Black-Scholes-Mertonintermediate

European Put Price

The BSM European put price is $P = K e^{-rT} N(-d_2) - S_0 N(-d_1)$ , where $d_1$ and $d_2$ are the same as in the call formula. The same result follows from put-call parity, $C - P = S_0 - K e^{-rT}$ , so $P = C - S_0 + K e^{-rT}$ once the call is known. Either route gives the same number for European options on a non-dividend stock (Hull, 2022, §15.8).

Why it matters

The put is the mirror image of the call. The probabilities flip from $N(d_1)$ and $N(d_2)$ to $N(-d_1)$ and $N(-d_2)$ , because the put pays off when the stock is below the strike rather than above it. $N(-d_2)$ is the risk-neutral probability the put finishes in the money. Put-call parity is usually the fastest way to get the put if you already have the call, since it avoids a second pair of table look-ups.

Formulas

BSM European put

P = K e^{-rT} N(-d_2) - S_0 N(-d_1)

d_1

and

d_2

are the same as in the call. Apply symmetry,

N(-x) = 1 - N(x)

, when reading the z-table.

From put-call parity

P = C - S_0 + K e^{-rT}

Holds for European options on a non-dividend stock and lets you reuse a known call price.

Worked examples

Scenario

Carry over the call example, $S_0 = K =$ A$50, $r = 5\%$ , $\sigma = 30\%$ , $T = 1$ , which gave $C \approx$ A$7.12, $d_1 = 0.3167$ , $d_2 = 0.0167$ .

Solution

By parity, $P = 7.12 - 50 + 50 e^{-0.05} = 7.12 - 50 + 47.561 \approx$ A$4.68. Direct formula, $N(-d_1) = 1 - 0.6242 = 0.3758$ and $N(-d_2) = 1 - 0.5067 = 0.4933$ . So $P = 50 \times 0.9512 \times 0.4933 - 50 \times 0.3758 = 23.460 - 18.790 \approx$ A$4.67. The 1-cent gap is rounding in the z-table look-ups.

Common mistakes

✗You must use the put formula directly. Put-call parity is often quicker once the call is known and uses fewer table look-ups. Either path is valid for European options.
✗A put is always cheaper than a call. Not in general. For deep in-the-money strikes or when $r$ is low and $q$ is high, the put can trade above the call. The order depends on parity, $C - P = S_0 e^{-qT} - K e^{-rT}$ .

Revision bullets

• $P = K e^{-rT} N(-d_2) - S_0 N(-d_1)$
•Or via parity, $P = C - S_0 + K e^{-rT}$
•Always $P \geq \max(K e^{-rT} - S_0,\, 0)$
•Cross-check call and put with parity
•Use $N(-x) = 1 - N(x)$ on the z-table

Quick check

The BSM put price can be derived from the call price via

Connected topics

PCP d1 & d2

In learning paths

Pricing Models Path

Sources

Hull (2022), §15.8
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Derives the BSM put formula and demonstrates how to recover it from the call via put-call parity.
Merton (1973), Bell J. Econ.
Merton, Robert C. "Theory of Rational Option Pricing." Bell Journal of Economics and Management Science 4, no. 1 (Spring 1973): 141-183.
Establishes the no-arbitrage bounds and parity relationships that make the put formula consistent with the call.
Stoll (1969), JF
Stoll, Hans R. "The Relationship Between Put and Call Option Prices." Journal of Finance 24, no. 5 (1969): 801-824.
Classic empirical and theoretical statement of put-call parity, the relation used to derive $P$ from $C$.

How to cite this page

Dr. Phil's Quant Lab. (2026). European Put Price. Derivatives Atlas. https://phucnguyenvan.com/concept/bsm-put

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