Methodology and limitations

Units and conventions

• Strikes, premiums and entry prices are per share. A position's amount is per-share value × contracts × contract multiplier (default 100). Stock legs are counted in shares.
• Time to expiration is calendar days ÷ 365 between the valuation date and the common expiration date. Intraday time is not modeled; on the expiration date options are worth their intrinsic value.
• Volatility, the risk-free rate and the dividend yield are annual percentages entered as e.g. 25 for 25%. Rates and yields are continuously compounded.
• Delta is in share-equivalents (P/L per +1 move in the underlying). Theta is per calendar day (annual rate ÷ 365). Vega is per 1 volatility point (25% → 26%). Rho is per 1 percentage point of the rate. All are instantaneous sensitivities.
• A net debit is cash paid to open the position; a net credit is cash received.
• Opening costs are two non-negative amounts in the workspace currency: a cost per opened option contract, charged for long and short contracts alike and counted over every leg (opposing legs do not cancel it; it is not multiplied by the contract multiplier), and additional opening costs charged once for the whole strategy, for example stock transaction costs or a fixed fee. Stock legs have no automatic cost. The net entry amount is the trade amount (premiums and stock prices, positive = debit) plus the opening costs; the cash flow at opening is its negative, so costs can turn a small credit into a net debit. Changing the currency converts nothing.

Pricing model

Options are valued with the Black-Scholes-Merton model for European exercise with a continuous dividend yield q:

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

Each leg uses its own implied volatility (or the base IV). Strategy values and Greeks are sums over legs, with short legs entering negatively. A scenario IV shift moves every leg's volatility by the same number of points.

With American exercise (selected under the workbench's market assumptions), options are valued numerically: the same Black-Scholes equation with the same continuous dividend yield is solved by finite differences (Crank-Nicolson, 400 price × 200 time steps), and at every step an option is worth at least what exercising it immediately would pay. Calculations are shared: each one values every underlying price at once, and nearby dates are grouped into the same calculation. Delta, gamma and theta come from these calculations; vega and rho from small volatility and rate changes on the same grids.

With discrete cash dividends (selected under the market assumptions instead of the continuous yield, which is then kept but not used) both exercise styles are valued with the same finite-difference method and q = 0: at each modeled dividend the price drops by the dividend, an American option may be exercised just before the drop, and the calculation continues from the lower price. The model label above the workbench's results names the exercise style and the dividend assumption.

Exercise diagnostics (American exercise)

The Exercise & assignment card describes, for each American option leg, the model economics of exercising now: at the valuation date with the entered IVs, or at the scenario date with its IV shift, both at the entered price. Every amount is per share and from the holder's side, also for legs the strategy is short.

• Exercise value E: what exercising immediately pays, max(S − K, 0) for a call and max(K − S, 0) for a put. It is exact.
• Model time value V − E, with V the American model value: what keeping the right to choose later is worth over exercising now (not the entered premium minus intrinsic value). Above its numerical uncertainty, holding is worth more under the model. Within it, holding and exercising now differ by no more than the model's accuracy; that is neither an exercise boundary nor proof that exercising now is optimal. Where E itself is within the uncertainty there is no material intrinsic value, and no exercise incentive.
• Early-exercise premium V_am − V_eu: the American value minus the value of the same option without the early-exercise right, on the identical numerical grid. It values the right over the remaining life; a positive premium does not mean that exercising now is better than holding.

These are model economics, not predictions of what the holder of an option you sold will do, and the workbench does not estimate assignment probabilities. A short American option can be assigned at any time before expiration, also when holding is worth more under the model, and each short leg can be assigned separately from the strategy's other legs: a long leg is not exercised with it. A long leg is your own right, not an assignment exposure. The card simulates nothing: every view, the expiration analysis included, keeps the legs as entered. With European exercise there is no early-exercise right, and the card says so.

