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# Imports
import numpy as np
from scipy.stats import norm
from .Base import OptionPricingModel
class BlackScholesModel(OptionPricingModel):
"""
Class implementing calculation for European option price using Black-Scholes Formula.
Call/Put option price is calculated with following assumptions:
- European option can be exercised only on maturity date.
- Underlying stock does not pay divident during option's lifetime.
- The risk free rate and volatility are constant.
- Efficient Market Hypothesis - market movements cannot be predicted.
- Lognormal distribution of underlying returns.
"""
def __init__(self, underlying_spot_price, strike_price, days_to_maturity, risk_free_rate, sigma):
"""
Initializes variables used in Black-Scholes formula .
underlying_spot_price: current stock or other underlying spot price
strike_price: strike price for option cotract
days_to_maturity: option contract maturity/exercise date
risk_free_rate: returns on risk-free assets (assumed to be constant until expiry date)
sigma: volatility of the underlying asset (standard deviation of asset's log returns)
"""
self.S = underlying_spot_price
self.K = strike_price
self.T = days_to_maturity / 365
self.r = risk_free_rate
self.sigma = sigma
def _calculate_call_option_price(self):
"""
Calculates price for call option according to the formula.
Formula: S*N(d1) - PresentValue(K)*N(d2)
"""
# cumulative function of standard normal distribution (risk-adjusted probability that the option will be exercised)
d1 = (np.log(self.S / self.K) + (self.r + 0.5 * self.sigma ** 2) * self.T) / (self.sigma * np.sqrt(self.T))
# cumulative function of standard normal distribution (probability of receiving the stock at expiration of the option)
# d2 = (d1 - (sigma * sqrt(T)))
d2 = (np.log(self.S / self.K) + (self.r - 0.5 * self.sigma ** 2) * self.T) / (self.sigma * np.sqrt(self.T))
return (self.S * norm.cdf(d1, 0.0, 1.0) - self.K * np.exp(-self.r * self.T) * norm.cdf(d2, 0.0, 1.0))
def _calculate_put_option_price(self):
"""
Calculates price for put option according to the formula.
Formula: PresentValue(K)*N(-d2) - S*N(-d1)
"""
# cumulative function of standard normal distribution (risk-adjusted probability that the option will be exercised)
d1 = (np.log(self.S / self.K) + (self.r + 0.5 * self.sigma ** 2) * self.T) / (self.sigma * np.sqrt(self.T))
# cumulative function of standard normal distribution (probability of receiving the stock at expiration of the option)
# d2 = (d1 - (sigma * sqrt(T)))
d2 = (np.log(self.S / self.K) + (self.r - 0.5 * self.sigma ** 2) * self.T) / (self.sigma * np.sqrt(self.T))
return (self.K * np.exp(-self.r * self.T) * norm.cdf(-d2, 0.0, 1.0) - self.S * norm.cdf(-d1, 0.0, 1.0)) |