def LSMC1(price_matrix, K, r, paths, T, dt, type):
# start timer
tic = time.time()
# total number of steps
N = T * dt
N = int(N)
# adjust yearly discount factor
r = (1 + r)**(1 / dt) - 1
# cash flow matrix
cf_matrix = np.zeros((N + 1, paths * 2))
# calculated cf when executed in time T (cfs European option)
cf_matrix[N] = payoff_executing(K, price_matrix[N], type)
# 1 if in the money, otherwise 0
execute = np.where(payoff_executing(K, price_matrix, type) > 0, 1, 0)
# execute = np.ones_like(execute) # use to convert to consider all paths
# end year = 35
ey = 25
# total - end
endloop = N - ey * dt + 1
for t in range(1, endloop):
# discounted cf 1 time period
discounted_cf = cf_matrix[N - t + 1] * np.exp(-r)
# slice matrix and make all out of the money paths = 0 by multiplying with matrix "execute"
X = price_matrix[N - t, :] * execute[N - t, :]
# +1 here because otherwise will loose an in the money path at T-t,
# that is out of the money in T-t+1(and thus has payoff=0)
Y = (discounted_cf + 1) * execute[N - t, :]
# mask all zero values(out of the money paths) and run regression
X1 = np.ma.masked_less_equal(X, 0)
Y1 = np.ma.masked_less_equal(Y, 0) - 1
if X1.count(
) > 0: # meaning all paths are out of the money, thus never optimal to exercise
regression = np.ma.polyfit(X1, Y1, 2)
# warnings.simplefilter('ignore', np.RankWarning)
# calculate continuation value
cont_value = np.zeros_like(Y1)
cont_value = np.polyval(regression, X1)
# update cash flow matrix
imm_ex = payoff_executing(K, X1, type)
cf_matrix[N - t] = np.ma.where(imm_ex > cont_value, imm_ex,
cf_matrix[N - t + 1] * np.exp(-r))
cf_matrix[N - t + 1:] = np.ma.where(imm_ex > cont_value, 0,
cf_matrix[N - t + 1:])
else:
cf_matrix[N - t] = cf_matrix[N - t + 1] * np.exp(-r)
# option value is average of continuation value in year 35
option_value = np.sum(cont_value) / (paths * 2) * np.exp(-r * dt * ey)
# st dev
st_dev = np.std(cont_value) / np.sqrt(paths)
# threshold value
threshold_price = option_value + K
# Time and print the elapsed time
toc = time.time()
elapsed_time = toc - tic
print('Total running time of LSMC: {:.2f} seconds'.format(elapsed_time))
print("Ran this with T: ", T, " and dt: ", dt, "\n")
print("Value of this", type, "option is:", option_value)
print("St dev of this", type, "option is:", st_dev, "\n")
print("Threshold price of the option is: ", threshold_price)
return option_value
def LSMC_Legendre(price_matrix, K, r, paths, T, dt, type, polydegree):
from numpy.polynomial.legendre import legval, legfit
# start timer
tic = time.time()
# total number of steps
N = T * dt
N = int(N)
# adjust yearly discount factor
r = (1 + r)**(1 / dt) - 1
# cash flow matrix
cf_matrix = np.zeros((N + 1, paths * 2))
# calculated cf when executed in time T (cfs European option)
cf_matrix[N] = payoff_executing(K, price_matrix[N], type)
# 1 if in the money, otherwise 0
execute = np.where(payoff_executing(K, price_matrix, type) > 0, 1, 0)
# execute = np.ones_like(execute) # use to convert to consider all paths
for t in range(1, N):
# discounted cf 1 time period
discounted_cf = cf_matrix[N - t + 1] * np.exp(-r)
# slice matrix and make all out of the money paths = 0 by multiplying with matrix "execute"
X = price_matrix[N - t, :] * execute[N - t, :]
# +1 here because otherwise will loose an in the money path at T-t,
# that is out of the money in T-t+1(and thus has payoff=0)
Y = (discounted_cf + 1) * execute[N - t, :]
# mask all zero values(out of the money paths) and run regression
#X1 = np.ma.masked_less_equal(X, 0)
#Y1 = np.ma.masked_less_equal(Y, 0) - 1
X1 = X[np.where(X > 0)]
Y1 = Y[np.where(X > 0)]
if np.count_nonzero(
X1
) > 0: # meaning all paths are out of the money, thus never optimal to exercise
regression = legfit(X1, Y1, polydegree)
# warnings.simplefilter('ignore', np.RankWarning)
# calculate continuation value
cont_value = np.zeros_like(X)
cont_value[np.where(X > 0)] = legval(X1, regression)
# update cash flow matrix
imm_ex = payoff_executing(K, X, type)
cf_matrix[N - t] = np.where(imm_ex > cont_value, imm_ex,
cf_matrix[N - t + 1] * np.exp(-r))
cf_matrix[N - t + 1:] = np.where(imm_ex > cont_value, 0,
cf_matrix[N - t + 1:])
else:
cf_matrix[N - t] = cf_matrix[N - t + 1] * np.exp(-r)
# obtain option value
cf_matrix[0] = cf_matrix[1] * np.exp(-r)
option_value = np.sum(cf_matrix[0]) / (paths * 2)
# st dev
st_dev = np.std(cf_matrix[0]) / np.sqrt(paths)
# Time and print the elapsed time
toc = time.time()
elapsed_time = toc - tic
print('Total running time of LSMC: {:.2f} seconds'.format(elapsed_time))
print("Ran this with T: ", T, " and dt: ", dt, "\n")
print("Value of this", type, "option is:", option_value)
print("St dev of this", type, "option is:", st_dev, "\n")
return option_value