# How to compute implied volatility in Python

1 min

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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