price_black_scholes_put: Black-Scholes European Put Option Price

In AsianOption: Asian Option Pricing with Price Impact

Black-Scholes European Put Option Price

Description

Computes the exact price of a European put option using the classical Black-Scholes (1973) analytical formula.

Usage

price_black_scholes_put(S0, K, r, sigma, time_to_maturity)

Arguments

S0	Initial stock price (must be positive)
K	Strike price (must be positive)
r	Continuously compounded risk-free rate (e.g., 0.05 for 5% annual rate)
sigma	Volatility (annualized standard deviation, must be non-negative)
time_to_maturity	Time to maturity in years (must be positive)

Details

The Black-Scholes formula for a European put option is:

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

where:

d_1 = \frac{\log(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}

d_2 = d_1 - \sigma\sqrt{T}

and N(\cdot) is the cumulative standard normal distribution function.

Alternatively, the put price can be derived from put-call parity:

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

Value

European put option price (numeric)

Put-Call Parity

The Black-Scholes put and call prices satisfy:

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

This relationship holds exactly for European options without dividends.

References

Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637-654. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1086/260062")}

See Also

price_black_scholes_call

Examples

price_black_scholes_put(S0 = 100, K = 100, r = 0.05, sigma = 0.2,
                        time_to_maturity = 1)

AsianOption documentation built on Dec. 23, 2025, 1:08 a.m.