figures, no backtests and no track records, because none of them would tell a reader what their own results would be.

It does not use a real underlying. Every figure comes from a stylised chain at fixed assumptions, chosen so the arithmetic is verifiable and the examples are comparable. Real chains have skew, term structure, dividends, non-zero rates and gaps — F2, F4 and F6 address each in turn, and every chapter notes where the simplification flatters the structure.

It does not claim the model is true. Black–Scholes is used because it is transparent, standard, and wrong in documented ways. Foundation F2 says so explicitly, and Modules 8 and 10 depend on the reader having taken that seriously.

Risk statement. Options trading involves substantial risk. Long option positions can expire worthless, losing the entire premium. Short option positions can lose amounts substantially greater than the premium received, and in some structures without limit. Multi-leg structures carry execution, liquidity, early-assignment and margin risks that no payoff diagram displays. Anyone applying this material is responsible for determining its suitability for their own circumstances, and for obtaining independent advice where appropriate.

Learning outcomes

• State the expiration payoff of a call, a put and a share as a piecewise function.
• Build any multi-leg payoff by adding leg payoffs at each spot price.
• Locate breakevens, plateaus and unbounded tails from the payoff array alone.
• Explain why a payoff diagram is a single-date slice and what it necessarily omits.

Every structure in this manual — all fifty-six, without exception — is built from three primitives and one operation.

The primitives are a long call, a long put and a long share. At expiration a call pays max(S − K, 0). A put pays max(K − S, 0). A share pays S. Short positions are the same functions multiplied by minus one. That is the entire vocabulary.

The operation is addition. To find the payoff of any combination, evaluate every leg at the same spot price and add the results. Do that at every spot price across a range and you have a payoff array. Plot the array and you have the diagram. There is no separate formula for a butterfly, an iron condor or a jelly roll; there is one loop over legs and a list of leg definitions. A reader who internalises this stops memorising fifty-six shapes and starts deriving them.

Reading the array. Four features carry almost all the information.

Kinks occur only at strikes. A payoff line changes slope where an option moves from worthless to intrinsic, and nowhere else. Count the kinks and you know how many distinct strikes are in the structure.

Slope between kinks is the net share-equivalent exposure over that interval. A slope of plus one means the position behaves like one long share there; minus two means two short shares. Read the slope at the far right and far left of the chart and you know what the position becomes in the extremes — which is the only place where structures actually hurt people.

Plateaus are intervals of zero slope, where profit or loss is constant. Their width is the range of outcomes that all return the same result, and for range structures that width is the product being sold.

Breakevens are the spot prices where the array crosses zero. There may be none, one, two or more. Solving for them algebraically is error-prone; scanning the array for sign changes is not, and it works identically for every structure.

The subtraction that matters. A payoff diagram shows the value of the position at expiration. Profit and loss is that value minus what you paid, or plus what you received. The entire diagram shifts vertically by the net premium. Two structures with identical shapes and different premiums are different trades, and the difference lives entirely in that vertical shift.

What the diagram cannot tell you. This is the most important paragraph in the unit.

An expiration payoff diagram is a slice through a valuation surface at one date. It says nothing about the path, and almost nothing about the position you will actually hold. It does not show time decay on the way there. It does not show what happens if implied volatility moves. It does not show margin requirements, which can force an exit long before expiration. It does not show early assignment, which is a real feature of American-style options and ends the structure without asking.

Traders who lose money on defined-risk positions usually did not misread the diagram. They read it correctly and then assumed it described the whole trade. Later units add back what it leaves out: pricing (F2), sensitivities (F3), volatility (F4), risk and sizing (F5) and mechanics (F6). Until all six are in place, a payoff diagram is a picture, not an analysis.

A note on the manual's conventions. All illustrations use one fixed set of assumptions: spot 100, volatility 25 percent, interest and dividends zero, a 30-day near expiry and a 90-day far expiry, on a strike ladder from 85 to 115. Every premium quoted anywhere in this book is drawn from the single option chain in Appendix A. That means any two examples can be compared directly, and any figure you doubt can be recomputed from the chain by hand.

Learning outcomes

• Decompose any option premium into intrinsic and extrinsic value.
• State the six inputs to a Black–Scholes price and the direction of each effect.
• Explain why the manual sets r = 0 and what that removes from the analysis.
• Use put–call parity to convert between calls, puts and shares.

A premium has exactly two parts. Intrinsic value is what the option is worth if exercised right now: max(S − K, 0) for a call, max(K − S, 0) for a put. It is arithmetic, and it is not a wager — it is capital you have parked. Extrinsic value is everything else. It is the entire content of the bet.

This split does more work than any other idea in the manual. It explains why a deeply in-the-money option is expensive without being aggressive, why the short guts collects a spectacular credit that is mostly an obligation (Chapter 29), and why comparing two positions by premium size is meaningless unless you strip the intrinsic component first.

The six inputs. Extrinsic value is a function of spot, strike, time to expiry, volatility, interest rates and dividends. Their directions are worth committing to memory:

• More time means more extrinsic value, but not proportionally — value decays roughly with

the square root of time remaining, so the last thirty days shed far more than the first.

• More volatility means more extrinsic value, in a near-linear way over normal ranges. This

is the input you are usually trading whether or not you intended to.

• Higher rates raise call values and lower put values, through the cost of carrying the

underlying.

• Dividends do the opposite, and they are the most common cause of early assignment on

short calls.

• Spot and strike set the intrinsic component and determine how much extrinsic value the

option can hold — at-the-money options carry the most.

Why this manual sets r = 0. Zero rates strip carry out of every example, which makes the payoff arithmetic clean and the parity relationships exact. It also has a consequence worth sitting with: at r = 0 a box spread (Chapter 55) is worth exactly its strike width and costs exactly its strike width, so its profit is precisely zero. That is not an error in the example. It is the point. A box contains nothing except interest, and when you set interest to zero, nothing is what remains. Every arbitrage structure in Module 9 behaves this way, and seeing them collapse to zero is the cleanest possible demonstration of what they actually are.

