Pricing A Barrier Option on FX Rate by Monte Carlo Simulation

To price a barrier option on FX rate by Monte Carlo simulation, you can follow these steps:

1. Define the basic parameters of the barrier option, including the spot FX rate, strike price, option expiration date, barrier level, barrier type (up/down), barrier monitoring frequency, volatility, and risk-free rate. Store these parameters in variables.

2. Generate the simulated FX rate paths. Use a random number generator (such as the normal distribution) to generate a set of random shocks to the FX rate at each time step. Use these random shocks to simulate a set of possible FX rate paths for the option's life. Store these paths in a matrix.

3. Determine if the barrier has been breached for each simulated path. At each monitoring frequency, check if the FX rate has crossed the barrier level. If it has, take note of the time and location of the first breach.

4. Calculate the payoff for each simulated path. If the FX rate breached the barrier before the option expiration, the option expires worthless. If the FX rate did not breach the barrier before the option expiration, the option payoff is the maximum of 0 and the difference between the FX rate and the strike price.

5. Discount the payoff to the present value. Use the risk-free rate and the option's time to expiration to calculate the discount factor and present value for each simulated path.

6. Calculate the option price. Take the average of all the present values calculated in step 5 to get the option price.

Here's an example of how to implement these steps in Python:

import numpy as np

# Define basic parameters
S0 = 1.2  # Spot FX rate
K = 1.3   # Strike price
T = 1     # Time to expiration
B = 1.1   # Barrier level
barrier_type = 'up'  # Barrier type
monitoring_freq = 10 # Barrier monitoring frequency
sigma = 0.2  # Volatility
r = 0.05    # Risk-free rate
N = 10000   # Number of simulations
dt = 1/252  # Time step

# Generate the simulated FX rate paths
ST = np.zeros((N, monitoring_freq+1))
ST[:,0] = S0
for i in range(N):
    for j in range(1, monitoring_freq+1):
        ST[i,j] = ST[i,j-1] * np.exp((r-sigma**2/2)*dt + sigma*np.sqrt(dt)*np.random.normal())

# Determine if the barrier has been breached for each simulated path
breached = np.zeros(N, dtype=bool)
time_to_breach = np.zeros(N)
for i in range(N):
    for j in range(1, monitoring_freq+1):
        if barrier_type == 'up':
            if ST[i,j] > B:
                breached[i] = True
                time_to_breach[i] = j
                break
        else:
            if ST[i,j] < B:
                breached[i] = True
                time_to_breach[i] = j
                break

# Calculate the payoff for each simulated path
payoff = np.zeros(N)
for i in range(N):
    if breached[i]:
        payoff[i] = 0
    else:
        payoff[i] = max(0, ST[i,-1] - K)

# Discount the payoff to the present value
df = np.exp(-r*T)
pv = payoff * df

# Calculate the option price
price = np.mean(pv)

This code generates random FX rate shocks at each time step to simulate possible FX rate paths for the option's life. It then checks if the barrier is breached for each path and calculates the payoff for each path. Finally, it discounts the payoff to the present value and calculates the option price as the average present value across all paths. Note that this is a simplified example and more advanced techniques may be required to get accurate option prices.