Advanced Algos: Outsmarting the Market, One Algorithm at a Time.: A Comprehensive Algorithmic Trading Guide For 2024 by Bissette Vincent & Strauss Johann & Van Der Post Hayden

Author:Bissette, Vincent & Strauss, Johann & Van Der Post, Hayden
Language: eng
Format: epub
Publisher: Reactive Publishing
Published: 2024-01-05T00:00:00+00:00

Implementing the Model in Python

In the following section, we transition from the theoretical to the practical, implementing the Black-Scholes model in Python to price European options. Python's simplicity and the power of its libraries allow for an elegant translation of mathematical models into computational algorithms that can be utilized in real-world financial analysis.

Building the Black-Scholes Function:

To price a European call or put option using the Black-Scholes formula, we first define a function that calculates the cumulative distribution function of the standard normal distribution, which is necessary for the model. This function, `norm.cdf`, is provided by the `scipy.stats` library, a powerful tool for statistical analysis.

We then create a function, `black_scholes`, which computes the present value of an option based on the Black-Scholes formula. The function will take inputs such as the asset price (`S`), the strike price (`K`), the time to maturity (`T`), the risk-free rate (`r`), and the volatility of the asset (`sigma`), and it will return the option's price.

```python

from scipy.stats import norm

def black_scholes(S, K, T, r, sigma, option_type='call'):

"""

Calculate the Black-Scholes option price for a call or put option.

Parameters:

S (float): Current asset price

K (float): Option strike price

T (float): Time to maturity (in years)

r (float): Risk-free interest rate (annual)

sigma (float): Volatility of the underlying asset (annual)

option_type (str): 'call' for call option, 'put' for put option

Returns:

float: Black-Scholes option price

"""

# Calculate d1 and d2 parameters

d1 = (np.log(S / K) + (r + 0.5 * sigma 2) * T) / (sigma * np.sqrt(T))

d2 = d1 - sigma * np.sqrt(T)

# Calculate option price based on type

if option_type == 'call':

option_price = (S * norm.cdf(d1) - K * np.exp(-r * T) * norm.cdf(d2))

elif option_type == 'put':

option_price = (K * np.exp(-r * T) * norm.cdf(-d2) - S * norm.cdf(-d1))

else:

raise ValueError("Invalid option type. Use 'call' or 'put'.")

return option_price

# Example usage:

option_price = black_scholes(100, 110, 1, 0.05, 0.25, 'call')

print(f"The Black-Scholes price for the call option is: {option_price:.2f}")

```

Applying the Model to Market Data:

To apply our `black_scholes` function to real market data, we can retrieve historical asset prices and implied volatility from financial databases or APIs. For the purposes of this example, let's assume we have historical data stored in a pandas DataFrame.

```python

import pandas as pd

# Sample market data (for illustration purposes only)

market_data = {

'Date': pd.date_range(start='2022-01-01', periods=5, freq='D'),

'Asset_Price': [100, 102, 104, 103, 105],

'Implied_Volatility': [0.2, 0.19, 0.21, 0.22, 0.2]

}

df_market = pd.DataFrame(market_data)

df_market['Option_Price'] = df_market.apply(lambda row: black_scholes(

row['Asset_Price'], 110, 1, 0.05, row['Implied_Volatility'], 'call'), axis=1)

print(df_market[['Date', 'Asset_Price', 'Implied_Volatility', 'Option_Price']])

```

Sensitivity Analysis:

A key aspect of options pricing is understanding how the price of an option changes in response to different factors. The `black_scholes` function can be used for sensitivity analysis, often referred to as 'Greek' calculations, which measure the sensitivity of the option's price to changes in underlying parameters like delta, gamma, and theta.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.