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Heston Stochastic Volatility Stochastic Process
import math
import numpy
import random
import decimal
import scipy.linalg
import numpy.random as nrand
import matplotlib.pyplot as plt
Note that this Gist uses the Model Parameters class found here -
def cox_ingersoll_ross_heston(param):
This method returns the rate levels of a mean-reverting cox ingersoll ross process. It is used to model interest
rates as well as stochastic volatility in the Heston model. Because the returns between the underlying and the
stochastic volatility should be correlated we pass a correlated Brownian motion process into the method from which
the interest rate levels are constructed. The other correlated process is used in the Heston model
:param param: the model parameters objects
:return: the interest rate levels for the CIR process
# We don't multiply by sigma here because we do that in heston
sqrt_delta_sigma = math.sqrt(param.all_delta) * param.all_sigma
brownian_motion_volatility = nrand.normal(loc=0, scale=sqrt_delta_sigma, size=param.all_time)
a, mu, zero = param.heston_a, param.heston_mu, param.heston_vol0
volatilities = [zero]
for i in range(1, param.all_time):
drift = a * (mu - volatilities[i-1]) * param.all_delta
randomness = math.sqrt(volatilities[i - 1]) * brownian_motion_volatility[i - 1]
volatilities.append(volatilities[i - 1] + drift + randomness)
return numpy.array(brownian_motion_volatility), numpy.array(volatilities)
def heston_model_levels(param):
NOTE - this method is dodgy! Need to debug!
The Heston model is the geometric brownian motion model with stochastic volatility. This stochastic volatility is
given by the cox ingersoll ross process. Step one on this method is to construct two correlated GBM processes. One
is used for the underlying asset prices and the other is used for the stochastic volatility levels
:param param: model parameters object
:return: the prices for an underlying following a Heston process
assert isinstance(param, ModelParameters)
# Get two correlated brownian motion sequences for the volatility parameter and the underlying asset
# brownian_motion_market, brownian_motion_vol = get_correlated_paths_simple(param)
brownian, cir_process = cox_ingersoll_ross_heston(param)
brownian, brownian_motion_market = heston_construct_correlated_path(param, brownian)
heston_market_price_levels = [param.all_s0]
for i in range(1, param.all_time):
drift = param.gbm_mu * heston_market_price_levels[i - 1] * param.all_delta
vol = cir_process[i - 1] * heston_market_price_levels[i - 1] * brownian_motion_market[i - 1]
heston_market_price_levels.append(heston_market_price_levels[i - 1] + drift + vol)
return numpy.array(heston_market_price_levels), numpy.array(cir_process)
def heston_construct_correlated_path(param, brownian_motion_one):
This method is a simplified version of the Cholesky decomposition method for just two assets. It does not make use
of matrix algebra and is therefore quite easy to implement.
:param param: model parameters object
:return: a correlated brownian motion path
# We do not multiply by sigma here, we do that in the Heston model
sqrt_delta = math.sqrt(param.all_delta)
# Construct a path correlated to the first path
brownian_motion_two = []
for i in range(param.all_time - 1):
term_one = param.cir_rho * brownian_motion_one[i]
term_two = math.sqrt(1 - math.pow(param.cir_rho, 2.0)) * random.normalvariate(0, sqrt_delta)
brownian_motion_two.append(term_one + term_two)
return numpy.array(brownian_motion_one), numpy.array(brownian_motion_two)
def get_correlated_geometric_brownian_motions(param, correlation_matrix, n):
This method can construct a basket of correlated asset paths using the Cholesky decomposition method
:param param: model parameters object
:param correlation_matrix: nxn correlation matrix
:param n: the number of assets i.e. the number of paths to return
:return: n correlated log return geometric brownian motion processes
assert isinstance(param, ModelParameters)
decomposition = scipy.linalg.cholesky(correlation_matrix, lower=False)
uncorrelated_paths = []
sqrt_delta_sigma = math.sqrt(param.all_delta) * param.all_sigma
# Construct uncorrelated paths to convert into correlated paths
for i in range(param.all_time):
uncorrelated_random_numbers = []
for j in range(n):
uncorrelated_random_numbers.append(random.normalvariate(0, sqrt_delta_sigma))
uncorrelated_matrix = numpy.matrix(uncorrelated_paths)
correlated_matrix = uncorrelated_matrix * decomposition
assert isinstance(correlated_matrix, numpy.matrix)
# The rest of this method just extracts paths from the matrix
extracted_paths = []
for i in range(1, n + 1):
for j in range(0, len(correlated_matrix)*n - n, n):
for i in range(n):
extracted_paths[i].append(correlated_matrix.item(j + i))
return extracted_paths
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