Put–call parity. For European options on a non-dividend-paying underlying:

C − P = S − K·e^(−rT)

which at r = 0 is simply C − P = S − K. Rearranged, it says a long call plus a short put at the same strike equals long stock. That single identity is the reason a synthetic long share (Chapter 51) draws a perfectly straight line, the reason conversions and reversals are flat, and the reason a call butterfly and a put butterfly at the same strikes are the same trade with different plumbing.

It also gives you a working tool. Any structure can be rewritten in calls, in puts, or in a mixture, and parity tells you the versions must price identically. When they do not, the difference is skew, dividends, borrow cost or an execution problem — never free money. Verify this against Appendix A: at every strike, the 30-day call minus the 30-day put equals 100 minus the strike, to the cent.

A caution about models. Black–Scholes is a pricing convention, not a description of reality. It assumes constant volatility, continuous trading, lognormal returns and no gaps. Markets provide none of these. The model is used throughout this manual because it is the common language, because it is transparent, and because the errors it makes are well documented. It is not used because it is true.

Learning outcomes

• Interpret net delta, gamma, theta and vega as a single description of a position.
• Predict the sign of each greek from a structure's legs without computing it.
• Explain why gamma and theta always carry opposite signs.
• Identify which greek dominates a position's day-to-day profit and loss.

The greeks are not a separate topic bolted on to strategy. They are the strategy, restated in a form that survives the passage of time. A payoff diagram tells you what happens on one day. The greeks tell you what is happening today.

Delta is the rate of change of position value with respect to spot — the share-equivalent exposure. It is the same number as the slope of the payoff line, measured at today's spot rather than at expiration. A net delta of +0.51 means the position currently behaves like 51 long shares per contract. Delta is also, loosely, the market's implied probability that an option finishes in the money, which is a useful intuition and a dangerous one to lean on.

Gamma is the rate of change of delta. It measures how quickly your directional exposure shifts as spot moves. Long options have positive gamma: the position gets longer as the market rises and shorter as it falls, which is a structurally favourable property. Short options have negative gamma, which is structurally unfavourable and is precisely what you are paid for when you sell premium.

Theta is the change in value per day of elapsed time. Long options have negative theta — they cost money to hold. Short options have positive theta.

Vega is sensitivity to implied volatility. Long options are long vega; short options are short vega.

The relationship that matters. Gamma and theta always carry opposite signs. There is no structure anywhere in the fifty-six that is long gamma and long theta at the same time, and anyone offering one is either confused or selling something. Positive gamma is paid for with negative theta: you own the ability to profit from movement and you pay rent daily for it. Negative gamma is compensated by positive theta: you collect rent and you own the obligation to lose from movement. Every position in this manual sits on one side of that trade or the other.

Reading a structure's signature without computing it. Count net long options versus net short options, weighted by how close they are to the money.

• Net long at-the-money options → long gamma, long vega, negative theta.
• Net short at-the-money options → short gamma, short vega, positive theta.
• Long the near expiry and short the far → short vega on balance, because the longer-dated

option carries far more vega.

• Balanced structures with equal counts on both sides — condors, butterflies, iron structures

— have signatures that flip sign depending on where spot sits relative to the strikes.

That last case is worth dwelling on. An iron condor is short gamma and positive theta while spot sits in the middle of its range, which is the comfortable state. As spot approaches a short strike, gamma turns sharply against you and the position stops behaving the way the diagram suggested. The diagram did not change. The position did.

Which greek is running your profit and loss. For short-dated at-the-money structures, gamma and theta dominate and vega is secondary. For long-dated structures and calendars, vega dominates and gamma is almost irrelevant. For deep in-the-money structures, delta dominates and everything else is noise. Knowing which regime you are in tells you what to watch, and watching the wrong greek is the most common form of being surprised by a position you understood perfectly on paper.

Every strategy chapter in this manual prints its net greek signature at entry, computed at spot 100 from the chain in Appendix A. Read it before you read the payoff diagram, not after.

Learning outcomes

• Distinguish realised, implied and forward volatility.
• Explain volatility skew and term structure, and read both off a chain.
• Identify which strategies are primarily volatility trades regardless of their appearance.
• Describe volatility crush and the conditions that produce it.

Most people who believe they are trading direction are trading volatility. This unit exists to make that visible before Module 4, where it becomes unavoidable.

Three volatilities. Realised volatility is what the underlying actually did, measured from price history. Implied volatility is what the option market is charging, backed out of current prices. Forward volatility is what is implied for a future period, extracted from the difference between two expiries — and it is the thing a calendar spread actually trades.

The gap between implied and realised is the volatility risk premium. It is usually positive: options tend to be priced above what the underlying subsequently delivers. This is the structural reason premium selling works often, and it is also the reason premium selling is crowded, which is a different problem.

Skew. Implied volatility is not constant across strikes. In equity markets, lower strikes almost always carry higher implied volatility than higher strikes. The practical consequences run through this entire manual:

• Puts are systematically more expensive than equidistant calls. A risk reversal (Chapter —)

typically opens for a credit purely because of this, and that credit is skew, not edge.

• Put-built structures and call-built structures with identical payoff shapes can price

differently. Parity guarantees the shapes match; it does not guarantee the volatilities do.

• Put backspreads (Chapter 19) buy two options on the expensive side of the curve and sell one

on the cheaper side, which is a headwind the payoff diagram never shows.

Term structure. Implied volatility also varies by expiry. In calm markets the curve usually slopes upward — longer options carry higher implied volatility. Around a known event it inverts sharply, with the expiry containing the event pricing far above its neighbours. Calendar and diagonal spreads (Module 8) are direct trades on this curve, whether or not the trader intends them to be.

