def get_path_multi_step(
t0: float, t1: float, parameters: Types.ndarray, f0: float, v0: float,
no_paths: int, no_time_steps: int,
type_random_number: Types.TYPE_STANDARD_NORMAL_SAMPLING, rnd_generator,
**kwargs) -> map:
nu_1 = parameters[0]
nu_2 = parameters[1]
theta = parameters[2]
rho = parameters[3]
rho_inv = np.sqrt(1.0 - rho * rho)
no_paths = 2 * no_paths if type_random_number == Types.TYPE_STANDARD_NORMAL_SAMPLING.ANTITHETIC else no_paths
t_i = get_time_steps(t0, t1, no_time_steps, **kwargs)
no_time_steps = len(t_i)
delta_t_i = np.diff(t_i)
x_t = np.empty((no_paths, no_time_steps))
v_t = np.empty((no_paths, no_time_steps))
int_variance_t_i = np.empty((no_paths, no_time_steps - 1))
x_t[:, 0] = np.log(f0)
v_t[:, 0] = v0
v_t_1_i_1 = np.empty(no_paths)
v_t_2_i_1 = np.empty(no_paths)
v_t_1_i_1[:] = v0
v_t_2_i_1[:] = v0
map_output = {}
for i_step in range(1, no_time_steps):
z_i = rnd_generator.normal(0.0, 1.0, no_paths, type_random_number)
z_sigma = rnd_generator.normal(0.0, 1.0, no_paths, type_random_number)
v_t[:, i_step] = get_variance_sampling(t_i[i_step - 1], t_i[i_step],
v_t_1_i_1, v_t_2_i_1, theta,
nu_1, nu_2, z_sigma)
int_variance_t_i[:, i_step - 1] = 0.5 * (
v_t[:, i_step - 1] * delta_t_i[i_step - 1] +
v_t[:, i_step - 1] * delta_t_i[i_step - 1])
x_t[:, i_step] = x_t[:, i_step - 1] - 0.5 * int_variance_t_i[:, i_step - 1] + \
AnalyticTools.dot_wise(np.sqrt(v_t[:, i_step - 1]),
(rho * z_sigma + rho_inv * z_i) * np.sqrt(delta_t_i[i_step - 1]))
map_output[Types.MIXEDLOGNORMAL_OUTPUT.SPOT_VARIANCE_PATHS] = v_t
map_output[Types.MIXEDLOGNORMAL_OUTPUT.TIMES] = t_i
map_output[
Types.MIXEDLOGNORMAL_OUTPUT.INTEGRAL_VARIANCE_PATHS] = int_variance_t_i
map_output[Types.MIXEDLOGNORMAL_OUTPUT.PATHS] = np.exp(x_t)
return map_output
def get_path_multi_step(
t0: float, t1: float, f0: float, no_paths: int, no_time_steps: int,
type_random_number: Types.TYPE_STANDARD_NORMAL_SAMPLING,
local_vol: Callable[[float, Types.ndarray],
Types.ndarray], rnd_generator, **kwargs) -> map:
no_paths = 2 * no_paths if type_random_number == Types.TYPE_STANDARD_NORMAL_SAMPLING.ANTITHETIC else no_paths
t_i = get_time_steps(t0, t1, no_time_steps, **kwargs)
no_time_steps = len(t_i)
t_i = np.linspace(t0, t1, no_time_steps)
delta_t_i = np.diff(t_i)
x_t = np.empty((no_paths, no_time_steps))
int_v_t = np.empty((no_paths, no_time_steps - 1))
v_t = np.empty((no_paths, no_time_steps))
x_t[:, 0] = np.log(f0)
v_t[:, 0] = local_vol(t0, x_t[:, 0])
sigma_i_1 = np.zeros(no_paths)
sigma_i = np.zeros(no_paths)
sigma_t = np.zeros(no_paths)
x_t_i_mean = np.zeros(no_paths)
map_output = {}
for i_step in range(1, no_time_steps):
z_i = rnd_generator.normal(0.0, 1.0, no_paths, type_random_number)
np.copyto(sigma_i_1, local_vol(t_i[i_step - 1], x_t[:, i_step - 1]))
np.copyto(x_t_i_mean,
x_t[:, i_step - 1] - 0.5 * np.power(sigma_i_1, 2.0))
np.copyto(sigma_i, local_vol(t_i[i_step], x_t_i_mean))
np.copyto(sigma_t, 0.5 * (sigma_i_1 + sigma_i))
v_t[:, i_step] = np.power(sigma_t, 2.0)
int_v_t[:, i_step - 1] = v_t[:, i_step] * delta_t_i[i_step - 1]
x_t[:, i_step] = np.add(
x_t[:, i_step - 1], -0.5 * v_t[:, i_step] * delta_t_i[i_step - 1] +
np.sqrt(delta_t_i[i_step - 1]) *
AnalyticTools.dot_wise(sigma_t, z_i))
map_output[Types.LOCAL_VOL_OUTPUT.TIMES] = t_i
map_output[Types.LOCAL_VOL_OUTPUT.PATHS] = np.exp(x_t)
map_output[Types.LOCAL_VOL_OUTPUT.SPOT_VARIANCE_PATHS] = v_t
