Write a Monte Carlo simulation in Python or R to price a European call option.?

Below is some Python code to perform Monte Carlo simulation to price a European call option. In this example, I’ll use the Black-Scholes model for stock price dynamics.

Assumptions for the Black-Scholes model are:

1. The stock price follows a Geometric Brownian Motion (GBM).

2. The risk-free rate is constant.

3. The stock’s volatility (sigma) is constant.

The equation for the stock price at time ‘t‘ following GBM is:

$$S_t = S_0 e^{(r - \frac{1}{2} \sigma^2) t + \sigma \sqrt{t} Z}$$

Where:

- S_t = stock price at time ‘t‘

- S₀ = initial stock price

- r = risk-free interest rate

- σ = stock’s volatility

- t = time period

- Z = standard Normal random variable

Now, let’s simulate to get the call option price. The call option payoff is:

Call Payoff = max(S_t − K, 0)

Where:

- K = strike price of the option

We will then calculate the expected payoff and discount it back to the present value, as the option price:

European Call Option Price = e^− rT * E[Call Payoff]

Here’s the Python code:

import numpy as np
import math

# Parameters
S0 = 100          # initial stock price
K = 110           # strike price
T = 1             # time to maturity (1 year)
r = 0.05          # risk-free interest rate
sigma = 0.20      # stock's volatility
num_simulations = 1000000  # number of Monte Carlo simulations

# Set a random seed for reproducibility
np.random.seed(1234)

# Generate stock prices at time T using GBM model
Z = np.random.normal(0, 1, num_simulations)
St = S0 * np.exp((r - 0.5 * sigma**2) * T + sigma * math.sqrt(T) * Z)

# Calculate the call option's payoff at time T
call_payoff = np.maximum(St - K, 0)

# Discount the expected payoff to obtain the option price
call_price = np.average(call_payoff) * np.exp(-r * T)

print("European Call Option Price:", call_price)

This code prices a European call option using Monte Carlo simulation based on the given parameters. You can change these parameters to get the option price under different conditions.