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delta_hedging_mc.py
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from math import log, sqrt, exp
import matplotlib.pyplot as plt
from scipy.stats import norm
import random
import pandas as pd
'''
k = strike price
s0 = initial stock price
dt = t/T = time to maturity
rf = remaining time
r = risk-less short rate
sig = volatility of stock value
m = the number of path nodes
n = the number of simulations
'''
class EuropeanCallOption:
def __init__(self, s0, k, r, sig, dt):
self.s0 = s0
self.k = k
self.r = r
d1_1 = self.d1(s0, k, r, sig, dt)
d2_1 = self.d2(d1_1, sig, dt)
self.dt = dt
self.value_1 = self.value(s0, k, r, dt, d1_1, d2_1)
self.delta_1 = self.delta(d1_1)
def d1(self, s0, k, r, sig, dt):
return (log(s0 / k) + (r + 0.5 * sig ** 2) * dt) / (sig * sqrt(dt))
def d2(self, d1, sig, dt):
return d1 - sig * sqrt(dt)
def value(self, s0, k, r, dt, d1, d2):
return s0 * exp(-r * dt) * norm.cdf(d1) - k * exp(-r * dt) * norm.cdf(d2)
def delta(self, d1):
return norm.cdf(d1)
class EuropeanPutOption:
def __init__(self, s0, k, r, sig, dt):
self.s0 = s0
self.k = k
self.r = r
d1_1 = self.d1(s0, k, r, sig, dt)
d2_1 = self.d2(d1_1, sig, dt)
self.dt = dt
self.value_1 = self.value(s0, k, r, dt, d1_1, d2_1)
self.delta_1 = self.delta(d1_1)
def d1(self, s0, k, r, sig, dt):
return (log(s0 / k) + (r + 0.5 * sig ** 2) * dt) / (sig * sqrt(dt))
def d2(self, d1, sig, dt):
return d1 - sig * sqrt(dt)
def value(self, s0, k, r, dt, d1, d2):
return k * exp(-r * dt) * norm.cdf(-d2) - s0 * exp(-r * dt) * norm.cdf(-d1)
def delta(self, d1):
return norm.cdf(d1) - 1
class EuropeanOption:
def __init__(self, s0, k, dt, r, sig, m):
self.stock_1 = self.stock_list(s0, dt, r, sig, m)
self.strike_1 = self.strike_list(k, m)
def stock_list(self, s0, dt, r, sig, m):
delta_t = dt / m # length of time interval
path = [s0]
for j in range(1, m):
path.append(path[-1] * exp((r - 0.5 * sig ** 2) * delta_t + (sig * sqrt(delta_t) * random.gauss(0, 1))))
stock = path
return stock
def strike_list(self, k, m):
strike = [k]
for i in range(1, m):
strike.append(k)
return strike
class Hedging:
def __init__(self, name, s0, k, dt, r, sig, m, towards, number, start_date, end_date):
sth = EuropeanOption(s0, k, dt, r, sig, m)
s_l = sth.stock_1
self.fixed_stock_path_1 = self.fixed_stock_path(s_l)
self.total_delta_1 = self.total_delta(name, s_l, k, r, sig, m, start_date, end_date)
total_delta_2 = self.total_delta(name, s_l, k, r, sig, m, start_date, end_date)
self.underlying_position_1 = self.underlying_position(total_delta_2, m, towards, number)
underlying_position_2 = self.underlying_position(total_delta_2, m, towards, number)
self.pol_1 = self.pol(s_l, underlying_position_2, m)
self.totaling_value_1 = self.totaling_value(name, s_l, k, r, sig, m, start_date, end_date)
def fixed_stock_path(self, s_l):
return s_l
def total_delta(self, name, s_l, k, r, sig, m, start_date, end_date):
# s_l = stock list
# name: "c"=call, "p"=put
delta_number = []
for i in range(0, m):
current_s = s_l[i]
s0 = current_s
t = pd.bdate_range(start_date, end_date)
dt = (len(t) / 250) * (1 - i / m)
if name == "c":
total = EuropeanCallOption(s0, k, r, sig, dt)
delta_number.append(total.delta_1)
else:
total = EuropeanPutOption(s0, k, r, sig, dt)
delta_number.append(total.delta_1)
return delta_number
def underlying_position(self, total_delta, m, towards, number):
# towards: buy=1, sell=-1
total_position = []
for i in range(0, m):
b = total_delta[i]
position = -round(b * number, 0) * towards
