Option Greeks Calculator (Δ Γ Θ ν ρ)
Option Greeks calculator. Returns delta, gamma, theta, vega, rho for European call and put options from the standard six Black-Scholes inputs. Educational only.
Option Greeks Calculator
| Greek | Call | Put | Interpretation |
|---|---|---|---|
| Price | — | — | Black–Scholes–Merton value |
| ΔDelta | — | — | $ change per $1 spot move |
| ΓGamma | — | Δ change per $1 spot move (same C&P) | |
| ΘTheta | — | — | $ change per calendar day |
| νVega | — | $ change per 1% vol move (same C&P) | |
| ρRho | — | — | $ change per 1% rate move |
How to use the Greeks calculator
Plug in the same six inputs as the pricer
Spot, strike, time-to-expiry, risk-free rate, volatility, dividend yield. Greeks are partial derivatives of the Black-Scholes price function with respect to each input — they describe HOW the option price moves when each input changes.
Read delta (Δ) as the directional exposure
Delta is the dollar change in option price per $1 change in spot. ATM calls have Δ ≈ 0.50; deep-ITM calls approach 1.0; deep-OTM approach 0. Puts are negative. Delta also tells you the hedge: a position of 100 deltas moves like 100 shares, so selling 100 shares offsets it for small moves.
Read gamma (Γ) as the second-order curvature
Gamma is how fast delta moves. Highest at ATM, near zero deep ITM/OTM. High-gamma positions need frequent re-hedging; that's why gamma is the "convexity tax" market makers charge for short-dated ATM options.
Read theta (Θ) as the daily time-decay drag
Theta is negative for long options — you LOSE money holding the position even if spot doesn't move. ATM options decay fastest near expiry. On a USD 100 stock at 20% volatility and a 5% rate, the at-the-money call loses about 4.5 cents a day with 30 days left, 8.6 cents with 7, and 22 cents on the last day.
Read vega (ν) as the volatility exposure
Vega measures dollar change per 1% volatility change. Long options are LONG vega — they appreciate if implied vol rises. Long-dated options have higher vega than short-dated; vega peaks roughly ATM. A position with high vega is essentially a bet on the volatility environment.
The Greeks — how options actually behave
The Greeks are the partial derivatives of an option's price with respect to its inputs. They tell traders the local sensitivity of the position — how much they make or lose for a small move in spot, time, vol, or rates. Where Black-Scholes gives the headline price, the Greeks give the risk profile. Every professional options desk reports a portfolio's Greeks in real time; they're the basic vocabulary of derivatives risk management. The five canonical Greeks are delta (Δ), gamma (Γ), theta (Θ), vega (ν), and rho (ρ) — but extended Greeks like vanna, volga, and charm matter for some strategies. This calculator surfaces the five canonicals from the same Black-Scholes-Merton inputs as the option pricer.
How traders use each Greek
Delta (Δ) is the directional exposure: the dollar change per $1 spot move. A call with Δ = 0.60 makes $0.60 if the underlying rises $1, loses $0.60 if it falls. The Options Industry Council notes that some traders read delta as the probability of finishing in the money. Under the model that probability is actually N(d₂), using the d₂ printed below the table, and it is lower than a call's delta: for a one-year at-the-money call on a USD 100 stock at 20% volatility and a 5% rate, delta is 0.64 and N(d₂) is 0.56. Traders run "delta-neutral" portfolios by hedging long options with short shares, neutralising the directional bet to isolate volatility exposure. Gamma (Γ) is the convexity: how fast delta moves. Long gamma positions benefit from large moves in either direction; short gamma positions get hurt. Whoever is short options is short gamma, and has to re-hedge as the price moves, which is a cost they build into the price they quote.
Theta is the rent on a long option. It is charged every day, whether or not the underlying moves, and the rent rises as expiry approaches.
Theta + vega — the time and volatility taxes
Theta (Θ) is time decay — option value decreases as expiry approaches, faster for ATM and short-dated options. Theta is negative for long positions, positive for short. The OIC calls the decay "not linear": on a USD 100 stock at 20% volatility, the at-the-money call loses about 4.5 cents a day with 30 days left, 8.6 cents with a week left and 22 cents on the last day, and it gives up about two-thirds of its value over the final 14 days. Theta works for the seller, which is why covered calls and credit spreads are built on it; the seller is also the one exposed to a large move. Vega (ν) is the implied volatility exposure. Long options are long vega — they gain when IV rises. Vega is largest for at-the-money and long-dated options: the one-year USD 100 call above gains about 37.5 cents per point of volatility, the 30-day call about 11 cents. Short straddles and iron condors are short vega, so they gain when volatility falls and lose when it jumps.
ASEAN-specific Greeks application
The formulas are the same whatever the market: a trader in Singapore pricing US-listed options, or one in Hong Kong pricing Hang Seng index options, changes only the inputs. Use the interest rate of the option's currency for r, and the underlying's dividend yield for q. Exchange-traded stock options in the US are American-style, so they can be exercised early, and this European-style model does not price that right; for index options that settle European-style it fits more closely. Multiply each Greek by the contract multiplier, often 100 shares, to get the position's figure: a theta of −0.045 is about −USD 4.50 a day per contract.
Delta as rough probability, where gamma peaks, and why theta accelerates near expiry
Delta is not quite the odds. The model's probability of finishing in the money is N(d₂): 0.56 for a one-year at-the-money call whose delta is 0.64.
Call delta ranges 0 to 1; put delta −1 to 0. ATM options have |Δ| ≈ 0.50.
Gamma is identical for call and put with the same strike/expiry. Peaks at ATM, decays toward OTM and ITM.
