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# author : S. Mandalia
# shivesh.mandalia@outlook.com
#
# date : March 19, 2020
"""
Path generator for underlying.
"""
import math
import random
from copy import deepcopy
from dataclasses import dataclass
from typing import List, Tuple
__all__ = ['PathGenerator']
@dataclass
class PathGenerator:
"""
Class for generating underlying prices using MC techniques.
Attributes
----------
S : Spot price.
r : Risk-free interest rate.
div : Dividend yield.
vol : Volatility.
net_r : Net risk free rate.
Methods
----------
generate(T)
Generate a random path {S_t1, S_t2, ..., S_tn}.
generate_antithetic(T)
Generate a random plus antithetic path
[{S_t1, S_t2, ..., S_tn}, {S'_t1, S'_t2, ..., S'_tn}].
Examples
----------
>>> from utils.path import PathGenerator
>>> path = PathGenerator(S=100., r=0.1, div=0.01, vol=0.3)
>>> print(path.generate(T=range(4)))
[100.0, 100.33539853588853, 122.76017088387074, 142.29540684005462]
>>> print(path.generate(T=range(4)))
[100.0, 73.03094019139712, 77.37310245438943, 66.54240939439934]
"""
S: float
r: float
div: float
vol: float
@property
def net_r(self) -> float:
"""Net risk free rate."""
return self.r - self.div
def generate(self, T: List[float]) -> List[float]:
"""
Generate a random path {S_t1, S_t2, ..., S_tn}.
Parameters
----------
T : Set of times {t1, t2, ..., tn} in years.
Returns
----------
spot_prices : Set of prices for the underlying {S_t1, S_t2, ..., S_tn}.
"""
# Calculate dt time differences
dts = [T[idx + 1] - T[idx] for idx in range(len(T) - 1)]
spot_prices = [0] * len(T)
spot_prices[0] = self.S
for idx, dt in enumerate(dts):
# Calculate the drift e^{(r - (1/2) σ²) Δt}
drift = math.exp((self.net_r - (1/2) * self.vol**2) * dt)
# Calculate the volatility term e^{σ √{Δt} N(0, 1)}
rdm_gauss = random.gauss(0, 1)
vol_term = math.exp(self.vol * math.sqrt(dt) * rdm_gauss)
# Calculate next spot price
S_t = spot_prices[idx] * drift * vol_term
spot_prices[idx + 1] = S_t
return spot_prices
def generate_antithetic(self, T: List[float]) -> Tuple[List[float],
List[float]]:
"""
Generate a random plus antithetic path
[{S_t1, S_t2, ..., S_tn}, {S'_t1, S'_t2, ..., S'_tn}].
Parameters
----------
T : Set of times {t1, t2, ..., tn} in years.
Returns
----------
prices_tuple : Set of prices for the underlying
[{S_t1, S_t2, ..., S_tn}, {S'_t1, S'_t2, ..., S'_tn}].
"""
# Calculate dt time differences
dts = [T[idx + 1] - T[idx] for idx in range(len(T) - 1)]
# Create data structures
spot_prices = [0] * len(T)
spot_prices[0] = self.S
a_spot_prices = deepcopy(spot_prices)
for idx, dt in enumerate(dts):
# Calculate the drift e^{(r - (1/2) σ²) Δt}
drift = math.exp((self.net_r - (1/2) * self.vol**2) * dt)
# Calculate the volatility term e^{σ √{Δt} N(0, 1)}
rdm_gauss = random.gauss(0, 1)
a_gauss = -rdm_gauss
vol_term = math.exp(self.vol * math.sqrt(dt) * rdm_gauss)
a_vol_term = math.exp(self.vol * math.sqrt(dt) * a_gauss)
# Calculate next spot price
S_t = spot_prices[idx] * drift * vol_term
a_S_t = a_spot_prices[idx] * drift * a_vol_term
# Add to data structure
spot_prices[idx + 1] = S_t
a_spot_prices[idx + 1] = a_S_t
return (spot_prices, a_spot_prices)
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