Volatility crush. After a scheduled event resolves, the uncertainty premium in the nearest expiry collapses, often within minutes of the open. This is the mechanism behind the single most common losing pattern in retail options: buying a straddle before earnings, being right about the direction of the move, and still losing money because implied volatility fell further than the price moved. The trade was not a directional bet. It was a bet that the realised move would exceed the priced move, and the priced move was larger.

Which strategies are volatility trades. More than most readers expect:

• Every long straddle, strangle, guts, strap and strip is explicitly long volatility.
• Every calendar and diagonal is a volatility trade wearing a neutral costume; the payoff

curve in Module 8 is drawn by a pricing model, and its height is set by the implied volatility of the surviving leg.

• Every credit structure is short volatility, including the ones sold as income.
• Backspreads and ratio spreads are long or short volatility depending on which side carries

more contracts, and their signs can invert as spot moves through the strikes.

A useful discipline: before entering any position, write down what implied volatility must do for the trade to work. If you cannot answer, you do not yet know what you are trading.

Learning outcomes

• Classify any structure as defined risk, bounded-but-large, or unbounded.
• Size a position from maximum loss rather than from premium or margin.
• Compute the reward-to-risk ratio of a credit structure and interpret it correctly.
• Explain why high win rate and positive expectancy are independent properties.

This is the unit that decides whether the rest of the manual helps you or hurts you.

Three risk classes. Every structure in this book falls into one of three categories, and the strategy chapters label each one explicitly.

Defined risk. Maximum loss is a fixed number known at entry, and it cannot be exceeded. Long options, debit spreads, butterflies, condors and all iron structures sit here. Note the word defined, not small — a defined loss is still a full loss.

Bounded but large. Maximum loss is finite only because the underlying cannot fall below zero. Short puts, covered calls, put ratio spreads and put ladders sit here. In practice these should be sized as though they were unbounded, because the bound is far outside the range in which you would still be solvent.

Unbounded. Maximum loss has no ceiling. Naked short calls, short straddles and strangles, call ratio spreads, ratio call writes and any structure with a net short call at the far right edge sit here. The chart clips these. The market does not.

Size from maximum loss. Not from premium collected, not from margin required, not from what the broker permits. Margin requirements are a lender's estimate of overnight risk, not a risk budget, and they are recalculated by someone whose interests are not yours. Decide the maximum currency loss you will accept on one position, look up the structure's maximum loss per contract, and divide. For unbounded structures, substitute a stress scenario — a move of several standard deviations — because there is no maximum to divide by.

The ratio, not the streak. Credit structures are sold on win rate, and win rate is the least informative statistic available. Take the iron condor from Chapter 42: it collects 1.52 and risks 3.48. That is a reward-to-risk ratio of roughly 0.44 to 1, which means it needs to win about 70 percent of the time simply to break even before costs. A strategy that wins 75 percent of the time at that ratio has a thin edge; the same strategy at 65 percent is a reliable way to lose money slowly while feeling successful most months.

Every credit structure in this manual prints its ratio. Compute the required win rate from it:

breakeven win rate = max loss ÷ (max loss + max gain)

Then ask honestly whether your actual hit rate clears it, using your own records rather than a backtest.

Correlation of positions. Ten iron condors on ten different names in the same index are not ten positions. They are one position in ten places, and they will all be tested in the same week. This is the mechanism by which a book of high-probability trades produces a single catastrophic month. Diversification across underlyings does not diversify a short volatility exposure.

Assignment and liquidity as risk. Two risks appear in no payoff diagram. Early assignment converts an option position into a share position overnight, usually on the leg you least want. Illiquidity means a four-leg structure with a defined maximum profit of 1.52 can cost a meaningful fraction of that to enter and exit, and can be impossible to close at any sensible price in the conditions that make you want to close it.

One line to carry out of this unit. Every structure in this manual can be made unrecoverable by size. None of them can be made safe by cleverness.

Learning outcomes

• Distinguish American and European exercise and identify which structures are exposed.
• State the two conditions under which early assignment is economically rational.
• Describe pin risk and how it arises at expiration.
• Account for the full transaction cost of a multi-leg structure.

The chapters that follow describe positions. This unit describes what a broker and a clearing house will actually do to them.

Exercise style. American-style options can be exercised on any business day up to expiry; European-style options only at expiry. Most single-name equity options are American. Most broad index options are European and cash-settled. This distinction decides whether early assignment is a live risk in a given structure, and it is the reason a box spread (Chapter 55) is a financing instrument on European options and a hazard on American ones.

When early assignment is rational. Two cases, and they cover nearly all of it.

Short calls before an ex-dividend date. If the dividend exceeds the remaining extrinsic value of the call, the holder captures more by exercising early and collecting the dividend. If you are short a call on a dividend-paying share and its extrinsic value is thin as the ex-date approaches, expect assignment.

Deep in-the-money short puts. When a put has almost no extrinsic value left, the holder can free up capital by exercising. At r = 0 this is weak, but in a positive-rate environment it is routine.

The general rule: extrinsic value is what prevents early exercise. When extrinsic value on a short leg approaches zero, assignment risk approaches certainty.

What assignment does to a structure. It dismantles it. A conversion (Chapter 53) assigned on its short call becomes a long put and no shares — a directional position you did not choose. An iron condor assigned on its short put becomes long shares against three remaining option legs, on margin, over a weekend. The defined-risk label described the structure that existed before assignment.

Pin risk. When the underlying settles almost exactly at a short strike, you do not know whether you will be assigned until after the market has closed. Butterflies and iron butterflies are most exposed, because their maximum profit sits precisely at the strike that produces the ambiguity. The scenario that pays best is the scenario that leaves you uncertain what you own on Monday.

Transaction cost, properly counted. For a four-leg structure the true cost is: four bid-ask spreads on entry, four on exit, plus commissions on eight contract legs, plus slippage, plus exercise or assignment fees where relevant. Against an iron condor with a maximum profit of 1.52, that is not a rounding error — it can be ten to twenty percent of the best case before anything goes wrong.

