How the options worked examples are built

What this page is for

The options library is a neutral course written for UK taxpayers: in pounds and pence, on the contracts UK investors can trade (ICE stock options, FTSE 100 options, US options through UK and US brokers), with the tax worked through to the SA108. No broker links, no affiliate fees, no trade ideas.

Behind that sits a promise about the numbers. Every premium, Greek, breakeven and probability on an options page comes from one engine, from inputs printed on the page, in one dated model market. This page publishes that market (the model sheet), the pricing method, the cost and rounding rules, the teaching conventions the strategy pages refer to, the dates on which the facts need checking again, and a way to rebuild any figure in a spreadsheet.

Named companies are model underlyings. BP, Tesco, GSK and the others appear because their options list on ICE and their numbers are easy to follow. Nothing here is a view on any of them, the trade dates are fixed in the past, and the conventions describe what a worked plan does, not what any reader ought to do.

Used on: covered call · bull call spread · iron condor · long straddle · calendar spread · jade lizard · LEAPS · FTSE 100 options, and every other page with a modelled example.

The model sheet: one dated market for every page

A worked example needs a share price, a volatility, an interest rate and a calendar. If each page chose its own, two pages could price the same BP option differently, and a reader could not tell a difference in strategy from a difference in inputs. So the library fixes one market, dated Monday 17 August 2026, and every page starts from it. A page that needs another date (a results day, a later roll) states that date and each input it changes.

The model levels are round numbers close to the real close, kept round so the arithmetic can be followed by hand. The real closes sit beside them (price data: Yahoo Finance, checked 26 September 2026). BP is the one wide gap: it is modelled at 530p against a 519.6p close, because about fifteen pages were built on 530p and rebasing would change every figure without teaching anything new. It is labelled illustrative wherever it appears.

Pages that model a share on another date say so; Tesco, for example, closed at 480.7p on Wednesday 5 August 2026, which is the level an early-August example would use. The 22 ICE mini options are listed on the exchange, but whether a UK broker offers a given mini, and whether it shows a two-way price, could not be confirmed, so every mini example carries that caveat. Contract terms come from the ICE product pages (for example BP and the Tesco mini); what they mean for a position is set out on the basics page.

The FTSE 100 volatility surface

Single-stock examples use one volatility per leg, printed on the page. FTSE 100 pages share a surface, because on several of them the skew (puts dearer than calls) is the lesson: IV(K) = 14.0% − 0.40 × ln(K ÷ 10,750), never below 5%. It was fitted to the quotes the FTSE pages already used (13.98% at the money, a slope of 0.395) and rounded. It is a model assumption, not market data. When the index moves, each strike keeps its volatility (sticky-strike), and stress rows add a stated parallel shift. The implied-volatility page explains skew and compares sticky-strike with sticky-delta.

How a figure is computed

All the maths lives in one file, /assets/tdx-options-engine.js. The site's calculators run it in the browser, and the writers' command-line calculator and the build checks run the same file, so a number on a page, in a tool and in the check that guards it cannot drift apart. It is plain JavaScript with no dependencies, readable in any browser. Which model prices an option depends on how the option can be exercised:

• European options (FTSE 100, SPX, XSP) can only be exercised at expiry, so the Black-Scholes-Merton formula prices them, with the FTSE 100's dividend yield entered as a continuous yield.
• American calls with no ex-dividend date before expiry also use Black-Scholes-Merton. Exercising such a call early throws away its time value, so its American value equals its European value.
• American puts, and American calls with an ex-dividend date in their life, use a Cox-Ross-Rubinstein binomial tree: the average of a 200-step and a 201-step tree, which cancels most of the odd-even wobble of a single tree. At every node the tree takes the better of holding and exercising, and no American value is shown below intrinsic value.
• Share dividends are discrete, on the model-sheet ex-dates, using the escrowed-dividend approximation: the present value of each dividend due before expiry comes off the share price, and the tree adds back the dividends still to come when it tests early exercise. Where no ex-date falls in an option's life, the dividend input is zero.

