Self-study and price China’s snowball option (雪球期权) based on what you learned from our course, and provide at least one numerical example.please show me the python code

Sure, I can help you with that.

First, let me briefly explain what a snowball option is. It is a type of option where the strike price is set at the beginning of the option period but the payoff is based on the cumulative return of an underlying asset over a fixed period of time. In other words, the payoff of a snowball option depends on the performance of the underlying asset over the option period.

To price a snowball option, we need to use a simulation approach. One common method is to use Monte Carlo simulation. The steps are as follows:

1. Generate random paths for the underlying asset based on a given stochastic process (e.g. geometric Brownian motion).
2. Calculate the cumulative return for each path over the option period.
3. Calculate the payoff of the option based on the cumulative returns.
4. Repeat steps 1-3 many times to obtain a distribution of payoffs.
5. Discount the expected payoff to obtain the option price.

Here is an example Python code to price a snowball option:

import numpy as np

# Parameters
S0 = 100 # initial stock price
T = 1 # option period (in years)
r = 0.05 # risk-free rate
sigma = 0.2 # volatility
K = 100 # strike price
n_sim = 100000 # number of simulations

# Simulation
dt = 1/252 # time step (in years)
n_steps = int(T/dt) # number of time steps
S = np.zeros((n_sim, n_steps+1))
S[:,0] = S0
for i in range(n_steps):
    dW = np.random.normal(0, np.sqrt(dt), size=n_sim)
    S[:,i+1] = S[:,i] * np.exp((r - 0.5*sigma**2)*dt + sigma*dW)

# Cumulative returns
R = np.cumprod(S, axis=1)[:,-1]

# Payoff
P = np.maximum(R - K, 0)

# Option price
C = np.exp(-r*T) * np.mean(P)

print("Option price:", C)

In this example, I have assumed a geometric Brownian motion process for the underlying asset. The simulation generates 100,000 random paths for the stock price over the option period, and calculates the cumulative return for each path. The payoff of the option is the maximum of the cumulative return minus the strike price, or zero if the cumulative return is less than the strike price. The option price is the discounted expected payoff.

Note that this is just a simple example, and there are many ways to modify or improve the simulation approach depending on the specific requirements of the snowball option.