如何解决如何计算具有对立变量的估计量的标准误差
我使用 Longstaff 和 Schwartz 最小二乘法为美式期权定价。
当使用以下 Python 代码时,我获得的价格和标准误差与 通过模拟评估美式期权几乎相同: Longstaff 和 Schwartz 的简单最小二乘法(2001 年):
import numpy as np
import numpy.random as npr
import warnings
warnings.simplefilter('ignore')
from numpy.polynomial.laguerre import lagfit,lagval
# Same parameters as in the original paper
class par: pass
par.S0 = 36
par.K = 40
par.r = 0.06
par.sigma = 0.2
par.T = 1.0
par.I = 100000
par.M = 50
def gen_sn(par,anti_path):
''' Function to generate random numbers for simulation.
Parameters
==========
M : int
number of time intervals for discretization
I : int
number of paths to be simulated
'''
np.random.seed(1)
if anti_path is True:
sn = npr.standard_normal((par.M + 1,par.I//2))
else:
sn = npr.standard_normal((par.M + 1,par.I))
return sn
def gbm_mcs_amer(par):
''' Valuation of American option in Black-Scholes-Merton
by Monte Carlo simulation by LS algorithm.
Parameters
==========
S0 : spot price
K : strike float
r : riskless interest rate
I : int,number of paths to be simulated
T : time to maturity,in years
M : int,number of time intervals for discretization
sigma : vol
Returns
=======
C0 : float
estimated present value of American call option
'''
dt = par.T / par.M
df = np.exp(-par.r * dt) # discount function
# Generation of underlying asset process
# Stock Price Paths
S = par.S0 * np.exp(np.cumsum((par.r - 0.5 * par.sigma ** 2) * dt
+ par.sigma * np.sqrt(dt) * sn,axis=0)) # by exponentiating the brownian motion
S[0] = par.S0 # Initiliazing underlying path
# put option pay-off
h = np.maximum(par.K - S,0)
# LS algorithm
V = np.copy(h)
for t in range(par.M - 1,-1):
reg = lagfit(S[t],V[t + 1] * df,10)
C = lagval(S[t],reg)
V[t] = np.where(C > h[t],h[t])
# MCS estimator
y_i = df * V[1]
C0 = np.mean(y_i)
SE = np.std(y_i) / np.sqrt(par.I)
return C0,SE
# Regular Estimate loop
sn = gen_sn(par,False)
print("Reg","T:",par.T,"sigma:",par.sigma)
for par.S0 in range(36,44+1,2):
print("S0:",par.S0,"Price,SE:",gbm_mcs_amer_reg(par)[0],gbm_mcs_amer_reg(par)[1])
然后,我尝试实施 Glasserman 提出的对立变量价格 Antithetic paths estimator 和 Boyle 和 Glasserman 提出的标准误差 Antithetic paths standard error:
def gbm_mcs_amer_AP(par):
dt = par.T / par.M
df = np.exp(-par.r * dt) # discount function
# Generation of underlying asset process
# Stock Price Paths
S = par.S0 * np.exp(np.cumsum((par.r - 0.5 * par.sigma ** 2) * dt
+ par.sigma * np.sqrt(dt) * sn,axis=0)) # by exponentiating the brownian motion
S[0] = par.S0
S1 = par.S0 * np.exp(np.cumsum((par.r - 0.5 * par.sigma ** 2) * dt
+ par.sigma * np.sqrt(dt) * -sn,axis=0)) # Antithetic paths
S1[0] = par.S0
# put option pay-off
h = np.maximum(par.K - S,0)
h1 = np.maximum(par.K - S1,0)
# LS algorithm
V = np.copy(h)
V1 = np.copy(h1)
for t in range(par.M - 1,-1):
reg = lagfit(S[t],10)
C = lagval(S[t],reg)
V[t] = np.where(C > h[t],h[t])
reg1 = lagfit(S1[t],V1[t + 1] * df,10)
C1 = lagval(S1[t],reg1)
V1[t] = np.where(C1 > h1[t],h1[t])
# MCS estimator
y_i = df * (V[1]+V1[1])/2 # avg. pairs
C0 = np.mean(y_i)
SE = np.std(y_i) / np.sqrt(par.I) # Sample std. dev. of avg. pairs
return C0,SE
# AP Estimate loop
sn = gen_sn(par,True)
print("AP",gbm_mcs_amer_AP(par)[0],gbm_mcs_amer_AP(par)[1])
对立变量的价格与 Longstaff 和 Schwartz 论文中的价格接近,但标准误差似乎太小了,因为它们比控制变量减少了更多的方差,并且在随机数生成中匹配了两个矩。
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