Theta accelerates near expiry. A 30-day at-the-money call on a USD 100 stock (20% vol, 5% rate) loses about two-thirds of its value in its last 14 days.
Vega is symmetric for call and put. Long options = long vega. Long-dated options have higher vega than short-dated.
Rho grows with time. On a USD 100 stock, an at-the-money call's rho is about 0.04 at 30 days and 0.53 at one year, per point of rates.
"Delta-neutral" portfolio = Σ deltas = 0. Re-hedged constantly by market makers; the basis of all volatility arbitrage strategies.
Charm = ∂Δ/∂t, vanna = ∂Δ/∂σ, volga = ∂ν/∂σ. Second-order Greeks matter for exotic options and risk reports.
Gamma is the "convexity tax" — short-gamma positions need rebalancing on every move; market makers price this risk into the spread.
Most professional trading platforms (thinkorswim, IBKR, Tastytrade) show Greeks per position and portfolio-aggregate in real time.
Frequently asked questions
Delta is your directional exposure now; gamma is how fast that exposure changes as the underlying moves. A call with delta 0.50 and gamma 0.05 has a delta of about 0.75 after a USD 5 rise, so the position is more directional than when you opened it. That is why option positions need re-hedging: the exposure you put on is not the exposure you hold after the market moves.
A long option loses time value as expiry approaches, all else equal, so its theta is negative; a short option has positive theta. This calculator quotes theta per calendar day. For an at-the-money call on a USD 100 stock at 20% volatility and a 5% rate with 30 days left, that is about −0.045, or −USD 4.50 a day on a 100-share contract.
Roughly as 1 ÷ √T for at-the-money options. In this model, the USD 100 call above decays about 4.5 cents a day at 30 days, 8.6 cents at 7 days and 22 cents on the last day. Pure √T scaling would predict 9.3 and 24.6 cents; the interest-rate term accounts for the difference. The Options Industry Council puts it plainly: theta "is not linear" and tends to rise as expiry approaches.
When the option is long-dated or rates move a lot. For an at-the-money call on a USD 100 stock at 20% volatility and a 5% rate, rho is about 0.04 at 30 days, so a one-point rise in rates adds four cents, and about 0.53 at one year, against a premium of about USD 10.45. For short-dated equity options it is usually the smallest of the five; for long-dated options and options on rate-sensitive underlyings it is worth reading.
Put-call parity. The difference between call and put prices is C − P = S·e^(−qT) − K·e^(−rT), which is linear in S and independent of σ. So the second derivative in S (gamma) and first derivative in σ (vega) of the difference is zero — meaning gamma_call = gamma_put and vega_call = vega_put. They're properties of the OPTIONALITY, not the call/put direction.
Holding positions whose deltas sum to zero, so small moves in the underlying roughly cancel out and what remains is exposure to volatility and time. A long at-the-money call with delta 0.50 plus 50 shares sold short is delta-neutral: it gains from a large move either way (long gamma and vega) and loses a little each day the price sits still (theta). It stays neutral only until delta changes, so it has to be re-balanced.
Yes, depending on long/short and call/put. Call delta is 0 to 1; put delta is −1 to 0. Theta is negative for long positions, positive for short. Vega is positive for long, negative for short. Rho is positive for long calls and short puts, negative for long puts and short calls. Gamma is always positive for long options regardless of call/put; negative for short.
They are exact for the Black–Scholes–Merton model with the inputs you give it, and are tested here against finite differences of the model's own price. Markets depart from the model: implied volatility differs by strike (the smile), US stock options can be exercised early, and dividends are discrete rather than a continuous yield. For positions you are about to trade, use your broker's figures, which account for those, and treat these as a way to understand them.
Likely an input bound. Check: T > 0 (can't price an expired option), σ > 0 (zero vol makes the formula degenerate), S and K > 0. Extremely small T (less than ~0.001 years = ~9 hours) can produce tiny numerical instabilities — round inputs to sensible precision (0.01-year T is plenty granular). Negative spot or strike is mathematically undefined and should produce NaN.
No. All math runs in your browser via JavaScript. Open DevTools → Network and confirm zero outbound requests when you click Calculate. Your spot, strike, vol, and position details never leave your device.
Method & sources
How it computes
Black-Scholes-Merton with continuous dividend yield q; normal CDF by Abramowitz and Stegun 7.1.26. Theta per calendar day (/365), vega and rho per percentage point (/100). RecaInvest.bsm: each Greek tested against a finite difference of the price, plus put-call parity and the 10.4506 benchmark call. Call and put prices now shown.
What this tool implements
- The page said delta is the probability of finishing in the money; the model's probability is N(d2), lower than call delta (0.56 vs 0.64 for a one-year ATM call at 20% vol, 5%). The OIC presents delta-as-probability as a view some traders hold, and that is how the page now puts it.
- Worked theta figures recomputed from the kernel for a USD 100 stock (30 days -0.045/day, 7 days -0.086, last day -0.216); the claim that a 30-day ATM call loses ~50% of its value in its last 14 days is ~66% in the model; rho cut-offs given as both 60 and 90 days replaced with computed values (0.04 at 30 days, 0.53 at one year).
- Removed as unsourced: SPY theta figures, VIX move ranges, a tail-hedge budget example that would burn its premium in 20 days, the 1-5% accuracy claim and the claim that successful traders are net theta-positive. Invalid inputs now clear the previous answer instead of leaving it on screen.
Sources
- The Options Industry Council. Delta. https://www.optionseducation.org/advancedconcepts/delta
- The Options Industry Council. Theta. https://www.optionseducation.org/advancedconcepts/theta
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