Discrete cash dividends: timing and cash

• Each schedule row is an ex-dividend date E and the announced cash amount per share, entered manually. Ex-date E is modeled as a jump at the end of calendar day E − 1, so a valuation on E is after it and the entered price on E is already ex-dividend. An ex-date tomorrow puts the jump at the valuation instant: the entered price is its known pre-dividend price. An ex-date on the expiration date is included; one after expiration, or on or before the valuation date, is kept and marked “not used”.
• This is an idealized convention, not a trading deadline: weekends, exchange holidays, broker cut-offs and the payment date are not modeled. Special dividends that adjust option contracts and other corporate actions are not supported.
• The modeled payment is the price drop: the announced amount, or the whole price if that is lower, in which case the stock is worthless afterwards (a price of 0 is final and nothing more is paid). If the price stays above 0, every dividend before it was paid in full.
• Stock legs receive (long) or pay (short) the modeled payment as cash on the ex-date; the interest between the ex-date and the payment date is ignored. Options receive no dividend; their values reflect the drop. Total P/L = price P/L + this cash, counted once.
• Scenario, matrix, time-decay and chart prices are the actual quoted prices on their dates; nothing is subtracted automatically. “Price unchanged” across an ex-date therefore assumes the stock recovered the dividend drop.
• At a price of exactly 0 after a later dividend, the stock's dividend income depends on when it became worthless, so the P/L there is a range (an end that is only approached is marked “not reached”). Where 0 cannot be reached, the value at 0 is the limit as the price falls. A dividend tomorrow at or above the entered price makes 0 the only later price: positive prices are then not states of the model.
• Dividend comparison (American exercise): for the next modeled dividend at or after the selected date and before expiration, the Exercise & assignment card compares exercising immediately before the drop (E_D, at the price just before it) with holding through it (C_D, the model value just after the drop, which keeps every later exercise right and dividend). For an ex-date tomorrow the dividend is at the valuation instant and the entered price is its known pre-dividend price; for a later dividend the entered price is an assumed price just before it, not a forecast. On the ex-date itself the dividend is already past. The modeled day follows the convention above and is not your last opportunity to exercise: actual exercise deadlines depend on business days and broker cut-offs. A continuous yield has no dated payment or ex-date, so it gives no dividend comparison; early exercise can still have model value with it, for example for puts when r > 0.
• Expiration figures are for the unchanged legs held to expiration. A maximum loss at a price of 0 can be a bound that is approached but not reached. Actual early exercise and assignment, which can change the dividends a position pays or receives, are not simulated.

Numerical model: accuracy and limits

Numerical values are approximations with a validated uncertainty of ε = 4·10⁻⁴·max(S, K)·σ√T + P·(1.5·10⁻⁶ + 1.5·10⁻⁵·((|r| + q)T)²), P = max(S, K, K·e^−rT): about $0.01 per share for a one-year at-the-money option at 25% IV. In thousands of comparisons with independent references no error exceeded ε.

The model is used only where that was validated: IV of at least 0.01%, σ√T ≤ 2 and |r − q|·T ≤ 2. Where early exercise can pay (puts with r > 0, calls with q > 0 or r < 0), the IV must also be at least |r − q|·√T / 1.5, e.g. 3.34% for r − q = 5% over one year. Below it the exercise boundary moves away from the strike faster than the grid can follow, and prices were outside the accuracy bound. Such inputs are rejected with a message, the scenario IV shift is limited the same way, and values the model cannot compute are shown as “—”: nothing falls back to Black-Scholes-Merton.

With cash dividends after the valuation date, each option also needs dividend ≤ 0.1 × IV × strike for every dividend and a total ≤ 0.5 × IV × strike, so larger dividends need a higher IV (shown on the leg's IV field). A put's dividends from the valuation date to expiration must total at most 0.15 × strike; no IV lifts that limit, so it is shown on the leg. A dividend tomorrow is exempt: it is settled from the entered price. While a modeled dividend lies after the date being valued (an ex-date more than one day later), a put is not valued at prices below 0.2 × strike (except exactly at 0), because the model's accuracy is not validated there; those prices are shown as “not computed” and charts leave a gap. Where an option's value has a kink caused by a dividend, a Greek at that price can be “not defined” or “not resolved”; it is shown as “—” with the reason, and the price and the other Greeks remain.

The exercise diagnostics classify a difference only beyond its uncertainty: ε of V at the selected date for V − E, ε at the dividend for E_D − C_D (none where the drop takes the whole price, which is exact), and the sum of both values' ε for the early-exercise premium. These are numerical error allowances, not probabilities or confidence intervals, and no difference is clipped. Where a value is outside the validated domain it is “not computed”: the time value, premium or comparison is then unknown, not zero, and the exact exercise value is still shown. After a dividend tomorrow at or above the entered price only a price of 0 can follow, so a positive price at a later date, the expiration date included, is “not a model state” and no value is shown for it.

The analysis values the right to exercise early but does not simulate exercise or assignment: legs stay as entered until expiration.

Expiration analysis

At expiration the P/L is a piecewise-linear function of the underlying price with kinks only at strikes. The engine evaluates it exactly at every strike and at a price of zero, and uses the slope above the highest strike to detect unlimited profit or loss. Maximum profit, maximum loss, break-evens and profitable price ranges are derived from those values for any combination of legs — nothing is hard-coded per strategy. “Unlimited” means the P/L keeps growing (or falling) as the underlying rises; the underlying cannot fall below zero, so downside amounts are always finite.