This is why the manual reports maximum profit and loss in gross terms and then insists you subtract your own costs. Two traders running an identical structure in identical size can have opposite expectancies purely on execution, and no chapter in this book can fix that for you.

Exit discipline. A position with four legs is four decisions to close. Closing legs separately — 'legging out' — introduces a directional exposure that was not part of the trade. Where liquidity permits, structures are entered and exited as a single order at a net price. Where liquidity does not permit that, the structure was too illiquid to enter.

Learning outcomes

• Derive every payoff in the manual from the four single-leg positions.
• State maximum profit, maximum loss and breakeven for each primitive from first principles.
• Explain why the long call's red segment is the cost of being early, not a drawing artefact.
• Classify each primitive by risk class and size it accordingly.

Four positions. Everything else in this manual is a combination of them, and a reader who can draw these four from memory can derive all fifty-two remaining structures without looking anything up.

What the four actually are. A long call is a right to buy with a deadline. A long put is a right to sell with a deadline. A short call is an obligation to deliver. A short put is an obligation to acquire. Note the asymmetry in the language, because it is the asymmetry in the risk: rights cost money and cannot lose more than they cost; obligations pay money and can lose far more than they pay.

The asymmetry, quantified. Using the manual's chain, a 30-day 105 call costs 1.08 and a 30-day 95 put costs 0.97. Selling either collects about a point. Buying either risks about a point. But the seller of the call has no ceiling on loss, and the seller of the put risks 94.03 per share down to zero. The premium is roughly symmetric. The exposure is not remotely so.

This is the single most consequential fact in options, and it is obscured by the fact that selling wins far more often than buying. Both statements are true at once: premium selling has a high hit rate, and premium selling carries the tail. Module 1 exists to make sure you never confuse the first with an argument against the second.

Doctrine for long primitives. You are paying for time and convexity. Three disciplines follow. First, buy more time than your thesis requires, because the last thirty days decay fastest and being right late is the same as being wrong. Second, check implied volatility before entry — buying a long option into elevated implied volatility means you need the move and you need volatility to hold. Third, size from the premium, because the premium is the whole loss, and treat every long option as capable of expiring worthless, because most do.

Doctrine for short primitives. You are selling time and convexity and collecting rent. Three disciplines follow. First, size from the maximum loss, not the credit — for a short 95 put that is 94.03 per share, not 0.97. Second, know your assignment exposure and check dividend dates. Third, accept that the loss distribution is not the win distribution: a long run of full credits followed by one full-width loss is the expected shape, not the unlucky one.

Where these appear later. The covered call in Module 2 is a short put wearing a costume. The bull put spread in Module 3 is a short put with a wing. The iron condor in Module 6 is two short primitives with two wings. Every credit structure in this manual inherits the tail behaviour of the short primitive at its core, softened but never removed. If Module 1 is understood, the rest of the book is mostly bookkeeping.

Learning outcomes

• Decompose any stock-plus-option position into a synthetic equivalent.
• Explain what a covered call sells and what it does not buy.
• Price protection as a recurring cost rather than a one-off purchase.
• Recognise a collar as a decision to convert a holding into a range.

This module adds one linear leg — a share position — to the primitives from Module 1. The maths barely changes. The framing changes completely, because these are the structures most often sold to people as conservative, and the word does a great deal of unexamined work.

The decomposition discipline. Every position in this module has a synthetic equivalent with fewer moving parts, and naming it is the fastest route to seeing the real exposure.

• Covered call = short put at the call strike, plus a fixed amount. Identical payoff shape.
• Protective put = long call at the put strike, plus a fixed amount.
• Collar = a bull call spread between the two strikes, plus a fixed amount.
• Covered put = short call at the put strike.

Anyone who would refuse to sell a naked put but happily writes covered calls is making a distinction the payoff diagram does not recognise. The distinction is real — it lives in capital treatment, cost basis and tax, not in risk shape — but it should be made knowingly.

What a covered call actually does. It sells the right tail to slightly soften the left. Using the chain: writing a 105 call against shares at 100 collects 1.08. Downside protection purchased: 1.08 points, or roughly one percent. Upside surrendered: everything above 106.08. That is the trade. It is defensible when you were going to sell at that level anyway. It is not risk reduction in any meaningful sense, and describing it as income obscures that you have sold an asset — optionality — for cash.

Protection is a subscription. A 95 put costs 0.97 for thirty days. Run continuously, that is roughly twelve percent of position value per year, against a deductible of five points before the floor engages. Protection bought after a fall costs considerably more, because implied volatility rises as markets decline — the insurance reprices precisely when you want it. Any protective programme has to be judged over a full cycle, not by the month in which it paid off.

The collar's real question. A collar converts a share holding into a bounded range. The arithmetic in Chapter 10 is straightforward. The question is not arithmetic: if you wanted a bounded range, was equity the right instrument in the first place? A collared share position has the return profile of a structured note and the tax and administrative burden of three positions. Sometimes that is exactly right — a concentrated holding under a lock-up, for instance. Often it is a way of holding something you no longer want to hold.

Doctrine. Name the synthetic before you enter. Price protection annually, not per trade. Decide the exit level before writing a call against shares, not after they are challenged. And never roll a short call up and out to avoid assignment without recognising that you have opened a new, larger position to avoid admitting the first one worked.

Learning outcomes

• Construct debit and credit verticals in both calls and puts and show they are equivalent.
• Compute the breakeven win rate of any credit vertical and use it as the entry filter.
• Choose between debit and credit construction on grounds other than payoff shape.
• Identify the unbounded tail in a ladder and explain why it is the trade's defining feature.

A vertical spread is the first genuinely two-dimensional trade in this manual. A single option bets on direction. A vertical bets on direction and distance, and the distance is usually the part traders get wrong.

