def Compound_Option_Pricing_Closed_Form(S0, X1, X2, r, T1, T2, sigma):
# Use fslove and Black Scholes formula to sovle I
f1 = lambda I: bsformula(1, I, X2, r, T2 - T1, sigma)[0] - X1[0]
f2 = lambda I: bsformula(-1, I, X2, r, T2 - T1, sigma)[0] - X1[1]
I = [fsolve(f1, S0), fsolve(f2, S0)]; I = np.array(I)
y1 = (np.log((S0) / I) + (r + sigma ** 2 / 2) * T1) / (sigma * np.sqrt(T1))
y2 = y1 - sigma * np.sqrt(T1)
z1 = (np.log((S0) / X2) + (r + sigma ** 2 / 2) * T2) / (sigma * np.sqrt(T2))
z2 = z1 - sigma * np.sqrt(T2)
rho = np.sqrt(T1 / T2)
# Compound option pricing
low = np.array([-np.inf, -np.inf]); M = mvn.mvndst # mvn.mvndst(low,upp,[0,0],rho)[1] if INFIN(I) = 0, integration range is (-infinity, UPPER(I)
CC = S0 * M(low, np.array([z1, y1[0]]), [0, 0], rho)[1] - \
X2 * np.exp(-r * T2) * M(low, np.array([z2, y2[0]]), [0, 0], rho)[1] - \
X1[0] * np.exp(-r * T1) * norm.cdf(y2[0])
PC = X2 * np.exp(-r * T2) * M(low, np.array([z2, -y2[0]]), [0, 0], -rho)[1] - \
S0 * M(low, np.array([z1, -y1[0]]), [0, 0], -rho)[1] + \
X1[0] * np.exp(-r * T1) * norm.cdf(-y2[0])
CP = X2 * np.exp(-r * T2) * M(low, np.array([-z2, -y2[1]]), [0, 0], rho)[1] - \
S0 * M(low, np.array([-z1, -y1[1]]), [0, 0], rho)[1] - \
X1[1] * np.exp(-r * T1) * norm.cdf(-y2[1])
PP = S0 * M(low, np.array([-z1, y1[1]]), [0, 0], -rho)[1] - \
X2 * np.exp(-r * T2) * M(low, np.array([-z2, y2[1]]), [0, 0], -rho)[1] + \
X1[1] * np.exp(-r * T1) * norm.cdf(y2[1])
return (CC, PC, CP, PP)
def newton(callput, S0, K, r, T, q, price, sigma, tolerance, it=100):
"""
:param callput: Indicates if the option is a Call or Put option
:param S0: Stock price
:param K: Strike price
:param r: Risk-free rate
:param T: Time to expiration
:param q: Dividend rate
:param price: Price of option
:param sigma: Initial Volatility
:param tolerance: The precision of the solution
:param it: Maximum number of iterations
:return: Implied volatility
"""
i = 0
vega_init = 0
while i < it:
bsret = bsformula(callput, S0, K, r, T, sigma, q)
optionValue = bsret[0]
vega = bsret[2]
if (vega == 0):
return sigma
vega_init = vega
sigma_est = float(sigma) - (float(optionValue) -
float(price)) / float(vega)
if (abs(sigma_est - sigma) < tolerance):
return sigma_est
sigma = sigma_est
i += 1
return float('NaN')
def bsimpvol(callput,
S0,
K,
r,
T,
price,
q=0.,
priceTolerance=0.01,
method='bisect',
reportCalls=False):
'''
:param callput: Callput is 1 for a call and -1 for a put
:param S0: The stock price at time zero
:param K: The strike price
:param r: Risk-free interest rate
:param T: The maturity time
:param price: The price of the option
:param q: A continuous return rate on the underlying
:param priceTolerance:
:param method: The method used to calculate sigma
:param reportCalls: The record of call bsformula
:return: sigma, numbers of calling bsformula
'''
y = lambda sigma: bsformula(callput, S0, K, r, T, sigma)[0]
dy = lambda sigma: bsformula(callput, S0, K, r, T, sigma)[2]
if method == 'bisect':
if bisect(price, y, bounds=[0.01, 1])[0] == None:
return None
item = bisect(price, y, bounds=[0.01, 1])[0]
elif method == 'newton':
if newton(price, y, dy, start=1, bounds=[0.01, 1])[0] == None:
