Characteristic Function Calculations

This module contains functions for calculating the characteristic functions of the log-stock price under the Black-Scholes and Heston models. These characteristic functions are used in Fourier methods for option pricing.

Functions

cf_BlackScholes(u, S0, r, q, sigma, T)

Computes the characteristic function of the log-stock price under the Black-Scholes model.

Parameters: - u (complex or float): Fourier transform variable. - S0 (float): Initial stock price. - r (float): Risk-free rate. - q (float): Dividend yield. - sigma (float): Volatility. - T (float): Time to maturity.

Returns: - (complex): Characteristic function value at u.

cf_Heston(u, S0, r, q, V0, kappa, eta, theta, rho, T)

Computes the characteristic function of the log-stock price under the Heston model. The Heston model allows for stochastic volatility, where volatility follows a mean-reverting process.

Parameters: - u (complex or float): Fourier transform variable. This is the variable for which the characteristic function is computed. - S0 (float): Initial stock price at time t = 0. - r (float): Risk-free rate. - q (float): Dividend yield. - V0 (float): Initial variance (starting value for the volatility). - kappa (float): Rate of mean reversion. Speed at which volatility reverts to its long-term mean (eta). - eta (float): Long-term mean of volatility. - theta (float): Volatility of volatility. - rho (float): Correlation between the stock price and its volatility. Must be between -1 and 1. - T (float): Time to maturity.

Returns: - (complex): Characteristic function value at u. This value is used in Fourier inversion to price the option.