The four verticals are two trades. Bull call spread and bull put spread at the same strikes draw the same line. So do bear call and bear put. Parity guarantees it — verify it in Chapters 12 and 13, where the computed maxima match to the cent. What differs is: whether you pay or receive at entry, which side of the volatility skew you traded, where assignment risk sits, and how much capital the broker holds. Those four things, not the shape, are the decision.

A working rule: construct with out-of-the-money options in whichever type keeps both strikes out of the money. That minimises assignment exposure and usually trades tighter. It is a mechanical preference, not an edge.

The credit vertical's real arithmetic. A bear call spread from 100 to 105 collects 1.78 and risks 3.22. Win rate needed to break even: 3.22 ÷ 5.00 = 64.4 percent. A bull put spread from 95 to 100 collects 1.89 and risks 3.11 — 62.2 percent needed. These numbers should be computed before every entry and compared against your own measured hit rate, not against a probability model. A structure that requires 64 percent and delivers 68 percent is a business. One that requires 64 percent and delivers 60 percent produces a long sequence of satisfying months and a negative account curve.

Width is a decision, not a default. Widening a spread increases both maximum profit and maximum loss, and shifts the breakeven. Narrowing does the reverse. The width should be chosen from the move you are forecasting — if your thesis is a three-point move, a five-point spread wastes premium on distance you do not expect to reach, and a two-point spread caps you below your own forecast. Most traders choose width by what looks affordable, which is choosing by irrelevance.

Ladders: where this module turns dangerous. Adding a third short strike converts a defined structure into an undefined one. A bull call ladder is a bullish position that loses without limit if the market rises far enough. A bear put ladder is a bearish position that cannot survive a crash. Both are, correctly understood, range trades with directional lean and a naked option attached.

The diagnostic is mechanical and appears in every chapter of this manual: read the slope of the payoff line at the far edge of the chart. If it is not zero, the structure is open there, regardless of how tidy the middle looks. Ladders are included in this module precisely because they look like verticals and are not.

Doctrine. Compute the breakeven win rate before entry. Choose construction on assignment and skew, never on shape. Set width from the forecast. Read the far edge. And treat any structure with more short contracts than long as belonging to Module 7, whatever it is called.

Learning outcomes

• Compute net contract exposure at both extremes of the payoff chart.
• Explain why a ratio spread's best case sits adjacent to its worst.
• Identify the trough in a backspread and state what market outcome produces it.
• Classify every structure in this module by risk class and size accordingly.

Every structure so far has had matched counts — as many long contracts as short. This module breaks that, and breaking it changes the character of the position entirely.

The single diagnostic. Read the net contract count at the far right and far left of the payoff chart. This manual prints it in every chapter as the edge slope. If it is not zero, the structure is open in that direction, and the friendly shape in the middle is irrelevant to how the position ends.

• Call ratio spread: net short one call above the upper strike. Unbounded.
• Put ratio spread: net short one put below the lower strike. Bounded at zero, and large.
• Call backspread: net long one call above the upper strike. Unbounded gain.
• Put backspread: net long one put below the lower strike. Large gain.
• Ratio call write: net short one call above the strike, despite owning shares. Unbounded.
• Ratio put write: unbounded above from short shares, large below from the doubled puts. The

only structure in the manual with both edges live.

Ratio spreads: the adjacency problem. A call ratio spread reaches maximum profit exactly at the strike where you are short two contracts. One point further and the position begins losing. Ten points further and it is losing steadily with no ceiling. The best outcome and the worst outcome are neighbours on the same axis. That is a structural property, not a risk of poor management, and it means being more right than expected is the losing case. Any position with that property should be sized as though the tail is the base case.

Backspreads: the trough problem. A backspread is the inverse and has the inverse pathology. Its worst outcome is a moderate move in the direction you forecast — spot finishing at the strike where you are long two contracts. You were right about direction and wrong about magnitude, and that combination is far more common than being wrong about direction outright. Backspreads are often opened for a credit, which creates a small profit shelf for doing nothing and disguises how expensive the trough is.

Ratio writes. These deserve a specific warning because they are frequently presented as an enhancement to covered calls. Selling two calls against one hundred shares is a covered call plus a naked call. The word covered describes the first contract only. Chapter — shows the payoff turning down above the strike and continuing; Chapter — is worse, with both edges open at once. Neither belongs in an account that is not explicitly running an unbounded-risk mandate with the margin and monitoring to match.

Doctrine. Read the edges before the middle. Size every structure in this module from a stress scenario rather than from margin. Treat 'more right than expected' as a live scenario in ratio spreads and 'right but not enough' as the base case in backspreads. And be sceptical of any presentation of these structures that leads with the credit received.

Learning outcomes

• Distinguish a ladder from the vertical it resembles using the edge slope alone.
• State the net contract exposure of a ladder beyond its outer strike.
• Explain why the improved entry price on a ladder is payment rather than advantage.
• Identify the condor as the structure that buys the open edge back.

Two structures, and they are a family of their own for one reason: they look like verticals and they are not.

Take a vertical from Module 3 and add a third strike, short. The middle of the payoff line barely moves. The risk class changes completely.

A bull call ladder — long one call, short two higher ones — is net short one call above the top strike. It is described as bullish, it reaches its maximum between the short strikes, and it loses without limit if the market keeps rising. A bear put ladder is the mirror: a bearish structure that cannot survive the decline it was built for, because two short puts below the market overwhelm the single long one.

Being more right than forecast is how these positions lose. That is not a management failure. It is the structure.

The diagnostic takes ten seconds and it is the reason the edge check is a ritual in this manual rather than a technique: read the slope at the far edge. If it is not zero, the structure is open there, however tidy the middle looks. Nothing about the profitable region of a ladder distinguishes it from a vertical, and no amount of studying that region will.

Why anyone opens one. The third short strike improves the entry price — sometimes converting a debit into a credit. That improvement is the market paying you to accept an unbounded tail, and it is priced accurately. Module 4 makes the same point about ratios and Module 6 about broken wings. A credit at entry is never an edge; it is a description of where the risk went.

