Generate a 3D plot
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from scipy.stats import norm
def black_scholes_merton(S, K, T, r, sigma, option_type):
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':
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
# Define ranges for stock prices and time to expiration
stock_prices = np.linspace(50, 150, 50) # Stock prices from 50 to 150
time_to_expiry = np.linspace(0.1, 2, 50) # Time to expiration from 0.1 to 2 years
# Create a grid of stock prices and time to expiration
S, T = np.meshgrid(stock_prices, time_to_expiry)
# Parameters
strike_price = 100
risk_free_rate = 0.05
volatility = 0.2
# Calculate call option prices for the grid of stock prices and time to expiration
call_option_prices = black_scholes_merton(S, strike_price, T, risk_free_rate, volatility, 'call')
# Calculate put option prices for the grid of stock prices and time to expiration
put_option_prices = black_scholes_merton(S, strike_price, T, risk_free_rate, volatility, 'put')
# Create 3D plot for Call Option Prices
fig = plt.figure(figsize=(12, 5))
# Plot for Call Option Prices
ax1 = fig.add_subplot(121, projection='3d')
ax1.plot_surface(S, T, call_option_prices, cmap='viridis')
ax1.set_title('Call Option Prices')
ax1.set_xlabel('Stock Price')
ax1.set_ylabel('Time to Expiry')
ax1.set_zlabel('Option Price (Call)')
# Plot for Put Option Prices
ax2 = fig.add_subplot(122, projection='3d')
ax2.plot_surface(S, T, put_option_prices, cmap='plasma')
ax2.set_title('Put Option Prices')
ax2.set_xlabel('Stock Price')
ax2.set_ylabel('Time to Expiry')
ax2.set_zlabel('Option Price (Put)')
plt.tight_layout()
plt.show()Output
The displayed 3D plots showcase the relationship between option prices, stock prices, and time to expiry using the Black-Scholes-Merton model for both call and put options.
Calculating Option Value with no Dividend Yield
import numpy as np
from scipy.stats import norm
def black_scholes_merton(S, K, T, r, sigma, option_type):
"""
Calculates the Black-Scholes-Merton option price for European call or put options.
Parameters:
S : float - Current stock price
K : float - Option's strike price
T : float - Time to expiration (in years)
r : float - Risk-free interest rate
sigma : float - Volatility of the underlying stock
option_type : str - 'call' or 'put' to specify the option type
Returns:
float - Option price
"""
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':
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:
stock_price = 100 # Current stock price
strike_price = 110 # Option's strike price
time_to_expiry = 1 # Time to expiration in years
risk_free_rate = 0.05 # Risk-free interest rate
volatility = 0.2 # Volatility of the underlying stock
# Calculate call option price
call_option_price = black_scholes_merton(stock_price, strike_price, time_to_expiry, risk_free_rate, volatility, 'call')
print(f"Call Option Price: {call_option_price:.2f}")
# Calculate put option price
put_option_price = black_scholes_merton(stock_price, strike_price, time_to_expiry, risk_free_rate, volatility, 'put')
print(f"Put Option Price: {put_option_price:.2f}")Output
Call Option Price: 6.04
Put Option Price: 10.68
Calculating Option Value with Dividend Yield
import numpy as np
from scipy.stats import norm
def black_scholes_merton_dividend(S, K, T, r, sigma, q, option_type):
d1 = (np.log(S / K) + (r - q + 0.5 * sigma**2) * T) / (sigma * np.sqrt(T))
d2 = d1 - sigma * np.sqrt(T)
if option_type == 'call':
option_price = S * np.exp(-q * T) * 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 * np.exp(-q * T) * norm.cdf(-d1)
else:
raise ValueError("Invalid option type. Use 'call' or 'put'.")
return option_price
# Example usage:
current_stock_price = 100 # Current stock price
strike_price = 110 # Option's strike price
time_to_expiry = 1 # Time to expiration in years
risk_free_rate = 0.05 # Risk-free interest rate
volatility = 0.2 # Volatility of the underlying stock
dividend_yield = 0.05 # Dividend yield
# Calculate call option price with dividend yield adjustment
call_option_price_with_dividend = black_scholes_merton_dividend(current_stock_price,strike_price,
time_to_expiry, risk_free_rate, volatility, dividend_yield, 'call')
# Calculate put option price with dividend yield adjustment
put_option_price_with_dividend = black_scholes_merton_dividend(current_stock_price, strike_price,time_to_expiry, risk_free_rate, volatility, dividend_yield, 'put')
print(f"Call Option Price with Dividend Yield Adjustment: {call_option_price_with_dividend:.2f}")
print(f"Put Option Price with Dividend Yield Adjustment: {put_option_price_with_dividend:.2f}")Output
Call Option Price with Dividend Yield Adjustment: 4.08
Put Option Price with Dividend Yield Adjustment: 13.59