return None
item = newton(price, y, dy, start=1, bounds=[0.01, 1])[0]
if item != None:
sigma = item[-1]
else:
sigma = item
return None
if reportCalls == True:
return (sigma, len(item))
else:
return sigma
def binomialAmerican(S0, Ks, r, T, sigma, q, callputs, M):
""" Computed values """
dt = T / M # Single time step, in years
df = np.exp(-(r - q) * dt) # Discount factor
u = np.exp(sigma * np.sqrt(dt))
d = 1. / u
p = (np.exp((r - q) * dt) - d) / (u - d)
q = 1 - p
# Initialize a 2D tree at T=0
STs = [np.array([S0])]
# Simulate the possible stock prices path
for i in range(M):
prev_branches = STs[-1]
st = np.concatenate((prev_branches * u, [prev_branches[-1] * d]))
STs.append(st) # Add nodes at each time step
payoffs = np.maximum(0, (STs[M] - Ks) if callputs == 1 else (Ks - STs[M]))
for j in reversed(range(M)):
# The payoffs from NOT exercising the option
payoffs = (payoffs[:-1] * p + payoffs[1:] * q) * df
early_ex_payoff = (STs[j] - Ks) if callputs == 1 else (Ks - STs[j])
payoffs = np.maximum(payoffs, early_ex_payoff)
price = bsformula(callput=callputs,
S0=S0,
K=Ks,
r=r,
T=T,
sigma=sigma,
q=q)[0]
error = price - payoffs[0]
return payoffs[0], error
def targetfunction(x):
return bsformula(callput, S0, K, r, T, x, q)[0]
S0, r, T1, T2 = [50.0, 0.025, 1.0, 2.0]; X2 = S0
# Different client's view
views = {'Mrs Smith': ([r + 0.03, 0.15], [r + 0.005, 0.3]), 'Mr Johnson': ([r - 0.03, 0.2], [r - 0.01, 0.18]),
'Ms Williams': ([r - 0.03, 0.18], [r + 0.03, 0.12]), 'Mr Jones': ([r + 0.02, 0.35], [r + 0.02, 0.1]),
'Miss Brown': ([r + 0.03, 0.15], [r - 0.05, 0.15])}
names = ('Mrs Smith', 'Mr Johnson', 'Ms Williams', 'Mr Jones', 'Miss Brown')
# in different views
'Question c1-Antithetic Sampling methods. ComOpt4 and ComOpt5 is the prices of compound options'
ComOpt4 = [0] * len(names); option = ('Call', 'Put')
ComOpt5 = [0] * len(names)
j = 0
for v in names:
mu1, sigma1, mu2, sigma2 = views[v][0][0], views[v][0][1], views[v][1][0], views[v][1][1]
# At the money
X1 = [bsformula(1, S0, X2, r, T2 - T1, sigma2)[0], bsformula(-1, S0, X2, r, T2 - T1, sigma2)[0]]
print()
print("Antithetic Sampling")
# For different underlying options
ComOpt4[j] = [0] * 2; i = 0
for underlying in option:
# Antithetic Sampling
start_c1 = time.clock()
ComOpt4[j][i] = [0] * 4
ComOpt4[j][i] = Comp_Opt_Var_Vol(S0, X1, X2, r, T1, T2, mu1, mu2, sigma1,
sigma2).antiSampling(underlying, int(10000), 777)
# Get enough size of number to satisfy The Central Limit Theorem
N = int(floor(std(ComOpt4[j][i][3]) ** 2.0 * 1.96 ** 2.0 * 1002001 /
(mean(ComOpt4[j][i][3]) ** 2.0)))
print("Using Antithetic Sampling, simulaiton number is", N)
def vega(sigma, callput, S0, K, r, T):
return bsformula(callput, S0, K, r, T, sigma, q=0.)[2]
def bsmvsact(sigma, callput, S0, K, r, T, price):
return bsformula(callput, S0, K, r, T, sigma, q=0.)[0] - price
def f_prime(x):
return bsformula(callput, S0, K, r, T, x, q)[2]
def f(x):
return bsformula(callput, S0, K, r, T, x, q)[0] - price
r=np.interp(T,tenors,rateCurve)
m=range(100,2010,100); n=range(1,101,1) # create dynamic M and N
print('Varied N is %s'%n,'Varied M is %s'%m)
'Vary N'
