Module mean_variance_hedge.black_scholes
Implementations of functions for Black-Scholes European Options Pricing
Expand source code
"""
Implementations of functions for
Black-Scholes European Options Pricing
"""
import numpy as np
from numpy.random import default_rng
from scipy.stats import norm
from scipy.optimize import brentq
def generate_GBM_paths(n_samples, S0, T, r, sigma, dt, seed=2021):
"""
Exact simulation of GBM under the risk-neutral measure Q
+ n_samples: Number of paths to simulate
+ S0: Initial Stock Price
+ T: Timesteps
+ r: risk free rate
+ sigma: volatility
+ dt: Time increment, e.g. dt = 1/250 years. Ensure that sigma and dt are of the same scale
+ seed: seed for reproducibilitys
Returns:
+ tis: timesteps
+ Sts: n_samples * (T + 1) numpy array , in which each row corresponds to a stock path
Note: prices are generated under Q such that the drift is the risk free rate r
"""
rng = default_rng(seed)
Zs = rng.standard_normal((n_samples, T)) # Brownian Motion increments
Zs = np.hstack([np.zeros((n_samples, 1)), Zs])
tis = np.arange(T + 1) #0, 1 .. T
tis = np.tile(tis, n_samples).reshape(n_samples, T + 1) # [[0, 1.. T], [0, 1.. T]...]
# Sample paths
Sts = S0 * np.exp((r - 0.5 * sigma ** 2) * tis * dt + sigma * np.sqrt(dt) * np.cumsum(Zs, axis=1))
return tis, Sts
def BlackScholes(St, K, r, sigma, tau, flag):
"""
Inputs:
+ St: Current Price of the stock at time t
+ K: Strike Price
+ r: risk-free rate
+ sigma: Black-Scholes implied volatility
+ tau: time-to-maturity. Ensure that sigma, tau are in the same scale.
+ flag: 1 if call, 0 if put
Outputs:
Returns the Black-Scholes price
"""
d1 = (np.log(St/K) + (r + 0.5 * sigma ** 2) * (tau)) / (sigma * np.sqrt(tau))
d2 = d1 - sigma * np.sqrt(tau)
if flag == 1:
return norm.cdf(d1) * St - np.exp(-r * (tau)) * norm.cdf(d2) * K
else:
return np.exp(-r * (tau)) * K * norm.cdf(-d2) - St * norm.cdf(-d1)
def delta(St, K, r, sigma, tau, flag):
"""
Inputs:
+ St: Current Price of the stock at time t
+ K: Strike Price
+ r: risk-fre rate
+ sigma: Black-Scholes implied volatility
+ tau: time-to-maturity. Ensure that sigma, tau are in the same scale.
+ flag: 1 if call, 0 if put
Outputs:
Returns the Black-Scholes Delta
"""
d1 = (np.log(St/K) + (r + 0.5 * sigma ** 2) * (tau)) / (sigma * np.sqrt(tau))
if flag == 1:
return norm.cdf(d1)
else:
return -norm.cdf(-d1)
def vega(St, K, r, sigma, tau, flag):
"""
Inputs:
+ St: Current Price of the stock at time t
+ K: Strike Price
+ r: risk-free rate
+ sigma: Black-Scholes implied volatility
+ tau: time-to-maturity. Ensure that sigma, tau are in the same scale.
+ flag: 1 if call, 0 if put
Outputs:
Returns the Black-Scholes Vega
"""
d1 = (np.log(St/K) + (r + 0.5 * sigma ** 2) * (tau)) / (sigma * np.sqrt(tau))
return St * norm.pdf(d1) * np.sqrt(tau) / 100
def bsinv(price, St, K, r, tau, flag):
"""
Inputs:
+ price: price of the liability
+ St: current stock price
+ K: strike price
+ r: risk-free rate
+ tau: time to maturity
+ flag: 1 if call, 0 if put
Outputs:
+ imp_vol: Black Scholes implied volatility
"""
error = lambda s: BlackScholes(St, K, r, s, tau, flag) - price
imp_vol = brentq(error, 1e-9, 1e+9)
return imp_volFunctions
def BlackScholes(St, K, r, sigma, tau, flag)-
Inputs:
- St: Current Price of the stock at time t
- K: Strike Price
- r: risk-free rate
- sigma: Black-Scholes implied volatility
- tau: time-to-maturity. Ensure that sigma, tau are in the same scale.
