price_black_scholes_call: Black-Scholes European Call Option Price

In AsianOption: Asian Option Pricing with Price Impact

Black-Scholes European Call Option Price

Description

Computes the exact price of a European call option using the classical Black-Scholes (1973) analytical formula. This is the continuous-time benchmark for comparison with discrete binomial models.

Usage

price_black_scholes_call(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 call option is:

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

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.

This formula assumes:

• Stock price follows geometric Brownian motion: dS_t = rS_t dt + \sigma S_t dW_t

• No dividends

• Constant risk-free rate and volatility

• Continuous trading with no transaction costs or price impact

Value

European call option price (numeric)

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")}

Examples

price_black_scholes_call(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.