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def price_options(S=100.0, K=100.0, sigma=0.25, r=0.05, days=260, paths=10000):
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"""
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Price European and Asian options using a Monte Carlo method.
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Parameters
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----------
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S : float
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The initial price of the stock.
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K : float
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The strike price of the option.
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sigma : float
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The volatility of the stock.
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r : float
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The risk free interest rate.
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days : int
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The number of days until the option expires.
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paths : int
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The number of Monte Carlo paths used to price the option.
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Returns
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-------
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A tuple of (E. call, E. put, A. call, A. put) option prices.
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"""
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import numpy as np
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from math import exp,sqrt
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h = 1.0/days
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const1 = exp((r-0.5*sigma**2)*h)
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const2 = sigma*sqrt(h)
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stock_price = S*np.ones(paths, dtype='float64')
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stock_price_sum = np.zeros(paths, dtype='float64')
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for j in range(days):
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growth_factor = const1*np.exp(const2*np.random.standard_normal(paths))
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stock_price = stock_price*growth_factor
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stock_price_sum = stock_price_sum + stock_price
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stock_price_avg = stock_price_sum/days
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zeros = np.zeros(paths, dtype='float64')
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r_factor = exp(-r*h*days)
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euro_put = r_factor*np.mean(np.maximum(zeros, K-stock_price))
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asian_put = r_factor*np.mean(np.maximum(zeros, K-stock_price_avg))
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euro_call = r_factor*np.mean(np.maximum(zeros, stock_price-K))
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asian_call = r_factor*np.mean(np.maximum(zeros, stock_price_avg-K))
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return (euro_call, euro_put, asian_call, asian_put)
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