How Options Payoff Builder works

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

1. 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.
2. Continuous dividend yield. Discrete dividends are not modelled; for single names with large cash dividends, use a dividend-adjusted model.
3. 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.
4. 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.
5. 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.
6. Greeks at current spot only. The aggregated Greeks are instantaneous sensitivities at S = spot now. As spot moves, all Greeks except the second-order approximation change.
7. 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.

Changelog

• 2026-04-20 — Initial release: 1–4 legs, 7 strategy presets, payoff + break-evens + aggregated Greeks.