Black-Scholes for Put Options
The Black-Scholes model gives the fair price of a European option before it expires, using five inputs: stock price, strike, time, interest rate, and volatility. For a put, you use the same d1 and d2 as for a call, plugged into a slightly different formula. Put-call parity lets you check the answer.
Black-Scholes for put options, explained simply
Black-Scholes prices a European put with the same d1 and d2 as a call: P = K × e^(−rT) × N(−d2) − S × N(−d1). Put-call parity (C + K × e^(−rT) = P + S) checks the answer. In the example (S = $40, K = $45, r = 4%, σ = 25%, T = 1), the put is worth about $5.96.
What this Black-Scholes put lesson covers
- The five inputs Black-Scholes needs and where they come from
- How to calculate d1 and d2 step by step
- The put formula: P = K e^(−rT) N(−d2) − S N(−d1)
- How to find N(−d1) and N(−d2) using N(−x) = 1 − N(x)
- How to check a put price with put-call parity
Black-Scholes put formulas, explained
- S
- current stock price
- K
- strike price
- r
- risk-free rate, continuously compounded, per year
- σ
- annual volatility of the stock's returns
- T
- time to expiration in years
- N(x)
- standard normal cumulative probability of x
- e^(−rT)
- continuous discount factor
- C
- European call price with the same strike and expiry
- P
- European put price
How to price a put with Black-Scholes step by step (worked example)
Price a European put with S = $40, K = $45, r = 4%, σ = 25%, T = 1 year (no dividends).
- d1 = [ln(40 ÷ 45) + (0.04 + 0.25² ÷ 2) × 1] ÷ (0.25 × 1) = (−0.1178 + 0.0713) ÷ 0.25 = −0.1861.
- d2 = −0.1861 − 0.25 = −0.4361.
- N(−d1) = N(0.1861) = 0.5738 and N(−d2) = N(0.4361) = 0.6686.
- PV of strike = 45 × e^(−0.04) = $43.2355.
- P = 43.2355 × 0.6686 − 40 × 0.5738 = 28.91 − 22.95 = $5.96.
- Parity check: C = P + S − PV(K) = 5.96 + 40 − 43.24 ≈ $2.72, matching the Black-Scholes call value.
Answer: The put is worth about $5.96 per share (normal probabilities computed exactly and shown to 4 decimals; price rounded to the cent).
What happens to the put price when an input rises
| Input rises | Put price | Why |
|---|---|---|
| Stock price (S) | Falls | The right to sell at $45 is worth less |
| Strike (K) | Rises | You can sell for more |
| Volatility (σ) | Rises | Bigger falls become more likely |
| Interest rate (r) | Falls | The strike you receive later is worth less today |
Harder Black-Scholes put questions
How much does volatility change the put price?
With the other inputs fixed: about $4.43 at 15% volatility, $5.96 at 25% and $7.54 at 35%.
How much of the $5.96 is time value?
Exercising now would pay $45 − $40 = $5, the intrinsic value. The remaining $0.96 is time value. A European put can even be worth less than its intrinsic value when rates are high, because you must wait to receive the strike.
Common Black-Scholes put questions
How do you calculate a put option price with Black-Scholes?
Compute d1 and d2 from the stock price, strike, rate, volatility, and time. Find N(−d1) and N(−d2), which equal 1 − N(d1) and 1 − N(d2). Then plug them into P = K e^(−rT) N(−d2) − S N(−d1).
What does N(−d2) mean in the Black-Scholes put formula?
N(−d2) is the risk-neutral probability that the put finishes in the money, meaning the stock ends below the strike. N(−d1) is linked to the put's delta: the put's delta equals −N(−d1), so it tells you how much the put price moves per $1 change in the stock.
Can Black-Scholes price American put options?
Not exactly. Black-Scholes assumes European options, which can only be exercised at expiry. An American put can be worth more because early exercise can pay off when the stock falls deep in the money. A binomial tree is the usual tool for American puts.
What is put-call parity?
Put-call parity links European calls and puts with the same strike and expiry: C + K e^(−rT) = P + S. If you know the call price, you can back out the put price, and vice versa. It's a handy way to check a Black-Scholes answer.