Option Pricing: Black-Scholes and Beyond

Module: Fintech AI | Difficulty: Advanced

Black-Scholes Formula

Greeks

Implied Volatility Surface

Heston Model

import numpy as np
from scipy.stats import norm

def black_scholes(S, K, T, r, sigma, option_type='call'):
    d1 = (np.log(S/K) + (r + sigma**2/2)*T) / (sigma*np.sqrt(T))
    d2 = d1 - sigma*np.sqrt(T)
    if option_type == 'call':
        return S*norm.cdf(d1) - K*np.exp(-r*T)*norm.cdf(d2)
    return K*np.exp(-r*T)*norm.cdf(-d2) - S*norm.cdf(-d1)

def implied_vol(price, S, K, T, r, option_type='call'):
    from scipy.optimize import brentq
    func = lambda sigma: black_scholes(S, K, T, r, sigma, option_type) - price
    return brentq(func, 0.01, 2.0)

Research Insight: The volatility smile reveals that Black-Scholes' constant volatility assumption is violated. Stochastic volatility models (Heston) and jump-diffusion models (Merton) capture this by allowing volatility to be random and adding jumps to the price process.