error_n=np.zeros((len(n),3))
checkpoints2 = [10,m[-1]]
num1_1=0
for k in n:
samples2 = rand(k, m[-1])
t2 = array([T / k] * k) # Monthly
num2_1 = 0
for u in ['standard', 'euler', 'milstein']: # calculate error in different integrator methods
test= MCOptionPrices(S0, K, T, rateCurve, sigma, t2, checkpoints2, samples2, u)
TV = test['TV']
error_n[num1_1,num2_1] = abs(TV[0] - bsformula(1, S0, K, r, T / 12., sigma)[0])
num2_1+=1
num1_1+=1
'Vary M'
error_m = np.zeros((len(m), 3))
t2 = array([T / n[-1]] * n[-1]) # Monthly
num1_2=0
for l in m:
samples2 = rand(n[-1], l)
checkpoints2 = [10,l]
num2_2 = 0
for u in ['standard', 'euler', 'milstein']: # calculate error in different integrator methods
test = MCOptionPrices(S0, K, T, rateCurve, sigma, t2, checkpoints2, samples2, u)
TV = test['TV']
error_m[num1_2, num2_2] = abs(TV[0] - bsformula(1, S0, K, r, T / 12., sigma)[0])
def fdAmerican(callput, S0, K, r, T, sigma, q, M, N, S_max, tol):
#dS = S_max / float(M)
dt = T / float(N)
i_values = np.arange(M+1)
j_values = np.arange(N+1)
boundary_conds = np.linspace(0, S_max, M+1)
if callput == 1:
payoffs = np.maximum(
boundary_conds[1:M]-K, 0)
past_values = payoffs
lower_boundary_values = np.maximum(S_max - K, 0) * \
np.exp(-r * dt * (N-j_values))
upper_boundary_values = 0* j_values
elif callput == -1:
payoffs = np.maximum(
K-boundary_conds[1:M], 0)
past_values = payoffs
upper_boundary_values = K * \
np.exp(-r * dt * (N-j_values))
lower_boundary_values = 0* j_values
alpha = 0.25*dt*((sigma**2)*(i_values**2) - r* i_values)
beta = -dt*0.5*((sigma**2)*(i_values**2) +r)
gamma = 0.25*dt*((sigma**2)*(i_values**2) +r*i_values)
M1 = -np.diag(alpha[2:M], -1) + \
np.diag(1-beta[1:M]) - \
np.diag(gamma[1:M-1], 1)
M2 = np.diag(alpha[2:M], -1) + \
np.diag(1+beta[1:M]) + \
np.diag(gamma[1:M-1], 1)
P, L, U = linalg.lu(M1)
aux = np.zeros(M-1)
new_values = np.zeros(M-1)
for j in reversed(range(N)):
aux[0] = alpha[1]*(upper_boundary_values[j] + upper_boundary_values[j+1])
aux[M-2]= gamma[M-1]*(lower_boundary_values[j]+ lower_boundary_values[j+1])
rhs = np.dot(M2, past_values) + aux
old_values = np.copy(past_values)
error = sys.float_info.max
while tol < error:
new_values[0] = \
max(payoffs[0],
old_values[0] +
1.0/(1-beta[1]) *
(rhs[0] -
(1-beta[1])*old_values[0] +
(gamma[1]*old_values[1])))
for k in range(M-2)[1:]:
new_values[k] = \
max(payoffs[k],
old_values[k] +
1.0/(1-beta[k+1]) *
(rhs[k] +
alpha[k+1]*new_values[k-1] -
(1-beta[k+1])*old_values[k] +
gamma[k+1]*old_values[k+1]))
new_values[-1] = \
max(payoffs[-1],
old_values[-1] +
1.0/(1-beta[-2]) *
(rhs[-1] +
alpha[-2]*new_values[-2] -
(1-beta[-2])*old_values[-1]))
error = np.linalg.norm(new_values - old_values)
old_values = np.copy(new_values)
past_values = np.copy(new_values)
values = np.concatenate(([upper_boundary_values[0]],
new_values,
[0]))
price = bsformula(callput = callput, S0 = S0, K = K, r = r, T = T, sigma = sigma, q = q)[0]
diff = price - np.interp(S0, boundary_conds, values)
return np.interp(S0, boundary_conds, values), diff
def optVega(x):
return bsformula(callput, S0, K, r, T, x, q)[2]
def optionValue(x):
return bsformula(callput, S0, K, r, T, x, q)[0]