Opening costs are charged once, at entry: every P/L — today, at the scenario date, in the matrix, the time decay and at expiration — is after them, and a later date neither charges them again nor removes them; they are not compounded or discounted. In the leg analytics each option leg's P/L is after its own per-contract cost and the additional costs are a separate strategy-level row. The expiration analysis runs on the P/L function after the costs, so they can move break-evens, merge them or remove them and the profitable ranges with them; finite maximum profit and loss move by the same amount, and unlimited stays unlimited.

Each break-even is also shown as its distance from the entered underlying price: (break-even ÷ entered price − 1) × 100%, negative below the entered price. A flat zero-P/L range gets the distance of each finite end. It only locates the break-even relative to the entered price: it is not a safety margin, an assignment threshold, a probability or a forecast.

The expiration figures keep every leg as entered until expiration. Early exercise and assignment are not simulated, also where the Exercise & assignment card shows a model incentive to exercise; an actual assignment would change the position and its P/L.

Implied volatility

“IV from premium” solves for the volatility at which the model reproduces the entered premium, using a bracketed Newton method on the out-of-the-money side of put-call parity. A result is reported only if it reprices the premium accurately. Premiums below the discounted intrinsic value, at or above the theoretical maximum, with no time value, or requiring more than 2,000% volatility are reported as having no valid implied volatility instead of returning a misleading number. Real quotes for American-style options, especially deep in-the-money puts, can sit above the European maximum because of early-exercise value; they have no European IV.

With American exercise the IV is the volatility at which today's model value, computed exactly as the report computes it, equals the premium, searched within the supported range. Because model values carry the uncertainty ε, each IV has an interval: it is shown to 0.1 point when the interval is narrower than that, as “≈” when it is up to 1 point wide, and otherwise as the interval itself, because the premium does not determine one volatility. Other results say why there is no number: the premium equals the exercise value (the model would exercise now), lies at or beyond the lowest or highest supported volatility, or outside the no-arbitrage bounds. “IVs from premiums” adopts only the first two kinds of result and keeps every other leg's IV.

With cash dividends the IV is “not computed” for puts with an ex-date after tomorrow, for American puts with r > 0, and after a dividend tomorrow at or above the price: there the model's price does not determine one volatility reliably, or does not depend on it.

Entered premiums are per share and exclude opening costs: the IV is implied by the premium alone, and “Prices from model” sets premiums to model values without any cost. With all entry prices at today's model values, today's P/L is therefore about minus the opening costs.

The IV is implied by the entered premium at today's inputs. It is the option's current market IV only if the premium is a current quote; a premium paid earlier, at another underlying price, implies a different number.

Probabilities

The probability of profit and the 68% range assume the underlying follows geometric Brownian motion (lognormal prices) with the base IV and the risk-neutral drift r − q. They describe the pricing model, not the real world: actual returns show fat tails, skew and an unknown expected return. They are rounded to whole percent and should be read as rough, model-dependent reference values, never as the chance that a trade will succeed. With American exercise they are unchanged and assume every leg is held to expiration; early exercise or assignment is not included. With discrete cash dividends the price at expiration is no longer lognormal under the model, so the probability of profit and the 68% range are not computed; the chart's automatic price range is a display choice, not a probability interval. Profit means a P/L above zero after the opening costs: the probability of profit is that of the profitable ranges after costs, and costs never change the price distribution or the 68% range.

What the model does not capture

• Exercise and assignment. Most listed US equity options are American-style. European exercise ignores the early-exercise right, which can make real prices higher (e.g. deep in-the-money puts). American exercise values that right, and the Exercise & assignment card shows its model economics, but neither model simulates an actual early exercise or assignment, which can close short legs unexpectedly. Whether and when holders exercise, and which short positions are assigned, depend on other market participants and the allocation process; they are not modeled. The workbench does not estimate assignment probabilities.
• Dividends are either a continuous yield or a manually entered schedule of cash dividends. With the yield, early exercise of calls just before an ex-dividend date is not captured and stock-leg P/L excludes dividends. With the schedule, stock legs receive or pay the modeled dividends; dividends are never estimated or fetched, and borrow costs are not included.
• Volatility smile, skew and dynamics. IV is constant over the option's life and, unless overridden per leg, equal across strikes.
• Trading frictions: only the opening costs you enter are included. Closing costs, bid-ask spreads, slippage, broker fee schedules, exercise and assignment fees, liquidity, margin requirements, financing of premiums and taxes are not.
• Settlement details such as pin risk, exercise cut-offs and corporate actions (adjusted deliverables) are not modeled.
• Scenario results change one assumption at a time and hold everything else constant.