total_position.append(position)
return total_position
def pol(self, s_l, underlying_position, m):
pol_in_stock = [0]
for i in range(1, m):
pol_in_stock.append(
(s_l[i] - s_l[i - 1]) * underlying_position[i - 1])
return pol_in_stock
def totaling_value(self, name, s_l, k, r, sig, m, start_date, end_date):
# name: "c"=call, "p"=put
total_value = []
for i in range(0, m):
s0 = s_l[i] # s0 = current s
t = pd.bdate_range(start_date, end_date)
dt = (len(t) / 250) * (1 - i / m)
if name == "c":
total = EuropeanCallOption(s0, k, r, sig, dt)
else:
total = EuropeanPutOption(s0, k, r, sig, dt)
total_value.append(total.value_1)
return total_value
def graph_show(name, s0, k, dt, r, sig, m, towards, number, start_date, end_date):
# graph simulation (underlying price monte carlo simulation)
one_path = Hedging(name, s0, k, dt, r, sig, m, towards, number, start_date, end_date)
list_stock = one_path.fixed_stock_path_1
fixed_strike = EuropeanOption(s0, k, dt, r, sig, m)
list_strike = fixed_strike.strike_1
plt.figure(figsize=(12, 6))
plt.grid(True) # 显示网格线
plt.xlabel('Time step')
plt.ylabel('price')
plt.legend()
plt.plot(list_stock)
plt.show()
# list show
list_value = one_path.totaling_value_1
list_delta = one_path.total_delta_1
list_position = one_path.underlying_position_1
list_pol = one_path.pol_1
my_dict = {'s': list_stock, 'k': list_strike, 'value': list_value, 'delta': list_delta,
'underlying_position': list_position,
'PoL_in_stock': list_pol}
total_list = pd.DataFrame(my_dict)
print(total_list)
def distribution(name, s0, k, dt, r, sig, m, n, towards, number, start_date, end_date):
# name: "c"=call, "p"=put
acc_value = []
acc_pol = []
for i in range(0, n):
# the final value - the initial value
one_list = Hedging(name, s0, k, dt, r, sig, m, towards, number, start_date, end_date)
list_stock = one_list.fixed_stock_path_1
s0 = list_stock[0]
t = pd.bdate_range(start_date, end_date)
dt = (len(t) / 250)
if name == "c":
total_0 = EuropeanCallOption(s0, k, r, sig, dt)
value_0 = total_0.value_1
else:
total_0 = EuropeanPutOption(s0, k, r, sig, dt)
value_0 = total_0.value_1
s0 = list_stock[-1]
t = pd.bdate_range(start_date, end_date)
dt = (len(t) / 250) * (1 / m)
if name == "c":
total_1 = EuropeanCallOption(s0, k, r, sig, dt)
value_1 = total_1.value_1
else:
total_1 = EuropeanPutOption(s0, k, r, sig, dt)
value_1 = total_1.value_1
acc_s = (value_1 - value_0) * number
acc_value.append(acc_s)
# accelerate PoL in stock
v = 0
for z in range(1, m):
a = one_list.underlying_position_1
b = (list_stock[z] - list_stock[z - 1]) * a[z - 1]
v = v + b
acc_pol.append(v)
# revenue = value - pol in stock
if len(acc_pol) == len(acc_value):
revenue = []
for x in range(0, n):
revenue_1 = acc_value[x] - acc_pol[x]
revenue.append(revenue_1)
my_dict_1 = {'revenue': revenue}
revenue_data = pd.DataFrame(my_dict_1)
plt.hist(revenue_data.revenue, bins=50, density=False)
plt.xlabel('Returns')
plt.ylabel('Frequency')
plt.title('Returns Best fit Distribution')
plt.show()
else:
print('please check acc_value and acc_pol')
pd.set_option('display.max_columns', None)
pd.set_option('display.max_rows', None)
# strike price = 100; initial stock price = 100; time to maturity = 20/250 = 0.08;
# risk-less short rate = 0.03; volatility of stock value = 0.2
# the number of path nodes = 20； the number of simulations = 100000
# type = call option; towards = buy; number of stock = 100
graph_show('c', 100, 100, 0.08, 0.03, 0.2, 20, 1, 100, "2021-7-05", "2021-7-31")
distribution('c', 100, 100, 0.08, 0.03, 0.2, 20, 100000, 1, 100, "2021-7-05", "2021-7-31")