Implied volatility is an estimate of the future volatility of a financial instrument, derived from the market price of options based on that instrument. It is a measure of market expectations of the future volatility of a security's price.

To compute implied volatility in Python, you can use the scipy.optimize module to minimize the difference between the market price of an option and its theoretical price. The theoretical price can be calculated using the Black-Scholes model. Here's an example:

import numpy as np
from scipy.optimize import minimize_scalar
from scipy.stats import norm

# Define the Black-Scholes model for pricing options
def black_scholes(s, k, t, r, sigma, option_type):
    # Calculate d1 and d2
    d1 = (np.log(s/k) + (r + 0.5*sigma**2)*t) / (sigma*np.sqrt(t))
    d2 = d1 - sigma*np.sqrt(t)
    # Return the option price based on the option type
    if option_type == 'call':
        return s*norm.cdf(d1) - k*np.exp(-r*t)*norm.cdf(d2)
    else:
        return k*np.exp(-r*t)*norm.cdf(-d2) - s*norm.cdf(-d1)

# Define the function to calculate implied volatility
def implied_volatility(market_price, s, k, t, r, option_type):
    # Define the error function as the absolute difference between the market price and theoretical price
    error_function = lambda sigma: np.abs(market_price - black_scholes(s, k, t, r, sigma, option_type))
    # Use the minimize_scalar method to find the value of sigma (implied volatility) that minimizes the error function
    implied_vol = minimize_scalar(error_function, bounds=(0, 2), method='bounded').x
    return implied_vol

n the code above, the black_scholes function calculates the option price based on the Black-Scholes model. The inputs to the function are the underlying asset price (s), strike price (k), time to expiration (t), risk-free interest rate (r), volatility (sigma), and option type ('call' or 'put').

The implied_volatility function calculates the implied volatility by minimizing the difference between the market price of an option and its theoretical price calculated using the black_scholes function. The minimize_scalar method from the scipy.optimize module is used to find the value of sigma (implied volatility) that minimizes the error function. The bounds for sigma are set to 0 and 2, and the optimization method used is the bounded method. The resulting value of sigma is the implied volatility.

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