The 530 call is the one to notice. Its European value is 27.13p, but the tree adds 2.20p: on some paths BP is far enough above 530p on Wednesday 11 November that exercising to collect the 6.39p dividend beats holding the call through the ex-date. The tree ignores stamp duty, which matches an options intermediary with SDRT relief. A private holder exercising would also pay 0.5% SDRT on the strike, 2.65p a share here, so the writer of the call is exposed to the intermediary's arithmetic rather than a private holder's; the assignment page works through both tests. The put's early-exercise value is small (0.09p) because a put is worth exercising early only when it is deep in the money and the interest on the strike outweighs its remaining time value. The early-exercise column is the difference before rounding, so it can differ by 0.01p from the gap between the two rounded values shown.

What the model leaves out

• Volatility that moves by itself. IV is held fixed unless a row states a shift. Only the FTSE 100 has a shared skew surface; share examples price each leg at the IV stated for it, so a page that models skew on a share gives each strike its own IV.
• Jumps. Prices follow a lognormal path with no gaps. Gaps appear only as stated instant moves in stress rows; the history is on the Level 3 page.
• Real quotes and liquidity. A model value is a mid-market estimate, not a price anyone was offered. It says nothing about whether a strike trades, which matters most for the ICE minis.
• Strike availability. ICE's product pages do not state stock-option strike intervals, and we found no published table (checked 26 September 2026), so the strikes used are plausible ones; a live chain may not list them.
• Other costs. Margin interest, interest earned on collateral, borrowing costs and US exchange and regulatory fees on top of the commission.
• Counterparty behaviour. The tree assumes early exercise happens exactly when it pays. Real assignment can come earlier, later or not at all.
• Tax beyond the stated lines. Only Capital Gains Tax at the rates below; no income-tax or trading treatment.

Costs, fills and rounding

The commissions are IBKR UK's published rates, used because they are published in full and cover ICE and US options. They are a reference point, not a comparison: other brokers charge differently, and the broker page maps them. Rates checked on 26 September 2026.

• Bid-ask. The cost of crossing the spread is half the quoted spread per leg, each way. FTSE 100 (ESX) examples use illustrative quotes, because no free ESX quote history was found: 3.0 points wide on the 45-day chain priced on 1 September (short straddle, short strangle and jade lizard pages) and 2.0 points on the 60-day chain priced on 17 August (ratio, backspread and broken-wing pages, and the spread-bet comparison).
• Stamp duty. 0.5% SDRT on the strike consideration when UK shares are delivered on exercise or assignment (HMRC STSM113030), paid by the buyer of the shares (Finance Act 1986 s91): the call holder who exercises, or the put writer who is assigned. Writing an option attracts none, nor do US shares or cash-settled index options. Who pays in each case is set out on the assignment page.
• Fills on the tick. 0.25p on standard BP, Barclays, HSBC, Rolls-Royce and Tesco options and on every mini; 0.5p on standard AstraZeneca, GSK, Shell and Unilever options; 0.5 of a point on FTSE 100 options.
• Model values are not rounded to the tick: they are shown to two decimal places, so a model value and a fill can differ by up to half a tick.
• Closing trades. Most pages fill a closing trade at the nearest tick, as they do an opening one. The calendar, diagonal and poor man's covered call pages fill a closing trade one tick against the holder where the model value falls between ticks (a buy-back rounded up, a sale down), a slightly more conservative choice.
• Breakevens are quoted before costs, with the figure after opening costs (the half-spread and commission paid to open) in brackets. Closing, exercise and SDRT costs are shown on their own lines.
• Maximum profit and loss include the opening commission, and a closing commission only where a closing trade exists (not on a lapse or a cash settlement).
• The strategy builder deducts the opening commission, and SDRT on shares bought, only. Closing commissions, the bid-ask spread and exercise or assignment costs are not deducted there, so its maximum profit and loss can differ from a strategy page's by those amounts.

One BP call, costed line by line

The BP 16 October 560 call from the table above, bought on the model date:

Dollar figures follow the same lines, then convert. A $2.50 premium on a 100-share US contract is $250, or £184.38 at the model rate of 1.3559. Each leg is converted at the rate on its own date and a net dollar gain is never converted in one step, as HMRC's manual sets out (CG78310); the tax examples show the two-date method.