And what to do instead. If you want the plateau without the tail, buy the outer strike back. That structure has a name and its own module: it is a condor, and Module 7 is where it lives.

Learning outcomes

• Express any volatility structure as an implied forecast of the size of the move.
• Convert a straddle's premium into the percentage move required to break even.
• Explain why direction can be correct and the position still lose.
• Distinguish long guts from a long strangle in capital terms despite identical payoffs.

Module 4 removes direction from the trade and leaves only magnitude. That sounds simpler. It is not, because magnitude is priced with far more sophistication than direction, and the person on the other side is usually a specialist.

Read the straddle as a forecast. A 30-day at-the-money straddle costs 5.72 against spot of

1. That is the market saying: I expect a move of at least 5.7 percent in thirty days, and I

have priced it. Buying it is not a bet that the market will move. It is a bet that the market will move more than 5.7 percent. Selling it is the opposite bet. Every long volatility position in this module should be restated in that form before entry, out loud, as a percentage. If the required move sounds implausible, you have found your answer.

Why correct direction still loses. Two reasons, and they compound. First, the move has to clear the premium, and the premium is largest exactly when large moves are expected. Second, implied volatility usually falls once the uncertainty resolves, which reduces the value of both legs regardless of where spot went. A trader who buys a straddle before an announcement, sees a four percent move, and loses money has not been unlucky. They have bought a 5.7 percent forecast and received a four percent outcome.

Straddle versus strangle. The straddle has one strike, a single low point, the tightest breakevens and the highest cost. The strangle has two strikes, a flat floor between them, wider breakevens and a lower cost. Cheaper is not better. The strangle costs 2.05 against the straddle's 5.72, but requires a move past 106.05 or 92.95 rather than past 105.72 or 94.28. You paid 64 percent less for a materially harder target — and the lower ticket price is exactly what tempts traders into three times the size, which converts a cheaper trade into a larger one.

Guts, and the intrinsic lesson. Long guts pays an enormous premium — 11.05 for the 95 call and 105 put — and has precisely the same payoff shape as a strangle. Ten points of that premium is intrinsic value: capital parked, not wagered. Strip it out and the wager is 1.05, close to the strangle's 2.05 for a tighter pair of breakevens. Guts exists in this manual for one reason: it is the clearest possible demonstration that premium size tells you nothing until you separate intrinsic from extrinsic. Chapter 29 makes the same point from the selling side, where a spectacular credit turns out to be mostly an obligation.

Doctrine for short volatility. Short straddles and strangles are the highest-hit-rate, highest-tail structures in the manual, and they carry two live tails simultaneously — a property shared with almost nothing else here. Size them from a stress scenario, not from margin. Understand that margin expands as the position moves against you, which can force an exit at the worst available price. And recognise that the probability band that makes the position look comfortable is generated by the same model that mispriced the last crisis.

Learning outcomes

• Build butterflies from calls, puts and iron construction, and show they are equivalent.
• Explain why butterflies are cheap and why cheapness is not probability.
• Locate the risk side of a broken wing butterfly.
• Describe how a butterfly's value develops over its life and why that shapes management.

A butterfly is a bet that price finishes at a specific level. It is the most precise instrument in this manual, it is usually the cheapest, and the relationship between those two facts is the subject of this module.

Construction. Three strikes, four contracts, weights of one, minus two, one. Build it in calls, in puts, or as an iron butterfly using both — parity makes all three the same trade. Chapters 30, 25 and 26 compute them side by side; the shapes match, the plumbing does not.

Why it is cheap, and what that means. The 95/100/105 call butterfly costs 1.34 and returns 3.66 net — a 2.74 to 1 ratio. That ratio is not generosity. It is the market's estimate that finishing within a narrow band around 100 is unlikely, and the market is usually right about that. Cheap structures with high maximum returns are cheap because the maximum is improbable, and the most common outcome for a butterfly is a total loss of a small amount.

This makes butterflies genuinely useful in two situations and misleading everywhere else. Useful: when you have a specific level thesis — a known settlement, a pinned expiry, a well-defined target — and when you want convex exposure to a level for a small ticket. Not useful: as a general neutral position, because the profit zone is narrow and the maximum requires precision you probably do not have.

The development curve. A butterfly is nearly worthless until close to expiry, then develops most of its value in the final days as the short strikes decay. This has a practical consequence that surprises people: for most of the position's life it will show little profit even when spot is sitting exactly where you wanted, and traders who monitor daily marks tend to exit before the structure has done anything. Butterflies reward patience and punish attention.

Broken wings. Move one wing further out and the structure stops being symmetric. It often opens for a credit, which makes it feel free, and it concentrates risk on the widened side. Chapter 36 works through the arithmetic. The doctrine is simple: find the wide wing, compute the loss on that side, and size against it. A credit at entry is not an edge — it is a description of where you moved the risk to.

Management. Four legs means eight transactions across the life of the trade, against a maximum profit that may be a few points. Enter and exit as a single order at a net price. Do not leg out. Accept that pin risk at the middle strike is the price of the best outcome (Foundation F6), and if you cannot tolerate an ambiguous assignment over a weekend, close before expiry and accept less than the maximum.

Learning outcomes

• Construct all four condor permutations and the two iron variants.
• Explain the trade between plateau width and maximum profit.
• Compute and apply the breakeven win rate for an iron condor.
• Describe why a book of condors is one position rather than many.

A condor is a butterfly with its apex pulled apart into a table top. Everything from Module 5 carries over, with one substitution: you have traded height for width.

The trade, quantified. The 95/100/105 butterfly can return 5.00. The 90/95/105/110 condor can return 5.00 as well, but across a five-point plateau rather than at a single point — and it costs more to enter, so its net maximum is lower. Wider profit zone, smaller maximum. That relationship is fixed and cannot be optimised away. Every condor decision is a position on that curve.