map_output[Types.LOCAL_VOL_OUTPUT.INTEGRAL_VARIANCE_PATHS] = int_v_t
return map_output
rng = RNG.RndGenerator(seed)
paths = fBM.cholesky_method(t0, t1, z0, rng, hurst_parameter, no_paths,
int(no_time_steps * t1))
# Time steps to compute
t = np.linspace(t0, t1, int(no_time_steps * t1))
# Compute mean
empirical_mean = np.mean(paths, axis=0)
# Compute variance
empirical_variance = np.var(paths, axis=0)
exact_variance = [fBM.covariance(t_i, t_i, hurst_parameter) for t_i in t]
# Compute covariance
no_full_time_steps = len(t)
empirical_covariance = np.zeros(shape=(no_time_steps, no_full_time_steps))
exact_covariance = np.zeros(shape=(no_time_steps, no_full_time_steps))
for i in range(0, no_time_steps):
for j in range(0, i):
empirical_covariance[i, j] = np.mean(AnalyticTools.dot_wise(paths[:, i], paths[:, j])) - \
empirical_mean[i] * empirical_mean[j]
exact_covariance[i, j] = fBM.covariance(t[i], t[j], hurst_parameter)
empirical_covariance[j, i] = empirical_covariance[i, j]
exact_covariance[j, i] = exact_covariance[i, j]
error_covariance = np.max(np.abs(empirical_covariance - exact_covariance))
error_variance = np.max(np.abs(exact_variance - empirical_variance))
error_mean = np.max(np.abs(empirical_mean))
def get_path_multi_step(
t0: float, t1: float, parameters: Vector, f0: float, v0: float,
no_paths: int, no_time_steps: int,
type_random_numbers: Types.TYPE_STANDARD_NORMAL_SAMPLING,
rnd_generator) -> ndarray:
k = parameters[0]
theta = parameters[1]
epsilon = parameters[2]
rho = parameters[3]
no_paths = 2 * no_paths if type_random_numbers == Types.TYPE_STANDARD_NORMAL_SAMPLING.ANTITHETIC else no_paths
t_i = np.linspace(t0, t1, no_time_steps)
delta_t_i = np.diff(t_i)
f_t = np.empty((no_paths, no_time_steps))
f_t[:, 0] = f0
delta_weight = np.zeros(no_paths)
gamma_weight = np.zeros(no_paths)
var_weight = np.zeros(no_paths)
inv_variance = np.zeros(no_paths)
ln_x_t_paths = np.zeros(shape=(no_paths, no_time_steps))
int_v_t_paths = np.zeros(shape=(no_paths, no_time_steps - 1))
v_t_paths = np.zeros(shape=(no_paths, no_time_steps))
ln_x_t_paths[:, 0] = np.log(f0)
v_t_paths[:, 0] = v0
map_out_put = {}
for i in range(1, no_time_steps):
u_variance = rnd_generator.uniform(0.0, 1.0, no_paths)
z_f = rnd_generator.normal(0.0, 1.0, no_paths, type_random_numbers)
np.copyto(
v_t_paths[:, i],
VarianceMC.get_variance(k, theta, epsilon, 1.5, t_i[i - 1], t_i[i],
v_t_paths[:, i - 1], u_variance, no_paths))
np.copyto(
int_v_t_paths[:, i - 1],
HestonTools.get_integral_variance(t_i[i - 1], t_i[i],
v_t_paths[:, i - 1],
v_t_paths[:, i], 0.5, 0.5))
HestonTools.get_delta_weight(t_i[i - 1], t_i[i], v_t_paths[:, i - 1],
v_t_paths[:, i], z_f, delta_weight)
HestonTools.get_var_weight(t_i[i - 1], t_i[i], v_t_paths[:, i - 1],
v_t_paths[:, i], z_f, var_weight)
inv_variance += HestonTools.get_integral_variance(
t_i[i - 1], t_i[i], 1.0 / v_t_paths[:, i - 1],
1.0 / v_t_paths[:, i], 0.5, 0.5)
k0 = -delta_t_i[i - 1] * (rho * k * theta) / epsilon
k1 = 0.5 * delta_t_i[i - 1] * (
(k * rho) / epsilon - 0.5) - rho / epsilon
k2 = 0.5 * delta_t_i[i - 1] * (
(k * rho) / epsilon - 0.5) + rho / epsilon
k3 = 0.5 * delta_t_i[i - 1] * (1.0 - rho * rho)
np.copyto(
ln_x_t_paths[:, i],
ln_x_t_paths[:, i - 1] + k0 + k1 * v_t_paths[:, i - 1] +
k2 * v_t_paths[:, i] + np.sqrt(k3) * AnalyticTools.dot_wise(
np.sqrt(v_t_paths[:, i - 1] + v_t_paths[:, i]), z_f))
