Black-Scholes Put Price Calculator
Price a European put option with Black–Scholes — enter spot, strike, volatility, rate and time to get the fair premium, delta and intrinsic vs time value.
How it works
P = K·e^(−rT)·N(−d2) − S·N(−d1) · d1 = [ln(S/K) + (r + σ²/2)T] ÷ (σ√T) · d2 = d1 − σ√T
The put mirrors the call: it pays off when the underlying finishes below the strike. Its delta, N(d1) − 1, is negative because the put gains value as the stock falls. When the risk-free rate is zero, an at-the-money put and call cost the same; a positive rate makes calls slightly dearer and puts slightly cheaper because holding the stock earns the carry.
Worked example
At spot $100, strike $100, 20% volatility, a 0% rate and one year: N(−d2) = 0.5398 and N(−d1) = 0.4602, so P = 100 × 0.5398 − 100 × 0.4602 = $7.97 — identical to the call, as expected at a zero rate.
Sources: Black–Scholes model (Wikipedia)
Frequently asked questions
Why does the put equal the call here?
Put–call parity says C − P = S − K·e^(−rT). With spot equal to strike and a 0% rate the right side is zero, so the call and put must be worth the same. Raise the rate and the call becomes worth more.
Can a put ever be worth more than its strike?
No. The most a put can pay is the strike (if the stock goes to zero), and that payoff is in the future, so the discounted value is always below the strike. This model respects that ceiling.
How do I price a protective put?
Price the put here, then compare its cost to the downside it removes. A put struck below spot is cheaper but only insures losses beyond that lower level.
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Educational estimate using a simplified Black–Scholes / options model — no dividends, no early-exercise, no fees or slippage. Options are high-risk. Not investment advice.