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import datetime
import math
import numpy as np
def sabr_lognormal(alpha, rho, nu, F, K, T):
A = 1 + (0.25 * (alpha * nu * rho) + nu * nu * (2 - 3 * rho * rho) / 24.0) * T
if F == K:
VOL = alpha * A
elif F != K:
nulogFK = nu * math.log(F / K)
z = nulogFK / alpha
x = math.log((math.sqrt(1 - 2 * rho * z + z ** 2) + z - rho) / (1 - rho))
VOL = (nulogFK * A) / x
return VOL
def sabr_normal(alpha, rho, nu, F, K, T):
if F == K:
V = F
A = (
1
+ (alpha * alpha / (24.0 * V * V) + nu * nu * (2 - 3 * rho * rho) / 24.0)
* T
)
VOL = (alpha / V) * A
elif F != K:
V = math.sqrt(F * K)
logFK = math.log(F / K)
z = (nu / alpha) * V * logFK
x = math.log((math.sqrt(1 - 2 * rho * z + z ** 2) + z - rho) / (1 - rho))
A = (
1
+ (
(alpha * alpha) / (24.0 * (V * V))
+ ((nu * nu) * (2 - 3 * (rho * rho)) / 24.0)
)
* T
)
logFK2 = logFK * logFK
B = 1 / 1920.0 * logFK2 + 1 / 24.0
B = 1 + B * logFK2
VOL = (nu * logFK * A) / (x * B)
return VOL
def sabr(alpha, beta, rho, nu, F, K, T):
if beta == 0.0:
return sabr_normal(alpha, rho, nu, F, K, T)
elif beta == 1.0:
return sabr_lognormal(alpha, rho, nu, F, K, T)
else:
if F == K: # ATM formula
V = F ** (1 - beta)
A = (
1
+ (
((1 - beta) ** 2 * alpha ** 2) / (24.0 * (V ** 2))
+ (alpha * beta * nu * rho) / (4.0 * V)
+ ((nu ** 2) * (2 - 3 * (rho ** 2)) / 24.0)
)
* T
)
VOL = (alpha / V) * A
elif F != K: # not-ATM formula
V = (F * K) ** ((1 - beta) / 2.0)
logFK = math.log(F / K)
z = (nu / alpha) * V * logFK
x = math.log((math.sqrt(1 - 2 * rho * z + z ** 2) + z - rho) / (1 - rho))
A = (
1
+ (
((1 - beta) ** 2 * alpha ** 2) / (24.0 * (V ** 2))
+ (alpha * beta * nu * rho) / (4.0 * V)
+ ((nu ** 2) * (2 - 3 * (rho ** 2)) / 24.0)
)
* T
)
B = (
1
+ (1 / 24.0) * (((1 - beta) * logFK) ** 2)
+ (1 / 1920.0) * (((1 - beta) * logFK) ** 4)
)
VOL = (nu * logFK * A) / (x * B)
return VOL
if __name__ == "__main__":
from analytics.option import BlackSwaption
from analytics import CreditIndex
from scipy.optimize import least_squares
underlying = CreditIndex("IG", 28, "5yr")
underlying.spread = 67.5
exercise_date = datetime.date(2017, 9, 20)
option = BlackSwaption(underlying, exercise_date, 70)
strikes = np.array([50, 55, 57.5, 60, 62.5, 65, 67.5, 70, 75, 80, 85])
pvs = np.array([44.1, 25.6, 18.9, 14, 10.5, 8.1, 6.4, 5, 3.3, 2.2, 1.5]) * 1e-4
strikes = np.array([50, 55, 57.5, 60, 62.5, 65, 67.5, 70, 75, 80, 85, 90, 95, 100])
pvs = (
np.array(
[
53.65,
37.75,
31.55,
26.45,
22.25,
18.85,
16.15,
13.95,
10.55,
8.05,
6.15,
4.65,
3.65,
2.75,
]
)
* 1e-4
)
def calib(x, option, strikes, pv, beta):
alpha, rho, nu = x
F = option.forward_spread
T = option.T
r = np.empty_like(strikes)
for i, K in enumerate(strikes):
option.strike = K
option.sigma = sabr(alpha, beta, rho, nu, F, K, T)
r[i] = option.pv - pv[i]
return r
prog = least_squares(
calib,
(0.3, 0.5, 0.3),
bounds=(np.zeros(3), [np.inf, 1, np.inf]),
args=(option, strikes, pvs, 1),
)
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