Value exotic options with Python

When you think of exotic options, what comes to mind? Lamborghini or Ferrari, palm trees or ski chalet, yacht or helicopter?

What about path-dependent barrier options?

Quants spend a lot of time selling complex derivatives like barrier options to institutional clients. Unfortunately, people like you and me can’t trade them.

The good news?

Even if you’re not a quant, you can still use Python to value exotic options. Most of the time, valuing exotic options is not hard—when you have a step-by-step guide to help you. Valuing them helps get an idea of the type of work quants do.

In today’s newsletter, I’m going to give you a step-by-step guide to value an Up-and-In European barrier call option with Python.

Up-and-In options are a type of exotic option. That’s because they are more complex than the normal American or European options that trade on exchanges. (If you’re interested in trading options, The 46-Page Ultimate Guide to Pricing Options and Implied Volatility With Python, can help.)

Up-and-In options have both a strike price and a barrier level. If the underlying price reaches the barrier, the buyer has a chance to make money. Otherwise, the option expires worthless.

Specialized quants sell exotic options, including barrier options, to institutional clients. These clients use barrier options to manage unique risks or speculate on the direction of the underlying.

You might not be a derivatives trader, but if you know Python, you can value barrier options too.

Imports and set up

Import NumPy for the math and Matplotlib for the charts.

import numpy as np
import matplotlib.pyplot as plt

Barrier options are path dependent. That means the value of the option is dependent on the price path of the underlying. To create the price path, we need to simulate stock prices. In a previous newsletter issue, you learned how to simulate stock prices with geometric Brownian motion. We’ll use a simplified version here.

def simulate_gbm(s_0, mu, sigma, T, N, n_sims=10**3, random_seed=1):
    """Simulate stock returns using Geometric Brownian Motion."""
    
    np.random.seed(random_seed)

    dt = T / N

    dW = np.random.normal(scale=np.sqrt(dt), size=(n_sims, N + 1))

    # simulate the evolution of the process
    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

This function returns 1,000 simulated price paths.

Build the simulation

Start with setting everything up.

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

BARRIER = 65
K = 60

The simulation starts with a stock price of $55 (S_0), a return of 6% (r), and volatility of 20% (sigma). T is the time frame of the simulation (in this case one year) and N is the time step (in this case 252 trading days). Setting T and N this way simulates one price per day for a year. The barrier is set to $65 and the strike price is $60.

Run the simulation and plot the price paths and barrier.

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

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

Change the drift and sigma term and see what happens to the price paths. Both affect the value of the barrier option as you can see next.

Value the barrier option

First, find the maximum simulated price for each price path. Since this is an Up-and-In option, you just need to know if the maximum price reached the barrier level at any time.

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

If the option reaches the barrier, the payoff is the same as a plain vanilla European call option.

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

You can use NumPy’s where method to find the payoffs. The first argument is an array of true or false depending on whether the maximum value of a price path exceeds the barrier. Where it’s true, you get the maximum value of the last stock price in the simulation minus the strike price, or zero. That’s the same payoff as a plain vanilla European call option.

To value the option, discount the average of the payoffs back to today.

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

Change the barrier and strike price. What happens to the value of the option? What about when you change the volatility of the simulated stock prices?

You don’t need to be a quant to value exotic options. You just need Python.