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Call option pricing using KDE estimate instead of the normality assumption
"""
MONTE CARLO PLAIN VANILLA OPTION PRICING
This script is used to estimate the price of a plain vanilla
option using the Monte Carlo method and assuming that returns
can be simulated using an estimated probability density (KDE estimate)
Call option quotations are available at:
http://www.google.com/finance/option_chain?q=NASDAQ%3AAAPL&ei=fNHBVaicDsbtsAHa7K-QDQ
Risk free* rates can be found here:
http://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?data=yield
In this script the following assumptions are made:
- Returns are not exactly normally distributed, but they following
a historical distribution that can be estimated from the historical prices.
Therefore, the price of the stock at time t+1
can be assumed to be:
s1 = s0 + s0*K
where: t=1 and K is a random variable whose density is described by the
empirical estimated density of the actual stock returns.
*I choose these as risk free (even though they are not), the concept of
risk free may be subjective.
"""
import pickle
import numpy as np
# Load the estimated probability density
pickle_in = open("KDE_","rb")
kde_dist = pickle.load(pickle_in)
pickle_in.close()
# Check that the total area under the probability density sums up to 1
print("Area below the curve from -10 to 10:",kde_dist.integrate_box(-10,10),'\n')
# Optional seed
#np.random.seed(12345678)
# Stocastic walk
# This function calculates the simulated price after periods
# and returns the final price.
def stoc_walk(p,periods):
w = kde_dist.resample(size=periods)[0] #compare to parametric: w = np.random.normal(0,1,size=periods)
for i in range(periods):
p += p*w[i] #compare to parametric: p += dr*p + w[i]*vol*p
return p
# Parameters
s0 = 114.64 # Actual price
t_ = 365 # Total periods in a year
r = 0.033 # Risk free rate (yearly)
days = 2 # Days until option expiration
N = 100000 # Number of Monte Carlo trials
zero_trials = 0 # Number of trials where the option payoff = 0
k=100 # Strike price
avg = 0 # Temporary variable to be assigned to the sum
# of the simulated payoffs
# Simulation loop
for i in range(N):
temp = stoc_walk(s0,days)
if temp > k:
payoff = temp-k
payoff = payoff*np.exp(-r/t_*days)
avg += payoff
else:
zero_trials += 1
# Averaging the payoffs
price = avg/float(N)
# Priting the results
print("MONTE CARLO PLAIN VANILLA CALL OPTION PRICING")
print("Option price: ",price)
print("Initial price: ",s0)
print("Strike price: ",k)
print("Total trials: ",N)
print("Zero trials: ",zero_trials)
print("Percentage of total trials: ",zero_trials/N*100,"%")
# Output
# Area below the curve from -10 to 10: 1.0
# MONTE CARLO PLAIN VANILLA CALL OPTION PRICING
# Option price: 15.1243063405
# Initial price: 114.64
# Strike price: 100
# Total trials: 100000
# Zero trials: 563
# Percentage of total trials: 0.563 %
# >>>
# Indeed 15.12 is closer to the real price of 14.65, we improved our model.
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