How to read
the options
pricing tool.
A working guide. What every input means, how the price and Greeks are computed, and what the spot/vol sensitivity grid is telling you. Read it once, then keep it open in a tab while you price.
Open the tool →The thirty-second workflow
Three steps. Pick a model from the index on the left, or open the palette with Cmd K (Ctrl K on Windows). Set your inputs in the ledger: spot, strike, time to expiry, rate, dividend yield, and volatility. Read the price, the Greeks, and the sensitivity grid on the right.
The ledger only surfaces the inputs the active model uses. Switching from Black–Scholes to Heston adds the variance, mean reversion, and correlation fields without losing the values you already typed.
Every input mutates the URL. Copy the address bar to share a setup, or use the permalink button under Model notes.
Inputs, by group
Inputs are grouped by purpose. Core inputs apply to almost every model. The other groups appear only when a model needs them.
Core
- Option type
- Call or put. Call pays max(S − K, 0) at expiry; put pays max(K − S, 0).
- Spot (S)
- Current price of the underlying. Anything positive.
- Strike (K)
- Contract strike. Same units as spot.
- Time to expiry (T)
- In years. 30 days is about 0.082; 90 days is about 0.247.
- Risk-free rate (r)
- Continuously compounded, per year, as a decimal. 4.5 percent is 0.045.
- Dividend yield (q)
- Continuous dividend or carry yield as a decimal. Used wherever the model accepts a carry adjustment.
- Volatility (σ)
- Annualized lognormal vol as a decimal. 25 percent is 0.25. This is the single input options pricing is most sensitive to.
Method (trees, FDM, Monte Carlo)
- Time steps
- Number of time slices in a tree or finite-difference grid. More steps means a finer lattice, slower compute, smoother convergence. 100 is a fine default; Leisen–Reimer wants an odd number.
- MC paths
- Sample size for Monte Carlo. Standard error falls as one over the square root of paths, so quadrupling paths halves the error.
- Seed
- Deterministic seed for the Monte Carlo generator. Same seed plus same inputs reproduces the price exactly.
Barrier
- Barrier (B)
- Price level at which the option knocks in or knocks out. Continuously monitored under the Reiner–Rubinstein closed forms.
- Rebate
- Cash paid out if the barrier kills the option (knock-out) or never triggers (knock-in). Zero by default.
Exotic payoff parameters
- Strike 2 / cash (K2)
- Second strike for gap and supershare payoffs, or the cash amount for cash-or-nothing digitals.
- Choice time (Tc)
- For a simple chooser, the time at which the holder commits to call or put. Must be less than T.
- Min observed (Mmin), Max observed (Mmax)
- Running min and max for lookback options. Seed with the realized extreme so far, or with spot if pricing at issue.
Heston (stochastic vol)
- Initial variance (v₀)
- Variance now, in decimal squared. v₀ = 0.04 corresponds to 20 percent vol.
- Mean reversion (κ)
- Speed at which variance pulls toward its long-run level. Larger κ means faster decay of vol shocks.
- Long-run variance (θ)
- The level variance reverts to. Often calibrated near v₀.
- Vol of vol (ξ)
- Volatility of the variance process. Drives the wings of the implied vol smile.
- Spot/vol correlation (ρ)
- Correlation between spot returns and variance changes. Equity markets typically run ρ around minus 0.5 to minus 0.7, which creates a left skew.
Jumps (Merton, Bates)
- Jump intensity (λ)
- Expected number of jumps per year. λ = 0.5 means roughly one jump every two years.
- Jump mean (μ_J)
- Mean log-jump size. Negative for crash-style jumps.
- Jump σ (σ_J)
- Standard deviation of the log-jump size.
Multi-asset and FX
- Foreign rate (r_f)
- Risk-free rate in the foreign currency for Garman–Kohlhagen FX pricing.
- Spot 2 (S₂), Vol 2 (σ₂), Yield 2 (q₂), Strike 2 (K₂)
- Second underlying for two-asset payoffs.
- Correlation (ρ₁₂)
- Correlation between the two underlyings.
Reading the price
The large number in the Output panel is the model price of the contract, in the same currency units as spot and strike. It is the present value of the expected payoff under the model’s risk-neutral dynamics.
For Monte Carlo models, the header shows SE ±x, the standard error of the price estimator. Roughly two thirds of the time the true model price sits within one SE of the printed value. To halve the SE, run four times the paths.
