Pricing

European Black–Scholes with a flat risk-free rate and no dividends. Each leg keeps its own days to expiry and IV, and a strategy is stored as its individual legs. A fill price entered for a whole combo is spread across its legs in proportion to their model prices, so the legs add up to your fill. A roll (changing a leg's strike or days) closes the old leg at its model mark, which realizes its P&L on the combo's subtotal and in the roll history, and opens the new leg at its model mark. Model marks stand in for fills; edit Entry to your real one. "Roll combo" on a subtotal row rolls every leg of that combo the same way (strikes shifted onto the strike grid, and/or a new expiry), one history record per leg. Removing a leg deletes it without realizing anything. Delta and gamma are in share-equivalents, vega is dollars per vol point and theta is dollars per calendar day.

Worked example

The textbook example from Hull's Options, Futures, and Other Derivatives: S = 42, K = 40, r = 10%, σ = 20%, T = 0.5 years. The engine's own numbers, which a test checks against this page:

• d1 = (ln(S/K) + (r + σ²/2)·T) / (σ·√T) = 0.7693, and d2 = d1 − σ·√T = 0.6278
• N(d1) = 0.7791, N(d2) = 0.7349
• Call S·N(d1) − K·e−rT·N(d2) = 4.76; put K·e−rT·N(−d2) − S·N(−d1) = 0.81 (Hull: 4.76 and 0.81)
• Call delta 0.7791, put delta −0.2209, gamma 0.0500 per share, vega 0.0881 per vol point, theta −0.0125 per calendar day for the call

Independent checks: a test compares price, delta, gamma, vega and theta with QuantLib on 288 calls and puts (agreement to about 1e-7 of the stock price, the accuracy of the normal CDF approximation used here).

European, not American

Every option is priced as European: exercised only at expiry, with no dividends. That is exact for SPX options, which are European and cash-settled. Options on SPY, QQQ and single stocks are American, and their underlyings pay dividends. We keep the European model on purpose: one closed-form formula gives exact Greeks; it is fast enough to reprice the whole book on every Monte Carlo path and every optimizer candidate in your browser; it is checked against QuantLib above; and an American model would need dividend dates, which illustrative quotes don't have.

What it costs, measured with QuantLib (S = 100, r = 4%, σ = 20%, a 2,000-step binomial tree for the American price): an American put is worth more than the European price shown here, about 1% more at the money with 30 days left, 2% more 10% in the money, and about 8% more 20% in the money with six months left. Without dividends an American call is worth the same as a European one; with dividends, exercising a call just before an ex-dividend date can pay. Leaving out a 1.5% dividend yield shifts a 30-day at-the-money price by about 3% (calls lower, puts higher). So expect short puts deep in the money, and short calls just before a dividend, to be assigned early more often than the Assignment exposure list suggests.

Beta and futures

Beta dollars are Σ Δ·S·β. "Gamma, 1% SPX move" is the quadratic term Σ ½·Γ·(S·β·1%)², an approximation that drifts from full repricing for short-dated or large books. ES (50 × SPX) and MES (5 × SPX) are booked as futures at the SPX level, with no basis, roll, margin or cash flows modelled. A one-point SPX rise pays 50 per ES contract.

Hedge sizing

The delta and gamma hedges can be sized without a Greek target. "% of the holding" hedges that share of the underlying's current delta: 50% of a 1,000-share holding sells 500 shares (SPX in MES futures). "Premium budget" buys as many whole ATM option contracts as the budget pays for at the model price, so the premium shown never exceeds it; a budget below one contract proposes nothing. Either one, when filled in, replaces the target.

Monte Carlo

A one-factor model: the market factor has SPX's IV, and each name's log return is β·market + own noise, with the noise sized so total volatility matches the name's IV. If β × SPX IV is above a name's own IV, its market loading is capped at IV ÷ SPX IV so it keeps its IV. Paths are risk-neutral: every price drifts at the risk-free rate, the same assumption the option prices use, so with IV held fixed a fairly priced option has zero expected P&L. Mean P&L and POP therefore carry no equity risk premium. Options are repriced with the time left at the horizon, and the horizon is capped at the nearest option expiry. IV stays fixed unless you set "IV change per 1% SPX move" in the Market card. That setting shifts every option's IV with each path's market move (−1 means IV rises 1 point per 1% drop). It is a single assumption, not a fitted volatility surface. Entry prices don't include it, so once it is set, mean P&L, POP and CVaR lean toward long-vega books (a long ATM straddle shows about +$170 mean P&L over 5 days at −5). Read them as a stress view, not a fair-value expectation. POP is the share of paths with a gain, CVaR 95 is the average loss over the worst 5% of paths, and Sortino is mean P&L over downside deviation at the horizon (target 0, not annualised).

Optimizer

Coordinate search on whole contracts, starting from no hedge and halving the step size. Weights below zero count as zero. Every candidate is scored on the same simulated paths, so differences between candidates are not sampling noise. The objective terms are relative to the unhedged book, where 1.0 means "unchanged".

Cash account

The Cash account card answers "how much cash does this book need?" for a cash account: no margin loan, no short selling. Each position's cash needed is what was paid, minus premium received, plus collateral held. Buying power left is the account cash you enter minus the total.

• Stock and long options are paid in full at their entry price: shares × price, or contracts × 100 × premium.
• Short put: cash-secured. Collateral is strike × 100 × contracts; the premium received offsets part of it.
• Short call: covered by a long call in the same combo, or by 100 shares of the same underlying per contract (each share covers one call only). Otherwise it is flagged "not allowed" and adds no collateral, because its risk has no limit; its premium does not count as cash either, so it never adds buying power. A spread entered one leg at a time pairs only if the second leg is added to the first one's combo ("Combo" in Add option leg).
• Defined-risk spreads: a short leg covered by a long leg in the same combo (same underlying and type, expiring no earlier) holds the width between the strikes × 100. A combo with one expiry holds at most its maximum loss at expiry, so a vertical credit spread needs width × 100 − net credit, an iron condor only one side's width, and a long butterfly only its debit. This assumes your broker allows spreads in this account. Legs entered one at a time are separate combos, so each is checked alone; legs with different expiries are not netted further.
• Futures (ES/MES): initial margin is your own per-contract figure (no default is assumed, so it counts as $0 until set). Futures are flagged because they need a futures-enabled margin account.
• Short stock: flagged "not allowed" in a cash account, with no cash figure.
• Assignment exposure: short legs in the money, or within 7 days of expiry, are listed with the cash (puts: strike × 100 × contracts) or shares (calls: 100 × contracts) that assignment would need.

Not modelled

• Margin accounts (Reg-T strategy margin or portfolio margin) and the probability of early exercise.
• American exercise and dividends: see "European, not American" above.
• Live market data. The quotes are examples you can edit.
• A volatility skew or smile by strike. The Monte Carlo only has the single spot-vol assumption above.
• Bid–ask spreads, beyond the optimizer's flat cost per contract.

Educational tool. Not investment advice.