Calculate European call and put prices and the Greeks (delta, gamma, theta, vega, rho) with the Black-Scholes-Merton model, including dividends.
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The call is worth $8.26 and the put $5.79. For each 1.00 rise in the stock price the call gains about 0.5909, and with nothing else changing one day of time takes about 0.0258 off its value.
| Measure | Call | Put |
|---|---|---|
| Price | 8.26 | 5.791 |
| Delta | 0.5909 | −0.4091 |
| Gamma | 0.022 | 0.022 |
| Theta per day | −0.0258 | −0.0124 |
| Vega per 1% volatility | 0.2747 | 0.2747 |
| Rho per 1% rate | 0.2541 | −0.2335 |
Theta is per calendar day (annual theta ÷ 365); vega and rho are per one percentage point. European exercise; the model assumes constant volatility and rates.
The Black-Scholes-Merton model prices a European option from the stock price S, the strike K, the time to expiry T in years, the volatility σ, the risk-free rate r and a continuous dividend yield q. A call is worth S·e^(−qT)·N(d₁) − K·e^(−rT)·N(d₂), where N is the standard normal distribution, and the put follows from the same terms. The Greeks measure sensitivity: delta to the stock price, gamma to delta itself, theta to the passing of time, vega to volatility and rho to the interest rate.
With the defaults, a stock at 100, a strike of 100, six months to expiry, 25% volatility and a 5% rate, the call is worth 8.26 and the put 5.79. The call's delta of 0.5909 means it gains about 0.59 for each 1 rise in the stock, and its theta shows it losing about 0.026 a day.
The model assumes constant volatility and rates, log-normal prices and exercise only at expiry. US stock and ETF options are American-style and can be exercised early; index options such as SPX are European and cash-settled.
A European call is worth S·e^(−qT)·N(d₁) − K·e^(−rT)·N(d₂), where N is the standard normal distribution. A stock at 100, a strike of 100, six months to expiry, 25% volatility and a 5% rate give a call worth 8.26 and a put worth 5.79.
With S = K = 100, T = 0.5 years, σ = 0.25, r = 0.05 and no dividends (q = 0):
The put uses the opposite tails: 97.53 × N(−d₂) − 100 × N(−d₁) = 46.70 − 40.91 = 5.79.
For European options on the same stock, strike and expiry, call − put = S·e^(−qT) − K·e^(−rT). Here that is 100 − 97.53 = 2.469, and the rounded prices give 8.26 − 5.79 = 2.47. Any put can be checked against its call this way.
The formula comes from Black and Scholes' 1973 paper in the Journal of Political Economy. The Black-Scholes-Merton form adds a continuous dividend yield, q.
These rows keep the stock, strike, time and rate fixed:
| Annual volatility (%) | Call value | Put value |
|---|---|---|
| 25 | 8.26 | 5.79 |
| 35 | 11.01 | 8.54 |
Both options gain 2.75. Parity fixes the gap between call and put, and that gap has no volatility term, so a change in volatility moves both by the same amount.
Market prices run the other way. The implied volatility is the σ at which the model matches a quoted premium: with the other inputs above, a call trading at 9.00 implies about 27.69%.
Each Greek is a local sensitivity with the other inputs held fixed. For the six-month at-the-money example:
Prices and Greeks are per share. A US equity option contract usually covers 100 shares, so the 8.26 call costs 826 a contract, and its 0.59 delta behaves like about 59 shares.
The model prices options that can be exercised only at expiry. US stock and ETF options are American-style, while index options such as SPX are European and settled in cash.
For a call on a stock that pays no dividends, early exercise is never worth it, so the model price is also the American price (Hull, Options, Futures, and Other Derivatives).
American puts, and calls just before a dividend, are worth more and need a binomial tree or a similar method.
The stock price, strike, time and volatility must all be above zero. At expiry an option is worth its intrinsic value: the larger of 0 and S − K for a call, or of 0 and K − S for a put.
For share positions and average cost, use the stock position calculator; for fixed-income cash flows, the bond calculator.
Checked against: Hull, Options, Futures, and Other Derivatives, Example 15.6: c = 4.76, p = 0.81, d1 = 0.7693, d2 = 0.6278; exact decimal arithmetic (prec 50, erf series) gives 4.7594224, 0.8085994
Checked against: Hull ch. 19 (S=49, K=50, 20 weeks): price 2.40, delta 0.522, gamma 0.066, theta −4.31/yr = −0.0118/day, vega 12.1 (0.121 per 1%), rho 8.91 (0.0891 per 1%); exact decimal arithmetic values to 6 dp
Checked against: Hull Example 17.1: c = 51.83, d1 = 0.5444; exact decimal arithmetic Merton formula gives 51.8329568
Checked against: With r = q = 0 and S = K, d1 = σ√T/2 = 0.1 and C = P = S(2N(0.1) − 1) = 7.9655675
C = S·e^(−qT)·N(d₁) − K·e^(−rT)·N(d₂), with d₁ = (ln(S/K) + (r − q + σ²/2)T) ÷ (σ√T) and d₂ = d₁ − σ√T. For the defaults, d₁ = 0.2298 and d₂ = 0.0530. Fischer Black and Myron Scholes published it in the Journal of Political Economy in 1973 (vol. 81, no. 3, pp. 637–654); Robert Merton and Scholes received the 1997 economics prize for the work, Black having died in 1995.
Delta is how much the option price changes for a 1 change in the stock price. The default call has a delta of 0.5909 and the put −0.4091; without dividends they always differ by exactly 1. One US equity option contract usually covers 100 shares, so a call with a delta of 0.59 moves roughly like 59 shares of the stock.
For European options with the same strike and expiry, call minus put equals S·e^(−qT) minus K·e^(−rT). Using full precision, the default difference is about 2.469009, which rounds to 2.47. Subtracting the displayed prices, 8.26 − 5.79, also gives 2.47; it does not give the extra unrounded digits. The relation uses consistent dividend, interest-rate and exercise assumptions. Actual trading comparisons must also account for transaction costs and contract terms.
Higher volatility raises both call and put prices, because it widens the range of prices the stock can reach before expiry. The default call's vega is 0.2747, so each extra 1% of volatility adds about 0.27; raising volatility from 25% to 35% lifts the call from 8.26 to 11.01 and the put from 5.79 to 8.54.
Only some. US stock and ETF options are American-style, so they can be exercised before expiry. For a call on a stock that pays no dividends, early exercise is never worth it, so the Black-Scholes price is also the American price (Hull, Options, Futures, and Other Derivatives). American puts, and calls just before a dividend, are worth more than the model shows and need a binomial tree or similar method.
For planning only. Lenders, tax authorities and markets apply their own rounding, fees and rules; confirm figures with them before you commit.
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