1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
|
import datetime
import math
import numpy as np
from numba import jit, float64
@jit(
float64(float64, float64, float64, float64, float64, float64),
cache=True,
nopython=True,
)
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
@jit(
float64(float64, float64, float64, float64, float64, float64),
cache=True,
nopython=True,
)
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
@jit(
float64(float64, float64, float64, float64, float64, float64, float64),
cache=True,
nopython=True,
)
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),
)
|
