quantitative-finance-guide

Quantitative Finance Guide

A rigorous skill for applying quantitative methods to financial research, covering derivatives pricing, portfolio optimization, risk modeling, and time series econometrics. Designed for academic researchers and quantitative analysts.

Derivatives Pricing

Black-Scholes-Merton Model

The foundational model for European option pricing:

import numpy as np
from scipy.stats import norm

def black_scholes(S: float, K: float, T: float, r: float,
                   sigma: float, option_type: str = 'call') -> dict:
    """
    Black-Scholes European option pricing.

    Args:
        S: Current stock price
        K: Strike price
        T: Time to maturity (years)
        r: Risk-free rate (annualized)
        sigma: Volatility (annualized)
        option_type: 'call' or 'put'
    """
    d1 = (np.log(S / K) + (r + 0.5 * sigma**2) * T) / (sigma * np.sqrt(T))
    d2 = d1 - sigma * np.sqrt(T)

    if option_type == 'call':
        price = S * norm.cdf(d1) - K * np.exp(-r * T) * norm.cdf(d2)
    else:
        price = K * np.exp(-r * T) * norm.cdf(-d2) - S * norm.cdf(-d1)

    greeks = {
        'delta': norm.cdf(d1) if option_type == 'call' else norm.cdf(d1) - 1,
        'gamma': norm.pdf(d1) / (S * sigma * np.sqrt(T)),
        'theta': -(S * norm.pdf(d1) * sigma) / (2 * np.sqrt(T)),
        'vega': S * norm.pdf(d1) * np.sqrt(T),
        'rho': K * T * np.exp(-r * T) * norm.cdf(d2) if option_type == 'call'
               else -K * T * np.exp(-r * T) * norm.cdf(-d2)
    }
    return {'price': price, 'greeks': greeks}

# Example: price a call option
result = black_scholes(S=100, K=105, T=0.5, r=0.05, sigma=0.20, option_type='call')
print(f"Call Price: ${result['price']:.2f}")
print(f"Delta: {result['greeks']['delta']:.4f}")

Monte Carlo Simulation

For path-dependent options and complex payoffs:

def monte_carlo_option(S0, K, T, r, sigma, n_paths=100000, n_steps=252):
    """Geometric Brownian Motion Monte Carlo pricer."""
    dt = T / n_steps
    Z = np.random.standard_normal((n_paths, n_steps))
    paths = np.zeros((n_paths, n_steps + 1))
    paths[:, 0] = S0

    for t in range(n_steps):
        paths[:, t + 1] = paths[:, t] * np.exp(
            (r - 0.5 * sigma**2) * dt + sigma * np.sqrt(dt) * Z[:, t]
        )

    payoffs = np.maximum(paths[:, -1] - K, 0)
    price = np.exp(-r * T) * np.mean(payoffs)
    std_err = np.exp(-r * T) * np.std(payoffs) / np.sqrt(n_paths)
    return {'price': price, 'std_error': std_err, '95_ci': (price - 1.96*std_err, price + 1.96*std_err)}