CreditRisk (version 0.1.7)

generalized_black_scholes: Generalized Black-Scholes Option Pricing Model

Description

This function calculates the price of a European call or put option using the generalized Black-Scholes formula, which extends the standard model to incorporate a continuous dividend yield.

Usage

generalized_black_scholes(TypeFlag = c("c", "p"), S, X, Time, r, b, sigma)

Value

Returns the price of the specified option (call or put).

Arguments

TypeFlag

A character vector indicating the type of option to be priced, either "c" for call options or "p" for put options.

S

Current stock price (scalar).

X

Strike price of the option (scalar).

Time

Time to expiration of the option (in years).

r

Risk-free interest rate (annualized).

b

Cost of carry rate, b = r - q for a dividend yield q.

sigma

Volatility of the underlying asset (annualized).

Details

The generalized Black-Scholes formula considers both the risk-free rate and a cost of carry, making it suitable for a wider range of financial instruments, including commodities and currencies with continuous yields.

The pricing formula for call and put options is determined by: $$C = S e^{(b-r)T} N(d_1) - X e^{-rT} N(d_2)$$ $$P = X e^{-rT} N(-d_2) - S e^{(b-r)T} N(-d_1)$$ where: $$d_1 = \frac{\log(S / X) + (b + \sigma^2 / 2) T}{\sigma \sqrt{T}}$$ $$d_2 = d_1 - \sigma \sqrt{T}$$ and \((N(\cdot))\) is the cumulative normal distribution function, estimated by the `cum_normal_density` function.

References

Haug, E.G., The Complete Guide to Option Pricing Formulas.

Examples

# Calculate the price of a call option
generalized_black_scholes("c", S = 100, X = 100, Time = 1, r = 0.05, b = 0.05, sigma = 0.2)
# Calculate the price of a put option
generalized_black_scholes("p", S = 100, X = 100, Time = 1, r = 0.05, b = 0.05, sigma = 0.2)