Black-Scholes Model

The Black-Scholes model tells us that if there are no arbitrage opportunities in the market and stock prices exhibit certain random fluctuations, we can calculate the ‘fair value’ of a European option.
European options are the most common type of option, where the two parties agree on an expiration date before settling profits or losses.
American options are more flexible and therefore usually more expensive than European options, especially when stock prices are highly volatile or there are dividends, making early exercise potentially more worthwhile.
The Black-Scholes model only applies to European options and can use a closed-form solution. It does not apply to American options, requiring other methods for calculation.
This model allows us to mathematically convert the randomness of future stock price movements into the fair value of options today (more intuitive).
Black-Scholes call and put options are derived from put-call parity:

\[C - P = S - K e^{-r(T-t)}\]

Put Option

\[P = K e^{-r(T-t)} \Phi(-d_2) - S \Phi(-d_1)\]

Call option

\[C = S \Phi(d_1) - K e^{-r(T-t)} \Phi(d_2)\]

where \(d_1 = \frac{\ln(S/K) + (r + 0.5\sigma^2)(T-t)}{\sigma\sqrt{T-t}}, \quad d_2 = d_1 - \sigma \sqrt{T-t}\)

# Import packages

import numpy as np

import matplotlib.pyplot as plt

from scipy.stats import norm

# Define call option function

def bs_call(S, K, T, r, sigma):

d1 = (np.log(S/K) + (r + 0.5 * sigma ** 2) * T) / (sigma * np.sqrt(T)) 
d2 = d1 - sigma * sigma * np.sqrt(T) 
return S * norm.cdf(d1) - K * np.exp(-r * T) * norm.cdf(d2)

# Define put option function
def bs_put(S, K, T, r, sigma): 
d1 = (np.log(S/K) + (r + 0.5 * sigma ** 2) * T) / (sigma * np.sqrt(T)) 
d2 = d1 - sigma * sigma * np.sqrt(T) 
return K * np.exp(-r * T) * norm.cdf(-d2) - S * norm.cdf(-d1)

# Draw Black-Scholes Call Option Price
S = np.linspace(50, 150, 100)
K=100
T=1
r = 0.05
sigma=0.2

prices = [bs_call(s, K, T, r, sigma) for s in S]

plt.plot(S, prices)
plt.xlabel("Stock Price S")
plt.ylabel("Call Option Price C")
plt.title("Black-Scholes Call Option Price")
plt.grid(True)
plt.show()

# Draw Black-Scholes Call Option Price
S = np.linspace(50, 150, 100)
K=100
T=1
r = 0.05
sigma=0.2

prices = [bs_put(s, K, T, r, sigma) for s in S]

plt.plot(S, prices, 'red')
plt.xlabel("Stock Price S")
plt.ylabel("Put Option Price C")
plt.title("Black-Scholes Put Option Price")
plt.grid(True)
plt.show()

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