Probabilities, bands and units

A probability on an options page is always written as a model probability (risk-neutral, lognormal, IV x%). It is what the pricing model implies at the stated volatility, with the share drifting at the interest rate less any dividend yield, not a forecast and not a "chance of profit". The conventions that follow from it:

• The one-standard-deviation band is lognormal, S × e^±σ√T, so it can never go below zero. For BP at 530p, IV 26% and 60 days it runs from 476.97p to 588.92p, and the model probability of finishing inside it is 68.3% (risk-neutral, lognormal, IV 26%). Over 30 days it is 491.93p to 571.02p.
• Delta is not a probability. The 560 call's delta is 0.34 (340 share-equivalents a contract), but the model probability of BP finishing above 560p is 30.3% (risk-neutral, lognormal, IV 26%). The Greeks page explains the gap.
• FTSE 100 probabilities come from the surface (the slope of the call price across strikes), not from a single flat volatility, because the skew changes them.
• The strategy builder opens with zero drift (a switch offers risk-neutral), so its probabilities differ slightly from the risk-neutral figures elsewhere: the same BP one-standard-deviation band reads 68.20% there against 68.27% on the planner and 68.3% here.
• Stress columns in Greeks tables mean an instant move at entry, at the IV named in the column header. Any other date is written in the header.

How each Greek behaves, and how position Greeks add up across legs, is on the Greeks page.

How tax lines are written

Tax figures on the strategy pages share four assumptions, so they can be read side by side: the £3,000 annual exempt amount is taken as used by other gains; both Capital Gains Tax rates are shown, 18% within the basic-rate band and 24% above it; the tax year is 2026/27 unless stated; and an example that crosses 5 April uses March 2027 as its opening month. A cross-year effect is described as timing (relief deferred to 2027/28, lost only if never used) unless relief is genuinely lost. How many computations a structure creates follows one counting rule, and the 5 April cases are worked on the tax examples page. The options CGT calculator applies the same rules to a list of trades a reader types in.

Teaching conventions: where they come from and what the evidence says

When a strategy page shows what happens after entry, it follows a small set of management conventions borrowed from US retail options education. They are published here once, so the strategy pages can say "the worked plan closes here" without restating them, and so their origin and evidence can be judged in one place. None is a rule, and none is advice.

One point frames all of them. Priced on the model and judged on the model's own (risk-neutral) probabilities, every option in the library is a fair bet before costs: the premium equals the discounted value of what the option is expected to pay. A convention can change when money is made or lost and how large the swings are; on its own it cannot change that average. Any edge has to come from the market's implied volatility proving different from the volatility that follows, which is a question about data, not about conventions; the implied-volatility page sets out what is known about implied and realised volatility.

What an IV band changes, in pounds

IV rank places today's implied volatility in its 12-month range: (today's IV − 12-month low) ÷ (12-month high − 12-month low) × 100. BP's model IV of 26% on the assumed 20% to 44% range gives IV rank 25. That range is a teaching assumption, not market data; for scale, BP's 20-day realised volatility ran from 14.9% to 57.5% in the 12 months to 17 August 2026 (price data: Yahoo Finance). The IV page compares IV rank and IV percentile. Here is one BP put priced across that range:

From the bottom of the range to the top, the same put is worth about 4.4 times as much. The movement the model expects grows with it: the one-SD range widens by more than 100p, and the model probability (risk-neutral, lognormal) of the put finishing in the money rises from 22.5% to 39.3%. At its own inputs every row is fairly priced. A band tells a reader where an example sat, which is what makes two pages comparable; it does not say whether a trade was worth doing. Strategy pages therefore give their band and what a lower or higher IV would have paid, never a band as a condition for entry.

Why 21 days, in numbers

Two claims about the last three weeks of an option's life circulate in options education, and only one survives the arithmetic. The chart follows two BP calls from 60 days to the last day, with BP held at 530p and IV at 26%.

The claim that fails is that time decay slows down in the final three weeks. For the at-the-money 530 call it does the opposite and keeps accelerating to the end, from £3.41 a day with 21 days left to £10.44 with two. Only the 560 call, 30p out of the money, peaks (at £2.38 a day, with about 17 days left) and then fades, because it has little value left to lose. Theta is the rate of decay at a given moment (the formula is in the box at the end of the page), so close to expiry it can differ from the loss over a whole day: with two days left the 560 call's theta reads £0.18 a day, but only £0.07 of value is left to lose.