map_out_put[HESTON_OUTPUT.PATHS] = np.exp(ln_x_t_paths)
map_out_put[HESTON_OUTPUT.INTEGRAL_VARIANCE_PATHS] = int_v_t_paths
map_out_put[
HESTON_OUTPUT.DELTA_MALLIAVIN_WEIGHTS_PATHS_TERMINAL] = np.multiply(
delta_weight, 1.0 / (np.sqrt(1.0 - rho * rho) * t1 * f0))
map_out_put[HESTON_OUTPUT.SPOT_VARIANCE_PATHS] = v_t_paths
map_out_put[HESTON_OUTPUT.TIMES] = t_i
HestonTools.get_gamma_weight(delta_weight, var_weight, inv_variance, rho,
t1, gamma_weight)
map_out_put[
HESTON_OUTPUT.GAMMA_MALLIAVIN_WEIGHTS_PATHS_TERMINAL] = np.multiply(
gamma_weight, 1.0 / ((1.0 - rho * rho) * np.power(t1 * f0, 2.0)))
return map_out_put
def get_path_multi_step(t0: float, t1: float, parameters: Vector, f0: float,
no_paths: int, no_time_steps: int,
type_random_number: TYPE_STANDARD_NORMAL_SAMPLING,
rnd_generator, **kwargs) -> map:
alpha = parameters[0]
nu = parameters[1]
rho = parameters[2]
rho_inv = np.sqrt(1.0 - rho * rho)
no_paths = 2 * no_paths if type_random_number == TYPE_STANDARD_NORMAL_SAMPLING.ANTITHETIC else no_paths
t_i = get_time_steps(t0, t1, no_time_steps, **kwargs)
no_time_steps = len(t_i)
delta_t_i = np.diff(t_i)
s_t = np.empty((no_paths, no_time_steps))
sigma_t = np.empty((no_paths, no_time_steps))
int_v_t_paths = np.zeros(shape=(no_paths, no_time_steps - 1))
s_t[:, 0] = f0
sigma_t[:, 0] = alpha
int_sigma_t_i = np.empty((no_paths, no_time_steps - 1))
sigma_t_i_1 = np.empty(no_paths)
sigma_t_i_1[:] = alpha
delta_weight = np.zeros(no_paths)
gamma_weight = np.zeros(no_paths)
var_weight = np.zeros(no_paths)
inv_variance = np.zeros(no_paths)
map_output = {}
for i_step in range(1, no_time_steps):
z_i = rnd_generator.normal(0.0, 1.0, no_paths, type_random_number)
z_sigma = rnd_generator.normal(0.0, 1.0, no_paths, type_random_number)
sigma_t_i = get_vol_sampling(t_i[i_step - 1], t_i[i_step], sigma_t_i_1,
nu, z_sigma)
sigma_t[:, i_step] = sigma_t_i
int_sigma_t_i[:, i_step - 1] = 0.5 * (
sigma_t_i_1 * sigma_t_i_1 * delta_t_i[i_step - 1] +
sigma_t_i * sigma_t_i * delta_t_i[i_step - 1])
diff_sigma = (rho / nu) * (sigma_t_i - sigma_t_i_1)
noise_sigma = AnalyticTools.dot_wise(
np.sqrt(int_sigma_t_i[:, i_step - 1]), z_i)
SABRTools.get_delta_weight(t_i[i_step - 1], t_i[i_step], sigma_t_i_1,
sigma_t_i, z_sigma, delta_weight)
SABRTools.get_var_weight(t_i[i_step - 1], t_i[i_step], sigma_t_i_1,
sigma_t_i, z_sigma, var_weight)
inv_variance += SABRTools.get_integral_variance(
t_i[i_step - 1], t_i[i_step], 1.0 / sigma_t_i_1, 1.0 / sigma_t_i,
0.5, 0.5)
np.copyto(
int_v_t_paths[:, i_step - 1],
SABRTools.get_integral_variance(t_i[i_step - 1], t_i[i_step],
sigma_t_i_1, sigma_t_i, 0.5, 0.5))
sigma_t_i_1 = sigma_t_i.copy()
s_t[:, i_step] = AnalyticTools.dot_wise(
s_t[:, i_step - 1],
np.exp(-0.5 * int_sigma_t_i[:, i_step - 1] + diff_sigma +
rho_inv * noise_sigma))
map_output[
SABR_OUTPUT.DELTA_MALLIAVIN_WEIGHTS_PATHS_TERMINAL] = delta_weight
map_output[SABR_OUTPUT.PATHS] = s_t
map_output[SABR_OUTPUT.INTEGRAL_VARIANCE_PATHS] = int_v_t_paths
map_output[SABR_OUTPUT.INTEGRAL_VARIANCE_PATHS] = int_sigma_t_i
map_output[SABR_OUTPUT.SIGMA_PATHS] = sigma_t
map_output[SABR_OUTPUT.TIMES] = t_i
SABRTools.get_gamma_weight(delta_weight, var_weight, inv_variance, rho, t1,
gamma_weight)
map_output[
SABR_OUTPUT.GAMMA_MALLIAVIN_WEIGHTS_PATHS_TERMINAL] = np.multiply(
gamma_weight, 1.0 / ((1.0 - rho * rho) * np.power(t1 * f0, 2.0)))
return map_output