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Options Greeks Explorer (Black-Scholes)
Black-Scholes options pricer + live Greeks visualizer. Drag spot, strike, vol, DTE, rate, dividend yield; see delta/gamma/theta/vega/rho. Free to use.
- Inputs
- Paste + configure
- Runtime
- 1–15 s
- Privacy
- Client-side · no upload
- API key
- Not required
- Methodology
- Open →
Inputs
Delta
+0.535
Long-call exposure — a $1 move in spot changes the option value by $0.535. ATM, balanced spot vs. vol risk.
Price $3.04 · Vega $0.114/1%σ · Theta $-0.054/day · CALL K=100 30d
Greeks
Price
$3.0418
CALL Black-Scholes
Delta
0.535
∂Price / ∂Spot
Gamma
0.0554
∂Delta / ∂Spot
Theta (per day)
$-0.0537
∂Price / ∂T
Vega (per 1%)
$0.1139
∂Price / ∂σ
Rho (per 1%)
$0.0415
∂Price / ∂r
Price today (emerald) vs payoff at expiry (dashed)
Spot ±40% · vertical = current spotBreakdown
Intrinsic
$0.0000
Extrinsic (time)
$3.0418
Put price (parity)
$2.6727
same inputs, flipped type
Moneyness
1.000
S / K
Model
Generalized Black-Scholes with continuous dividend yield q:
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)
Reported Greeks are per-day for theta, per 1% vol point for vega, per 1% rate for rho. See methodology for derivation + limitations.
How to use
Step-by-step
- 1
Enter underlying price, strike, time to expiration (days), implied volatility, risk-free rate, and dividend yield (or carry).
- 2
Pick option type (call/put) and exercise style (European/American).
- 3
Read all five Greeks at once: Delta, Gamma, Theta, Vega, Rho. Each scales differently with the inputs.
- 4
Use the strike-sweep view to see Greeks across the strike grid — Delta moves smoothly, Gamma peaks ATM, Vega peaks ATM.
- 5
Sweep one input (e.g., IV) to see how Vega exposure changes. This is essential for sizing volatility positions.
For agents
Use in an agent
Same math, same result shape as the UI above — as a static ES module. No HTTP request, no auth, no rate limit.
import { compute } from "https://aifinhub.io/engines/options-greeks-explorer.js";Contract: /contracts/options-greeks-explorer.json Full agent guide →
Glossary references
Terms used by this tool
Questions people ask next
FAQ
Which pricing model does the tool use?
Black-Scholes for European options, with cost-of-carry adjustment for index/currency/dividend-paying assets. American options use the Bjerksund-Stensland 2002 closed-form approximation, which is within 0.1% of binomial-tree pricing for most strikes. The methodology page documents both.
What are the Greeks measuring?
Delta is sensitivity of option price to a $1 underlying move. Gamma is the rate of change of Delta. Theta is daily time-decay (premium lost per day, holding all else constant). Vega is sensitivity to a 1-percentage-point implied volatility change. Rho is sensitivity to a 1-percentage-point interest-rate change.
Why does Vega look high near at-the-money?
Vega is maximized at-the-money and falls toward zero as the option moves deep in-the-money or out-of-the-money. This is mechanical — at-the-money options have the most convexity to vol changes. The tool plots Vega across strikes so you can see the shape directly.
Does the tool include American early-exercise premium?
Yes for puts on dividend-paying assets, where early exercise can be optimal. The Bjerksund-Stensland model handles this. For non-dividend calls, early exercise is never optimal in Black-Scholes, so American = European pricing — the tool indicates this with a flag.
What if I want to price exotics?
The tool only handles vanilla puts and calls. Barriers, Asians, lookbacks, and digitals require Monte Carlo or specialized closed-form models the tool doesn't ship. For exotics, the methodology page links to QuantLib and to Wystup's textbook treatments.
Related deep dive
All articles →Read further
Long-form context behind the tool output.
- Tutorial · Runnable·10 min
Options Greeks for LLM-Driven Trading
Options Greeks for LLM-driven trading: delta, gamma, theta, vega, rho — what each costs, three rules, plus a prompt template for multi-leg positions.
Read - Methodology · Opinion·10 min
Why LLMs Fail Options Greeks
LLMs misfire on theta sign, vega-vs-gamma conflation, and ITM-vs-ATM gamma ranking. The three reproducible error categories, plus a verifier fix.
Read - Pillar · Guide·11 min
Options Greeks: 30-DTE OTM Call, Worked End to End
Engine returns delta 0.301, gamma 0.0217, theta −$0.10/day, vega $0.20/IV-point for a 30-DTE 5% OTM call on $200 spot at 28% IV — the LLM-confounder case.
Read
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