Black_Scholes {qrmtools}

Black–Scholes formula and the Greeks

Description

Compute the Black–Scholes formula and the Greeks.

Usage

Black_Scholes(t, S, r, sigma, K, T, type = c("call", "put"))
Black_Scholes_Greeks(t, S, r, sigma, K, T, type = c("call", "put"))

Arguments

t

initial or current time t (in years).

S

stock price at time t.

r

risk-free annual interest rate.

sigma

annual volatility (standard deviation).

K

strike.

T

maturity (in years).

type

character string indicating whether a call (the default) or a put option is considered.

Details

Note again that t is time in years. In the context of McNeil et al. (2015, Chapter 9), this is \tau_t = t/250.

Value

Black_Scholes() returns the value of a European-style call or put option (depending on the chosen type) on a non-dividend paying stock.

Black_Scholes_Greeks() returns the first-order derivatives delta, theta, rho, vega and the second-order derivatives gamma, vanna and vomma (depending on the chosen type) in this order.

Author(s)

Marius Hofert

References

McNeil, A. J., Frey, R., and Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques, Tools. Princeton University Press.