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barrier_option.py
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barrier_option.py
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#-*-conding:utf-8-*-
import math
from scipy.stats.distributions import norm
import random
import numpy as np
def bsm_barrier_option(k, s, h, t, r, sig, rebate, payoff, n, oi):
# ================================================================================================
# equations from the complete guide to option pricing formulas, second edition. pp.125-126
# author and year: Espen Gaarder Haug (2018)
# ================================================================================================
"""
Parameters:
k = strike price
s = stock price
h = barrier price
t = t/T = time to maturity
r = risk-less short rate
sig = volatility of stock value
rebate = 退款
payoff = "in" or "out" options
oi: call = 1 put = -1
n：down = 1 up = -1
"""
miu = (- (sig ** 2) / 2) / (sig ** 2)
la = math.sqrt(miu ** 2 + 2 * r / (sig ** 2))
z = math.log(h / s, math.e) / (sig * math.sqrt(t)) + la * sig * math.sqrt(t)
x1 = math.log(s / k, math.e) / (sig * math.sqrt(t)) + (1 + miu) * sig * math.sqrt(t)
x2 = math.log(s / h, math.e) / (sig * math.sqrt(t)) + (1 + miu) * sig * math.sqrt(t)
y1 = math.log((h ** 2) / (s * k), math.e) / (sig * math.sqrt(t)) + (1 + miu) * sig * math.sqrt(t)
y2 = math.log(h / s, math.e) / (sig * math.sqrt(t)) + (1 + miu) * sig * math.sqrt(t)
A = oi * s * math.exp(-r * t) * norm.cdf(oi * x1) - oi * k * math.exp(-r * t) * norm.cdf(
oi * x1 - oi * sig * math.sqrt(t))
B = oi * s * math.exp(-r * t) * norm.cdf(oi * x2) - oi * k * math.exp(-r * t) * norm.cdf(
oi * x2 - oi * sig * math.sqrt(t))
C = oi * s * math.exp(-r * t) * (h / s) ** (2 * miu + 2) * norm.cdf(n * y1) - oi * k * math.exp(-r * t) * ((
h / s) ** (2 * miu)) * norm.cdf(n * y1 - n * sig * math.sqrt(t))
D = oi * s * math.exp(-r * t) * ((h / s) ** (2 * miu + 2)) * norm.cdf(n * y2) - oi * k * math.exp(-r * t) * ((
h / s) ** (2 * miu)) * norm.cdf(n * y2 - n * sig * math.sqrt(t))
E = rebate * math.exp(-r * t) * (norm.cdf(n * x2 - n * sig * math.sqrt(t)) - ((h / s) ** (2 * miu)) * norm.cdf(
n * y2 - n * sig * math.sqrt(t)))
F = rebate * (((h / s) ** (miu + la)) * norm.cdf(n * z) + ((h / s) ** (miu - la)) * norm.cdf(
n * z - 2 * n * la * sig * math.sqrt(t)))
value = 0
if payoff == "in":
if n == 1 and oi == 1: # down-and-in call options
if k > h:
value = C + E
else:
value = A - B + D + E
elif n == -1 and oi == 1: # up-and-in call options
if k > h:
value = A + E
else:
value = B - C + D + E
elif n == 1 and oi == -1: # down-and-in put options
if k > h:
value = B - C + D + E
else:
value = A + E
elif n == -1 and oi == -1: # up-and-in put options
if k > h:
value = A - B + D + E
else:
value = C + E
else:
if n == 1 and oi == 1: # down-and-out call options
if k > h:
value = A - C + F
else:
value = B - D + F
elif n == -1 and oi == 1: # up-and-out call options向上敲出认购
if k > h:
value = F
else:
value = A - B + C - D + F
elif n == 1 and oi == -1: # down-and-out put options
if k > h:
value = A - B + C - D + F
else:
value = F
elif n == -1 and oi == -1: # up-and-out put options
if k > h:
value = B - D + F
else:
value = A - C + F
return value
def mc_barrier_option(s0, t, r, k, sig, m, n, h, rebate, towards, w, x):
"""
Parameters:
s0 = initial stock price
t = t/T = time to maturity
r = risk-less short rate
k = strike price
sig = volatility of stock value
m = the number of path nodes
n = the number of simulation
h = barrier price
rebate = 退款
towards: 1=up -1=down
w: 1=call； -1=put
x: 1=in； -1=out
"""
delta_t = t / m # length of time interval
list_price = []
for i in range(0, n):
path = [s0]
for j in range(0, m):
path.append(path[-1]*math.exp((r-0.5*sig**2)*delta_t+(sig*math.sqrt(delta_t)*random.gauss(0, 1))))
# it is based on the definition of standard barrier option
if towards == -1:
if (x == 1 and min(path) <= h) or (x == -1 and min(path) > h):
price = max(math.exp(-r * delta_t)*(w * path[-1] - w * k), 0)
else:
price = rebate
else:
if (x == 1 and max(path) >= h) or (x == -1 and max(path) < h):
price = max(math.exp(-r * delta_t)*(w * path[-1] - w * k), 0)
else:
price = rebate
list_price.append(price)
value = np.average(list_price)
return value
'''
trial:
analytical_solution = bsm_barrier_option(100, 100, 105, 0.08, 0.03, 0.2, 0, "out", -1, 1)
arithmetic_solution = mc_barrier_option(100, 0.08, 0.03, 100, 0.2, 100, 100000, 105, 0, 1, 1, -1)
print("analytical_solution = ", analytical_solution)
print("arithmetic_solution = ", arithmetic_solution)
'''