strategy-lab/to_explore/pyquantnews/31_BarrierOptionPricing.ipynb

This code simulates stock returns using Geometric Brownian Motion (GBM) and calculates the premium for a barrier option. It defines a function to simulate GBM paths based on initial stock price, drift, volatility, and other parameters. The code then generates multiple simulated price paths, checks if the maximum value of each path exceeds a barrier, and calculates the option payoff accordingly. Lastly, it discounts the payoffs to present value and computes the average premium. This is useful for pricing exotic financial derivatives and analyzing risk.

In [ ]:

import numpy as np
import matplotlib.pyplot as plt

In [ ]:

def simulate_gbm(s_0, mu, sigma, T, N, n_sims=10**3, random_seed=42):
    """
    Function used for simulating stock returns using Geometric Brownian Motion.

    Parameters
    ------------
    s_0 : float
        Initial stock price
    mu : float
        Drift coefficient
    sigma : float
        Diffusion coefficient
    T : float
        Length of the forecast horizon, same unit as dt
    N : int
        Number of time increments in the forecast horizon
    n_sims : int
        Number of simulation paths
    random_seed : int
        Random seed for reproducibility

    Returns
    -----------
    S_t : np.ndarray
        Matrix (size: n_sims x (T+1)) containing the simulation results.
        Rows represent sample paths, while columns point of time.
    """

    # Set random seed for reproducibility
    np.random.seed(random_seed)

    # Calculate time increment
    dt = T / N

    # Generate normally distributed random values for the Wiener process
    dW = np.random.normal(scale=np.sqrt(dt), size=(n_sims, N + 1))

    # Simulate the evolution of the process using GBM formula
    S_t = s_0 * np.exp(np.cumsum((mu - 0.5 * sigma**2) * dt + sigma * dW, axis=1))
    S_t[:, 0] = s_0

    return S_t

Define initial stock price, drift, volatility, time horizon, and number of increments

In [ ]:

S_0 = 55
r = 0.06
sigma = 0.2
T = 1
N = 252

Define barrier level and strike price for the option

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BARRIER = 65
K = 60

Generate GBM simulations for the given parameters

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gbm_sims = simulate_gbm(s_0=S_0, mu=r, sigma=sigma, T=T, N=N)

Plot the simulated price paths and the barrier level

In [ ]:

plt.axhline(y=BARRIER, color='r', linestyle='-')
plt.xlim(0, N)
plt.plot(gbm_sims.T, linewidth=0.25)

Calculate the maximum value per path to determine if the barrier was breached

In [ ]:

max_value_per_path = np.max(gbm_sims, axis=1)

Calculate the payoff of the barrier option based on the barrier breach condition

In [ ]:

payoff = np.where(
    max_value_per_path > BARRIER, 
    np.maximum(0, gbm_sims[:, -1] - K), 
    0
)

Calculate the discount factor and the average premium for the option

In [ ]:

discount_factor = np.exp(-r * T)
premium = discount_factor * np.mean(payoff)
premium

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