Methodology · Playground · Last updated 2026-04-20
How Options Payoff Builder works
How the Options Payoff Builder tool actually works — pricing, aggregation, payoff computation, limitations.
Scope
Computes the at-expiry payoff diagram and current-spot Greeks for a user-constructed 1–4 leg option strategy. Each leg is a long or short call or put at a specified strike, sized by integer contract count. Pricing uses generalised Black-Scholes with continuous dividend yield.
Per-leg pricing
d1 = [ln(S/K) + (r − q + σ²/2)·T] / (σ·√T)
d2 = d1 − σ·√T
call = S·e^(−qT)·Φ(d1) − K·e^(−rT)·Φ(d2)
put = K·e^(−rT)·Φ(−d2) − S·e^(−qT)·Φ(−d1) S is spot, K strike, σ annualised
volatility, T years to expiry (minimum 0.5/365 to avoid
singularities), r risk-free rate, q continuous
dividend yield. Φ uses the Abramowitz & Stegun 26.2.17
rational approximation.
Per-leg Greeks
- Delta: call
e^(−qT)·Φ(d1), put−e^(−qT)·Φ(−d1). - Gamma:
e^(−qT)·φ(d1) / (S·σ·√T), same for call + put. - Vega (per 1% vol):
S·e^(−qT)·φ(d1)·√T / 100. - Theta (per day): annualised BS theta divided by 365.
- Rho (per 1% rate): call
K·T·e^(−rT)·Φ(d2) / 100, put negative.
Multi-leg aggregation
For each leg i with side s_i ∈ {+1, −1} (long / short)
and contract count n_i:
Greek_total = Σ_i s_i · n_i · Greek_i
net_premium = Σ_i s_i · n_i · price_i
payoff_at_expiry(S) =
Σ_i s_i · n_i · max(0, (type_i == call ? S − K_i : K_i − S))
net_profit_at_expiry(S) = payoff_at_expiry(S) − net_premium
Break-even points are zero crossings of net_profit_at_expiry(S),
found by linear interpolation across a 200-point grid centred on
spot (±50% by default).
Presets
The strategy presets seed a starting point; you can edit any leg afterwards. Strikes scale from spot:
- Long call: 1 long call @ spot.
- Long straddle: long call + long put @ spot.
- Long strangle: long 1.05× OTM call + long 0.95× OTM put.
- Bull call spread: long spot call + short 1.05× call.
- Bear put spread: long spot put + short 0.95× put.
- Iron condor: short 1.05× call + long 1.10× call + short 0.95× put + long 0.90× put.
- Long call butterfly: long 0.95× call + 2 short spot calls + long 1.05× call.
Assumptions + limitations
- European exercise. Black-Scholes assumes no early exercise. For American equity options this is an approximation; for index options (SPX, RUT) it is exact in structure.
- Continuous dividend yield. Discrete dividends are not modelled; for single names with large cash dividends, use a dividend-adjusted model.
- Constant volatility + rate. No term structure, no skew/smile. For multi-leg strategies that depend on skew (risk reversals), this can materially under-price the structure. Use a local-vol or SABR model for production.
- Log-normal underlying. Fat-tailed equity returns cause Black-Scholes to under-price OTM options. For deep OTM legs in your structure, treat the price as an optimistic lower bound.
- No transaction costs, no bid-ask. Net premium is theoretical. Live bid-ask can cost 2–20% of theoretical on low-liquidity strikes, especially for multi-leg combos.
- Greeks at current spot only. The aggregated Greeks are instantaneous sensitivities at
S = spotnow. As spot moves, all Greeks except the second-order approximation change. - Integer contracts. Fractional-contract strategies are not supported.
Privacy
All pricing, payoff computation, and aggregation run in the browser. No inputs leave the page.
References
- Black, F., & Scholes, M. (1973). "The Pricing of Options and Corporate Liabilities." Journal of Political Economy 81(3).
- Merton, R. C. (1973). "Theory of Rational Option Pricing." Bell Journal of Economics 4(1).
- Hull, J. C. (2017). Options, Futures, and Other Derivatives, 10th ed., §15.
- McMillan, L. G. (2012). Options as a Strategic Investment, 5th ed. — classic reference for multi-leg strategies.
Connects to
- Options Greeks Explorer — single-leg Greeks with more granular visualisation.
- Returns Distribution Analyzer — test whether the underlying's distribution is anywhere near log-normal before trusting model prices.
External resources
Changelog
- 2026-04-20 — Initial release: 1–4 legs, 7 strategy presets, payoff + break-evens + aggregated Greeks.