Compute theoretical European option price and all five Greeks.
Call price
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Put price
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Greeks (call)
Delta
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Gamma
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Theta/day
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Vega/1%
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Rho/1%
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Your breakdown
Updates live as you type| Output | Value |
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What this model actually answers
The Black-Scholes-Merton formula gives the theoretical fair value of a European call or put: an option that can only be exercised at expiration. You feed it six things, the spot price, strike, time to expiry, volatility, the risk-free rate, and any dividend yield, and it returns a price plus the five Greeks that describe how that price moves. The Greeks are the part traders and finance teams use most, because they translate an abstract value into sensitivities you can hedge or manage.
The model rests on a few load-bearing assumptions: the underlying follows a lognormal random walk, volatility and the interest rate are constant, and there are no transaction costs. Real markets violate all of these to some degree, which is why traders speak of an implied volatility smile rather than a single flat number. Still, Black-Scholes remains the common language of option pricing and the engine behind most 409A and ASC 718 stock-compensation valuations.
Reading d1, d2, and the Greeks
Inside the formula, two intermediate values d1 and d2 position the option against its strike in standard-deviation terms. From them you get delta, the change in option price per $1 move in the stock; gamma, the rate at which delta itself changes; theta, the daily decay as time runs out; vega, the price change per one percentage point of volatility; and rho, the sensitivity to interest rates. Delta doubles as a rough probability the option finishes in the money.
An at-the-money one-year call on a $100 stock
Price a one-year option with the stock and strike both at $100, volatility of 30%, a 4.5% risk-free rate, and no dividend. Here d1 works out to 0.30 and d2 to exactly 0.00. The call is worth about $13.99 and the put about $9.59. The call is more expensive than the put even though both are struck at the money, because the risk-free rate pushes the forward price of the stock above $100, tilting value toward the call. Delta on the call is 0.62, meaning it gains about $0.62 for the next dollar the stock rises.
Volatility is the input that matters most
Of the six inputs, two are observable on day one, the spot price and the strike, and one is nearly fixed, the risk-free rate. Time decays predictably. That leaves volatility as the single assumption that drives most of the disagreement over an option's value, and it is the one you cannot read off a screen. Vega here is about $0.38, so every percentage point you add to the 30% volatility assumption raises the call by roughly $0.38. Double-check a quote that looks cheap or rich, and the culprit is almost always the volatility number behind it.
The most common mistake when valuing employee stock options is feeding this European model a US employee option and stopping there. Standard US stock options are American-style and, more importantly, carry vesting conditions, early-exercise behavior, and forfeiture risk that Black-Scholes ignores. For financial-reporting and 409A work, the accepted practice is to use the expected term rather than the full contractual term, or to move to a lattice or binomial model that can handle early exercise. Treat the output here as a clean theoretical benchmark, not a substitute for a formal valuation.
Why is theta shown as a tiny daily number?
Theta is the option's time decay, and this calculator reports it per calendar day by dividing the annual figure by 365. For a one-year option that daily slice is small, well under a dollar, because most of the decay is still far off. Theta accelerates as expiration nears, so the same option with a week left would show a much larger daily decay. If you want the annual decay, multiply the per-day figure back by 365.
Can I use this for put-call parity checks?
Yes, and it is a good sanity test. Put-call parity says the call price minus the put price should equal the discounted forward, the spot adjusted for dividends minus the present value of the strike. With the call at $13.99 and the put at $9.59, the $4.40 gap reflects exactly that interest-rate-driven forward premium on an at-the-money one-year option. If you ever see a call and put that violate parity by more than transaction costs, the prices, not the model, are off.