- flag: 1 if call, 0 if put
Outputs:
Returns the Black-Scholes price
Expand source code
def BlackScholes(St, K, r, sigma, tau, flag): """ Inputs: + St: Current Price of the stock at time t + K: Strike Price + r: risk-free rate + sigma: Black-Scholes implied volatility + tau: time-to-maturity. Ensure that sigma, tau are in the same scale. + flag: 1 if call, 0 if put Outputs: Returns the Black-Scholes price """ d1 = (np.log(St/K) + (r + 0.5 * sigma ** 2) * (tau)) / (sigma * np.sqrt(tau)) d2 = d1 - sigma * np.sqrt(tau) if flag == 1: return norm.cdf(d1) * St - np.exp(-r * (tau)) * norm.cdf(d2) * K else: return np.exp(-r * (tau)) * K * norm.cdf(-d2) - St * norm.cdf(-d1) def bsinv(price, St, K, r, tau, flag)-
Inputs:
- price: price of the liability
- St: current stock price
- K: strike price
- r: risk-free rate
- tau: time to maturity
- flag: 1 if call, 0 if put
Outputs:
- imp_vol: Black Scholes implied volatility
Expand source code
def bsinv(price, St, K, r, tau, flag): """ Inputs: + price: price of the liability + St: current stock price + K: strike price + r: risk-free rate + tau: time to maturity + flag: 1 if call, 0 if put Outputs: + imp_vol: Black Scholes implied volatility """ error = lambda s: BlackScholes(St, K, r, s, tau, flag) - price imp_vol = brentq(error, 1e-9, 1e+9) return imp_vol def delta(St, K, r, sigma, tau, flag)-
Inputs:
- St: Current Price of the stock at time t
- K: Strike Price
- r: risk-fre rate
- sigma: Black-Scholes implied volatility
- tau: time-to-maturity. Ensure that sigma, tau are in the same scale.
- flag: 1 if call, 0 if put
Outputs:
Returns the Black-Scholes Delta
Expand source code
def delta(St, K, r, sigma, tau, flag): """ Inputs: + St: Current Price of the stock at time t + K: Strike Price + r: risk-fre rate + sigma: Black-Scholes implied volatility + tau: time-to-maturity. Ensure that sigma, tau are in the same scale. + flag: 1 if call, 0 if put Outputs: Returns the Black-Scholes Delta """ d1 = (np.log(St/K) + (r + 0.5 * sigma ** 2) * (tau)) / (sigma * np.sqrt(tau)) if flag == 1: return norm.cdf(d1) else: return -norm.cdf(-d1) def generate_GBM_paths(n_samples, S0, T, r, sigma, dt, seed=2021)-
Exact simulation of GBM under the risk-neutral measure Q
- n_samples: Number of paths to simulate
- S0: Initial Stock Price
- T: Timesteps
- r: risk free rate
- sigma: volatility
- dt: Time increment, e.g. dt = 1/250 years. Ensure that sigma and dt are of the same scale
- seed: seed for reproducibilitys
Returns:
- tis: timesteps
- Sts: n_samples * (T + 1) numpy array , in which each row corresponds to a stock path
Note: prices are generated under Q such that the drift is the risk free rate r
Expand source code
def generate_GBM_paths(n_samples, S0, T, r, sigma, dt, seed=2021): """ Exact simulation of GBM under the risk-neutral measure Q + n_samples: Number of paths to simulate + S0: Initial Stock Price + T: Timesteps + r: risk free rate + sigma: volatility + dt: Time increment, e.g. dt = 1/250 years. Ensure that sigma and dt are of the same scale + seed: seed for reproducibilitys Returns: + tis: timesteps + Sts: n_samples * (T + 1) numpy array , in which each row corresponds to a stock path Note: prices are generated under Q such that the drift is the risk free rate r """ rng = default_rng(seed) Zs = rng.standard_normal((n_samples, T)) # Brownian Motion increments Zs = np.hstack([np.zeros((n_samples, 1)), Zs]) tis = np.arange(T + 1) #0, 1 .. T tis = np.tile(tis, n_samples).reshape(n_samples, T + 1) # [[0, 1.. T], [0, 1.. T]...] # Sample paths Sts = S0 * np.exp((r - 0.5 * sigma ** 2) * tis * dt + sigma * np.sqrt(dt) * np.cumsum(Zs, axis=1)) return tis, Sts def vega(St, K, r, sigma, tau, flag)-
Inputs:
- St: Current Price of the stock at time t
- K: Strike Price
- r: risk-free rate
- sigma: Black-Scholes implied volatility
- tau: time-to-maturity. Ensure that sigma, tau are in the same scale.
- flag: 1 if call, 0 if put
Outputs:
Returns the Black-Scholes Vega
Expand source code
def vega(St, K, r, sigma, tau, flag): """ Inputs: + St: Current Price of the stock at time t + K: Strike Price + r: risk-free rate + sigma: Black-Scholes implied volatility + tau: time-to-maturity. Ensure that sigma, tau are in the same scale. + flag: 1 if call, 0 if put Outputs: Returns the Black-Scholes Vega """ d1 = (np.log(St/K) + (r + 0.5 * sigma ** 2) * (tau)) / (sigma * np.sqrt(tau)) return St * norm.pdf(d1) * np.sqrt(tau) / 100