Parallel Monto-Carlo options pricing¶

%matplotlib inline
import matplotlib.pyplot as plt

import sys
import time
from IPython.parallel import Client
import numpy as np

price = 100.0  # Initial price
rate = 0.05  # Interest rate
days = 260  # Days to expiration
paths = 10000  # Number of MC paths
n_strikes = 6  # Number of strike values
min_strike = 90.0  # Min strike price
max_strike = 110.0  # Max strike price
n_sigmas = 5  # Number of volatility values
min_sigma = 0.1  # Min volatility
max_sigma = 0.4  # Max volatility

strike_vals = np.linspace(min_strike, max_strike, n_strikes)
sigma_vals = np.linspace(min_sigma, max_sigma, n_sigmas)

print "Strike prices: ", strike_vals
print "Volatilities: ", sigma_vals

def price_option(S=100.0, K=100.0, sigma=0.25, r=0.05, days=260, paths=10000):
    """
    Price European and Asian options using a Monte Carlo method.

    Parameters
    ----------
    S : float
        The initial price of the stock.
    K : float
        The strike price of the option.
    sigma : float
        The volatility of the stock.
    r : float
        The risk free interest rate.
    days : int
        The number of days until the option expires.
    paths : int
        The number of Monte Carlo paths used to price the option.

    Returns
    -------
    A tuple of (E. call, E. put, A. call, A. put) option prices.
    """
    import numpy as np
    from math import exp,sqrt
    
    h = 1.0/days
    const1 = exp((r-0.5*sigma**2)*h)
    const2 = sigma*sqrt(h)
    stock_price = S*np.ones(paths, dtype='float64')
    stock_price_sum = np.zeros(paths, dtype='float64')
    for j in range(days):
        growth_factor = const1*np.exp(const2*np.random.standard_normal(paths))
        stock_price = stock_price*growth_factor
        stock_price_sum = stock_price_sum + stock_price
    stock_price_avg = stock_price_sum/days
    zeros = np.zeros(paths, dtype='float64')
    r_factor = exp(-r*h*days)
    euro_put = r_factor*np.mean(np.maximum(zeros, K-stock_price))
    asian_put = r_factor*np.mean(np.maximum(zeros, K-stock_price_avg))
    euro_call = r_factor*np.mean(np.maximum(zeros, stock_price-K))
    asian_call = r_factor*np.mean(np.maximum(zeros, stock_price_avg-K))
    return (euro_call, euro_put, asian_call, asian_put)

%timeit -n1 -r1 print price_option(S=100.0, K=100.0, sigma=0.25, r=0.05, days=260, paths=10000)

rc = Client()

view = rc.load_balanced_view()

async_results = []

%%timeit -n1 -r1

for strike in strike_vals:
    for sigma in sigma_vals:
        # This line submits the tasks for parallel computation.
        ar = view.apply_async(price_option, price, strike, sigma, rate, days, paths)
        async_results.append(ar)

rc.wait(async_results)  # Wait until all tasks are done.

len(async_results)

results = [ar.get() for ar in async_results]

prices = np.empty(n_strikes*n_sigmas,
    dtype=[('ecall',float),('eput',float),('acall',float),('aput',float)]
)

for i, price in enumerate(results):
    prices[i] = tuple(price)

prices.shape = (n_strikes, n_sigmas)

plt.figure()
plt.contourf(sigma_vals, strike_vals, prices['ecall'])
plt.axis('tight')
plt.colorbar()
plt.title('European Call')
plt.xlabel("Volatility")
plt.ylabel("Strike Price")

plt.figure()
plt.contourf(sigma_vals, strike_vals, prices['acall'])
plt.axis('tight')
plt.colorbar()
plt.title("Asian Call")
plt.xlabel("Volatility")
plt.ylabel("Strike Price")

plt.figure()
plt.contourf(sigma_vals, strike_vals, prices['eput'])
plt.axis('tight')
plt.colorbar()
plt.title("European Put")
plt.xlabel("Volatility")
plt.ylabel("Strike Price")

plt.figure()
plt.contourf(sigma_vals, strike_vals, prices['aput'])
plt.axis('tight')
plt.colorbar()
plt.title("Asian Put")
plt.xlabel("Volatility")
plt.ylabel("Strike Price")