tiny_pricing_utils Documentation

Overview

tiny_pricing_utils is a Python package for option pricing and financial modeling. It includes implementations of the Black-Scholes and Heston models, Monte Carlo simulations, and characteristic functions for Fourier-based pricing.

Working examples can be found at: https://github.com/MichaelCarloH/Option-Pricing/tree/main/Option%20pricing

Key Features: - Black-Scholes Model: European option pricing. - Heston Model: Class-based implementation with FFT pricing methods. - Monte Carlo Simulations: Simulates stock price paths under different models. - Characteristic Functions: Fourier-based characteristic functions for pricing using BS and Heston.

Aknowledgements: Part of the Theory and code snippets are based on the course “Financial Engineering” by Peter Leon and Wim Schoutens and course collaborators at KU Leuven.

Installation

To install the required dependencies, use:

pip install numpy scipy

Modules

Contents:

API Reference

class tiny_pricing_utils.HestonModel(S0, r, q, V0=0.05, kappa=0.5, eta=0.05, theta=0.2, rho=-0.75, T=1, alpha=1.5, N=4096, eta_cm=0.25, b=None, strikes=None)

Bases: object

calculate_auxiliary_variables()

Calculate log_strikes, v, u, and rho_cm based on model parameters.

compute_fft(rule='rectangular', simpson_weights=None)

Compute the FFT based on the specified integration rule.

compute_fft_rectangular()

Computes option prices using FFT with the rectangular rule.

compute_fft_simpson(simpson_weights)

Computes option prices using FFT with Simpson’s rule.

get_N()
get_S0()
get_T()
get_V0()
get_alpha()
get_b()
get_eta()
get_eta_cm()
get_kappa()
get_q()
get_r()
get_rho()
get_specific_strikes()
get_theta()
interpolate_prices()

Interpolate option prices for specific strikes.

price_options(rule='rectangular', simpson_weights=None)

Price options based on selected integration rule.

set_N(value)
set_S0(value)
set_T(value)
set_V0(value)
set_alpha(value)
set_b(value)
set_eta(value)
set_eta_cm(value)
set_kappa(value)
set_q(value)
set_r(value)
set_rho(value)
set_specific_strikes(value)
set_theta(value)
tiny_pricing_utils.black_scholes_price(S, K, T, r, q, sigma, option_type)

Computes the Black-Scholes price for European call and put options with dividends.

Parameters:

  • S (float) – Current stock price

  • K (float) – Strike price

  • T (float) – Time to maturity (years)

  • r (float) – Risk-free interest rate

  • sigma (float) – Volatility of the underlying asset

  • option_type (int) – 0 for call option, 1 for put option

  • q (float) – Dividend yield (default is 0, no dividend)

Returns:

Option price

Return type:

float

tiny_pricing_utils.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:

Characteristic function value at u

Return type:

complex

tiny_pricing_utils.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. This is the rate of return on a risk-free asset.

  • q (float) – Dividend yield. This is the annual dividend yield of the stock.

  • V0 (float) – Initial variance. This is the starting value for the volatility.

  • kappa (float) – Rate of mean reversion. This is the speed at which volatility reverts to its long-term mean (eta).

  • eta (float) – Long-term mean of volatility. This is the value to which volatility reverts over time.

  • theta (float) – Volatility of volatility. This determines how much volatility fluctuates over time.

  • rho (float) – Correlation between the stock price and its volatility. A value between -1 and 1.

  • T (float) – Time to maturity. This is the time in years until the option expires.

Returns:

Characteristic function value at u. This value is used in the Fourier inversion to price the option.

Return type:

complex

tiny_pricing_utils.geometric_brownian_motion(S0, r, q, sigma, T, N, M)
tiny_pricing_utils.heston_euler(S0, r, q, v0, kappa, eta, theta, rho, T, N, M)
tiny_pricing_utils.heston_milstein(S0, r, q, v0, kappa, eta, theta, rho, T, N, M)
tiny_pricing_utils.price_asian_call(S0, K, r, q, sigma, T, N, M)
tiny_pricing_utils.price_european_call(S0, K, r, q, sigma, T, N, M)
tiny_pricing_utils.price_european_put(S0, K, r, q, sigma, T, N, M)
tiny_pricing_utils.price_heston_euler_call(S0, K, r, q, v0, kappa, eta, theta, rho, T, N, M)
tiny_pricing_utils.price_heston_euler_put(S0, K, r, q, v0, kappa, eta, theta, rho, T, N, M)
tiny_pricing_utils.price_heston_milstein_call(S0, K, r, q, v0, kappa, eta, theta, rho, T, N, M)
tiny_pricing_utils.price_heston_milstein_put(S0, K, r, q, v0, kappa, eta, theta, rho, T, N, M)
tiny_pricing_utils.price_up_and_in_put(S0, K, H, r, q, sigma, T, N, M)
tiny_pricing_utils.price_up_and_out_put(S0, K, H, r, q, sigma, T, N, M)
tiny_pricing_utils.sum_squared_diff(sigma, S0, K, T, r, q, market_prices, option_type)

Computes the sum of squared differences between Black-Scholes prices and market prices with dividends.

Parameters:

  • sigma (float) – Implied volatility to be calibrated

  • S0 (float) – Current stock price

  • K (array-like) – Strike prices

  • T (array-like) – Time to maturities

  • r (array-like) – Risk-free interest rates

  • q (array-like) – Dividend yields

  • market_prices (array-like) – Observed market option prices

  • option_type (array-like) – Option types (0 for call, 1 for put)

Returns:

Sum of squared differences

Return type:

float