The iron condor, and why it dominates retail. The iron condor collects a credit rather than paying a debit, requires no capital beyond the margin on its widest side, and wins whenever price stays inside a range — which, most of the time, it does. Chapter 42 gives the numbers on the manual's chain: collects 1.52, risks 3.48, requires a 69.6 percent hit rate before costs.

That last number is the entire discussion. Sixty-nine point six percent is achievable. It is not comfortably achievable, and it does not leave much room for four legs of transaction cost. A trader who runs iron condors without computing the required rate for each entry is running a strategy whose expectancy they have never established.

Adjustment: the honest position. A great deal of published material describes adjusting a tested condor — rolling the challenged side, adding a spread on the other side, converting to an iron butterfly. Here is the assessment this manual takes: most adjustments move risk rather than reduce it, add transaction cost, and extend the duration of a losing position. Rolling a challenged short strike out in time collects more credit and increases the total exposure. It converts a defined loss into a longer, larger position, and it feels like management.

The defensible adjustments are narrow: closing the untested side to remove residual risk once its value is negligible, and closing the whole structure at a predetermined loss. Both are decisions to reduce exposure. Neither requires new legs.

Correlation, again. This is where Foundation F5's warning becomes concrete. Ten iron condors on ten large-cap names are ten expressions of one view: that nothing unusual happens. They are tested simultaneously, they gap simultaneously, and their margin requirements expand simultaneously. A book of them is not diversified. It is concentrated in a factor that does not appear on any position screen.

Doctrine. Compute the required win rate at every entry. Choose plateau width from a distribution you have measured, not one you assume. Predetermine the exit loss before entry. Do not adjust into a larger position. Count your condors as one position for sizing purposes.

Learning outcomes

• Explain why a calendar's profit curve requires a pricing model rather than arithmetic.
• Identify the volatility exposure in a calendar and state its sign.
• Distinguish a diagonal's three simultaneous exposures.
• Describe how term structure and skew change a time spread's value.

Every structure up to this point could be drawn with addition and a straight edge. This module cannot, and understanding why is the point of it.

Why the line curves. In a calendar spread, one leg expires before the other. At the near expiry the short leg settles to intrinsic value — that is arithmetic. But the long leg is still alive. It has time remaining, so it has a price, and that price has to be computed from a model at every spot on the axis. The resulting chart is not a payoff diagram at all. It is a valuation surface sliced at one date, and its height depends on assumptions you supplied.

Change the implied volatility of the surviving leg and the whole curve rises or falls. Nothing about the position changed. The assumption changed. Any chart in this module should be read with that in mind, and any tool that does not let you vary the assumption is hiding the main variable.

What a calendar actually trades. Three things at once, in descending order of importance:

Forward volatility. You are long the volatility of the back month and short the volatility of the front. If the term structure flattens against you, the position loses regardless of where spot goes. This is the dominant exposure in almost every calendar, and it is the one least often named by the trader holding it.

Time decay differential. The front-month option decays faster than the back-month, which is the mechanism the position is usually sold on. It is real, and it is the smallest of the three.

Spot position. The curve peaks at the strike, so there is a directional component even in a 'neutral' calendar.

Where parity does not fail, and what does. It is tempting to say that call-built and put-built structures finally diverge here. They do not. Parity binds within each expiry, and subtracting one expiry from another leaves a put calendar cheaper than a call calendar by exactly K(e^(-rT2) - e^(-rT1)) - a pure carry term, independent of either leg's volatility. At this manual's zero rate the two are identical to the cent, whatever volatilities you assume. What changes in Module 8 is not the identity but the drawability: this is the first family whose chart cannot be produced by arithmetic at all. And that carry term is not a nuisance - it is precisely what the jelly roll in Chapter 56 isolates and trades.

Diagonals. Different strikes and different expiries at once. The position is now directional, volatility-sensitive and time-sensitive simultaneously, and traders routinely manage one of the three while the other two determine the outcome. The discipline is to name all three exposures at entry, in writing, and to check which one is currently dominant before making any adjustment.

Rolling. The short leg of a calendar or diagonal expires, and the natural response is to sell another. Each roll is a new trade with new assumptions, new transaction costs and a new volatility environment. A position that has been rolled six times is not one position with a long history; it is the seventh trade in a sequence, and its cumulative result should be tracked as a campaign rather than a holding.

Learning outcomes

• Apply put–call parity to convert freely between calls, puts and shares.
• Explain why conversions, reversals and boxes have flat payoff lines.
• State what determines the profit of a flat-line structure.
• Describe the specific hazard American-style exercise poses to a box spread.

This module contains no directional trades. Every structure in it has a payoff line that is flat, or nearly so, and the point is what remains when direction is removed.

Parity, restated as a tool. C − P = S − K at zero rates. Rearranged five ways:

• Long call + short put = long shares (synthetic long, Chapter 51)
• Short call + long put = short shares (synthetic short, Chapter 52)
• Long shares + long put = long call (protective put, Chapter 8)
• Short shares + long call = long put (protective call, Chapter 9)
• Long shares + long put + short call = flat (conversion, Chapter 53)

These are not curiosities. They are the reason a single payoff engine handles all fifty-six structures without knowing any of their names, and they are the practical route to trading around an assignment, a borrow constraint or a liquidity problem in one leg.

Why flat is interesting. When a payoff line is horizontal, price has been removed from the outcome. What is left is carry: interest on the capital, dividends on the shares, borrow cost on any short position, and the quality of your execution across three or four legs. That is the entire content of a conversion or a reversal, and it is why these are professional structures with thin margins rather than retail opportunities.

This manual sets r = 0, which makes the point vividly: every arbitrage structure in this module computes to exactly zero profit. Chapter 55's box spread is worth 10.00 and costs 10.00. The structure contains nothing but interest, and at zero interest it contains nothing. Reintroduce a positive rate and the numbers separate — which tells you precisely what you were trading all along.