The claim that holds is about proportion. For both calls, the cost of an adverse 10p move measured against the premium still to be earned climbs steeply: for the 560 call from 38% at 45 days to 68% at 21 days and 199% at 7 days. That is the defensible reason for a time stop. Its price is what it leaves behind: at 21 days the 560 call still holds £37.43 (about 59% of its 45-day value has already gone), and a writer who holds on keeps that £37.43 if BP stays below 560p. A 21-day stop swaps that £37.43 for stepping off the steepest part of the risk curve. The numbers do not say which is better, and the balance shifts with the strike, the volatility and the size of the position.

Dated facts and review hooks

The model date stays at 17 August 2026: the worked examples are historical models, not live prices, and they do not change when the market does. What does change are the facts about the future that the examples lean on, such as a dividend not yet declared or a rule not yet in force. Each is re-checked on the date below; when one moves, the fact register is updated, the pages that use it are recomputed and re-checked, and the change is logged under the corrections policy. Every page also lists its own sources with the date each was checked.

Assumptions that are still open

These could not be confirmed from a primary source (checked 26 September 2026). Pages label them as assumptions and they are the first things re-checked:

• BP's third-quarter dividend (6.39p assumed) and its February 2027 ex-date (18 February assumed); 6.39p dividends assumed ex on Thursday 13 May and Thursday 12 August 2027 (long call and income pages).
• Tesco's 2026/27 interim dividend and ex-date, and a 9.70p final assumed ex on Thursday 13 May 2027 (wheel page); Shell's third-quarter ex-date; HSBC's third interim, assumed at US$0.10 ex on Thursday 5 November 2026 (bull put spread page).
• Rolls-Royce dividends of 5.0p (22 April 2027), 6.0p (5 August 2027) and 5.0p (20 April 2028), and a twelve-month IV range of 26% to 46% (LEAPS page); a GSK implied volatility of 22% on every strike (long butterfly page); GBP/USD of 1.3450 on 16 October 2026 (the wheel page's US turn).
• ICE's strike intervals for stock options, and whether BP lists serial expiries out to two years.
• Whether UK brokers offer the ICE minis and the Mini FTSE 100 daily options to private clients, and at what commission.
• The FTSE 100 close on 14 August 2026 from FTSE Russell or the London Stock Exchange (secondary data shows 10,750.1), which is why 10,750 is labelled a model level.
• The in-the-money threshold ICE Clear Europe applies when it exercises equity options automatically at expiry, and whether SDRT is collected in practice on secondary trades in ICE stock options.

How the figures are checked

• Engine tests. More than 340 checks run on every build of the site, and any failure stops the build: textbook values, the test cases in the tools specification, put-call parity, the American rules, edge cases such as expiry day and zero volatility, and date arithmetic.
• Example checks. Each options page lists its headline figures, with their inputs, in a file the build recomputes with the engine. A figure that moves outside its tolerance, or no longer appears on the page, is reported.
• Wording and fact checks. The build also scans options pages for advice-style phrasing and for facts known to be out of date, such as superseded dates and contract sizes.
• A second recompute. Each rewritten page is recomputed by a separate reviewer with its own script, from the inputs printed on the page, before it is published.

The site-wide approach to sources and review is on the methodology page and in the editorial standards.

Rebuilding a figure yourself

1. Read the example's "Model inputs" line: share price, strike, days, IV for each leg, rate, dividends and contract.
2. Turn days into years by dividing by 365: 60 days is 0.16438.
3. For a European option, or an American call with no ex-date before expiry, use the formulas in the box below. A spreadsheet does it in one row.
4. Multiply the per-share or per-point value by the contract: £10 per 1p on a standard ICE contract, £1 per 1p on a mini, £10 a point on the FTSE 100 (£1 on the daily mini), $100 per $1 on US contracts, then convert dollars at the rate the page states.
5. For an American put, or a call across an ex-date, a spreadsheet gives the European value. The tree's early-exercise value sits on top: 0.09p on the 95-day BP put above, 2.20p on the 530 call across the 12 November ex-date.
6. For interactive checks, the Greeks page, the planner and the strategy builder (for positions with several legs) run the same engine in the browser.

If a figure on any options page does not rebuild from the inputs it states, it is an error. Please report it through the contact page; corrections are dated and listed under the corrections policy.