The Note line carries model-specific context (for example, the perpetual American option reports its optimal exercise boundary). When a Note is present, read it first.
Reading the Greeks
The Greeks are first-order sensitivities of the price to the model inputs. Analytic Greeks are emitted where the kernel can derive them in closed form; the rest are computed by bumping an input and re-pricing. Bumped Greeks inherit any noise in the underlying pricer, so Monte Carlo Greeks can be choppy at low path counts.
A dash in the value column means the active model does not produce that Greek (for instance, perpetual American has no Theta because there is no time to expiry).
Reading the sensitivity grid
The Sensitivity grid is a 7 by 5 panel of prices. The default layout has Spot on the X-axis (seven columns) and Vol on the Y-axis (five rows). Each cell re-runs the active model with the row and column values substituted into the inputs, holding everything else fixed.
The two toggles in the header swap the axes between Spot, Vol, and Time. Range is the input value plus or minus 30 percent for Spot and Vol, plus or minus 50 percent for Time, evenly spaced.
Shading is tonal (ink only, no color). Darker cells are higher prices. The footer prints the grid min and max so you can read the dynamic range.
What to look for
- Spot row, call. Prices should rise left to right. The rate of rise is delta; the change in that rate is gamma.
- Spot row, put. Prices should fall left to right.
- Vol column. Both calls and puts should rise with σ for vanilla payoffs. The slope is vega. If a row is flat, vega is near zero (very deep ITM or OTM, or very short dated).
- Time column. With Spot on X and Time on Y, the column shows the term structure of the option price at a fixed spot. Long-dated options are worth more for vanillas; for some exotics the relationship reverses.
- Discontinuities. For barriers and digitals you may see a hard step where the spot crosses the barrier or the strike. That is the payoff structure, not a bug.
- Dashes in a cell. Means the model failed to converge under those shocked inputs. Try fewer shocks (move the center toward the failing cell) or switch models.
The grid is meant for one quick check: does the surface look the way the payoff says it should? If it does, your inputs are sane and the model is telling a consistent story. If a slice runs the wrong way, that is a signal to revisit the inputs before quoting from the number.
Model families at a glance
The full index sits behind the Search models button (Cmd K). The eleven families, in one line each:
- Closed-form European
- Analytic prices and Greeks. Fastest. Use these whenever the payoff is European and the dynamics are vanilla Black–Scholes.
- American / Early exercise
- Closed-form approximations (Barone-Adesi–Whaley, Bjerksund–Stensland) for speed. Pair with FDM or Longstaff–Schwartz for cross-checks.
- Trees and lattices
- CRR, Jarrow–Rudd, Tian, Leisen–Reimer, Trigeorgis, Boyle trinomial. Convergent in step count. Leisen–Reimer with odd steps has the smoothest convergence.
- Finite difference
- Crank–Nicolson on a log-spaced grid. Good for American options when smooth Greeks matter.
- Monte Carlo
- GBM Europeans, arithmetic Asians, Longstaff–Schwartz Americans, and Heston Europeans. Standard error reported in the output header.
- Barrier
- Eight Reiner–Rubinstein closed forms covering up/down by in/out by call/put. Continuous monitoring.
- Asian and Lookback
- Geometric Asian (closed form), Turnbull–Wakeman (arithmetic Asian), floating- and fixed-strike lookbacks.
- Digital and Exotic payoffs
- Cash-or-nothing, asset-or-nothing, gap, supershare, simple chooser.
- Multi-asset
- Two-asset cash-or-nothing with correlated underlyings.
- Stochastic vol and Jumps
- Heston (semi-analytic), Bates (Heston plus Merton jumps), Merton jump-diffusion.
- Currency, Commodity, Bond
- Garman–Kohlhagen FX, American FX binomial, American futures option.
Permalinks and reproducibility
The URL is the single source of truth for a pricing setup. The model id lives in m, and every input the active model surfaces is serialized as a query parameter. Open the same URL again, anywhere, and you get the same ledger and the same price.
For Monte Carlo models, reproducibility also depends on the Seed input. Same model, same inputs, same seed produces an identical price down to the last decimal.
The Copy permalink button under Model notes copies the URL to the clipboard. Paste it into a note, a chat, or a code review.
You have the whole interface now. The fastest way to make the rest of this stick is to open the tool, switch through three or four models on the same inputs, and watch how the price and the grid respond.
Open the tool →