Conversion and reversal as boundaries. Together these define the no-arbitrage bounds inside which every other option price in this manual must sit. They are more useful as a checking tool than as a trade: if a structure prices outside those bounds in your data, your data is wrong before the market is. This manual uses them exactly that way, and Appendix B's verification table is built on them.

The box spread hazard. A box on European options is a synthetic zero-coupon loan with a known settlement value. A box on American options is not, because either short leg can be assigned early, and a broken box is a directional position you did not choose in a size you did not select. There is well-documented history of retail accounts taking severe losses on boxes that were understood as risk-free. The instrument was not the problem. The exercise style was.

The jelly roll. Chapter 56 is the most obscure structure in the manual and the best explanation of what an option costs. It pairs a call calendar against a put calendar, cancels the direction and most of the volatility, and leaves the pure cost of carry between two expiries. Every option you have ever bought contained that component. The jelly roll is where it can be seen priced on its own.

Spot 100.00 · volatility 25 percent · interest 0 · dividend 0 · expiries at 30 and 90 days. Prices are per share; one contract covers 100 shares. Values are Black–Scholes, rounded to two decimals for display and carried at full precision in every computation.

The final column verifies put–call parity: at zero rates, the call less the put must equal spot less strike at every strike. It does, exactly. That column is the check this manual runs against every structure it prints.

How to read the extrinsic column. It is the wager. The intrinsic column is capital parked. A 15.03 premium on the 85-strike call contains 15.00 of parked capital and 0.03 of actual bet; a 2.86 premium at the 100 strike is 2.86 of pure bet. Premium size and exposure are unrelated, and Chapters 21 and 22 exist to make that unforgettable.

Every value below was produced by one payoff engine from the chain in Appendix A, then checked against the closed-form expression for that structure. Where a structure has no closed form — the calendars, the diagonals, the jelly roll — the value is the model price of the surviving leg at the near expiry, and it is marked model-dependent in its chapter.

Sort by the win-rate column and read down. It is the most useful page in this manual.

Everything in this manual reduces to the expressions below. K is a strike, S spot, D a net debit, C a net credit, W a spread width.

Payoff primitives at expiration

Critical values by family

Pricing

d₁ = [ ln(S/K) + (r − q + σ²/2)·T ] / (σ√T) d₂ = d₁ − σ√T Call = S·e^(−qT)·N(d₁) − K·e^(−rT)·N(d₂) Put = K·e^(−rT)·N(−d₂) − S·e^(−qT)·N(−d₁)

Put–call parity

C − P = S·e^(−qT) − K·e^(−rT) which at r = q = 0 is C − P = S − K

Risk arithmetic

reward : risk = max profit ÷ |max loss| breakeven win rate = |max loss| ÷ ( |max loss| + max profit ) position size = risk budget ÷ |max loss per contract| ÷ 100

For unbounded structures there is no max loss to divide by. Substitute a stress move of at least three standard deviations and size from that instead.

Diagnostics used throughout this manual

kinks occur only at strikes slope between kinks = net share-equivalent exposure over that interval plateau an interval where the first difference of the payoff array is zero breakevens sign changes in the payoff array edge slope net contract exposure at the extremes — the risk-class test

Assignment

The obligation on a short option holder to fulfil the contract when the long holder exercises. Converts an option position into a share position, usually on the leg you least want. See F6.

Bounded but large

A risk class in this manual: maximum loss is finite only because the underlying cannot fall below zero. Short puts, covered calls and put ladders sit here, and should be sized as though unbounded.

Breakeven

A spot price at which the position's profit and loss is exactly zero. Found by scanning a payoff array for sign changes; there may be none, one, two or more.

Credit

Net premium received on entry. Not income, and not profit, until the position has run its course.

Debit

Net premium paid on entry. For long structures it is the maximum loss.

Defined risk

Maximum loss is a fixed number known at entry and cannot be exceeded. Note: defined, not small.

Delta

Rate of change of position value with respect to spot; the share-equivalent exposure. Equals the slope of the payoff line, measured today rather than at expiry.

Early exercise

Exercise of an American-style option before expiry. Rational mainly when a dividend exceeds a short call's remaining extrinsic value, or when a deep in-the-money put's extrinsic value is near zero.

Edge slope

The slope of the payoff array at the far left and far right of the chart. The diagnostic used throughout this manual to classify risk; a non-zero edge slope means the structure is open in that direction.

Extrinsic value

Premium in excess of intrinsic value. The entire content of the wager, and the part that decays.

Gamma

Rate of change of delta. Long options are long gamma; short options are short gamma. Always carries the opposite sign to theta.

Intrinsic value

What an option is worth if exercised immediately. Capital parked, not capital wagered.

Leg

One contract line within a structure, defined by type, quantity, strike and expiry. The only per-strategy data a payoff engine requires.

Legging out

Closing the legs of a structure separately. Introduces a directional exposure that was not part of the trade.

Pin risk

Uncertainty over assignment when the underlying settles almost exactly at a short strike. Butterflies are most exposed, because maximum profit sits at the strike that creates the ambiguity.

Plateau

An interval of spot prices over which profit or loss is constant. Its width is the range of outcomes a structure treats identically.

Put–call parity

C − P = S − K at zero rates. The identity that lets any structure be rewritten in calls, puts or shares, and the reason one payoff engine handles all fifty-six.

Skew

The variation of implied volatility across strikes. In equity markets, lower strikes carry higher implied volatility, which systematically favours put sellers and taxes put buyers.

Term structure

The variation of implied volatility across expiries. What a calendar spread actually trades.

Theta

Change in position value per day of elapsed time. Long options have negative theta; short options positive.

Vega

Sensitivity to implied volatility, per volatility point. Long options are long vega.

Volatility crush

The collapse in implied volatility after a scheduled event resolves. The mechanism behind losing on a long straddle despite a correct directional call.

Width

The distance between two strikes. Sets the maximum loss on a credit spread and the maximum profit on a debit spread.