This document contains a mix of text, numbers, formulas and code snippets related to quantitative finance modeling. It discusses topics like derivative pricing under negative interest rates, the SABR model, implied volatility, and mean reversion. Various formulas are presented for pricing derivatives, calculating implied volatility, and estimating parameters for models like SABR.

1. This picture was .aken by Aaron Davis.
# Black Swan

2. # British Exit
This picture was .aken by Paul Lloyd.

3. The Nominal Short Rate Cannot Be Negative.
/* Fischer Black, 1995 */

4. /* 2008 ~ 2015 */

5. - %Hello world;)

6. 2016/06/30

7. Derivative Pricing Under Negative
Interest Rate environment- %Hello world;)

8. - %Hello world;)

9. - %Hello world;)
- %Hello world;)

13. 1 2 3 4 5 6 7 8
0.20
9 10
0.12
30
0.79
0.55 0.08 0.43 0.72 0.98 1.51
0.05 0.35 0.50 0.63 0.83 0.94 0.95 1.58
0.01 0.18 0.28 0.41 0.58 0.73 0.88 1.56
0.02 0.13 0.25 0.31 0.42 0.68 0.78 0.92 1.56
0.15 0.34 0.48 0.68 0.88 1.04 1.20 2.03
0.19 0.34 0.54 0.75 0.90 1.06 1.21 2.00
0.06 0.22 0.45 0.62 0.81 0.93 1.07 1.66
0.19 0.31 0.51 0.74 0.88 0.88 1.02 1.71

15. log(F0/K)
Implied

17. - %Hello world;)
- %Hello world;)

18. dFt = FtdWt
Ft = F0exp( Wt
2
t
2
)
Vc
= P(0, T)[F0 (d1) K (d2)]

21. Vp
(0) = P(0, T)[K ( d2) F0 ( d1)]
log(
F0
K
)K 0
d1 =
log(F0/K) + (
2
2 )T
T
d2 = d1 T
d1
d2
( d1) 0
( d2) 0
Vp
= 0

22. Vp
(0) = P(0, T)[K ( d2) F0 ( d1)]
log(
F0
K
)K 0
d1 =
log(F0/K) + (
2
2 )T
T
d2 = d1 T
d1
d2
( d1) 0
( d2) 0
Vp
= 0

23. f( ) = P(0, T)[K ( d2) F0 ( d1)]
Vp
black
Vp
market = 0

25. Implied(K, T)

28. dFt = LV(t, Ft)FtdWt
LV(T, K) =
2 C(T,K)
T
K2
2C(T,K)
2K

29. dFt = LV(t, Ft)FtdWt
LV(T, K) =
2 C(T,K)
T
K2
2C(T,K)
2K
c
LV
Vc
LV
F
=
B
F
+
B
B
B
F
= B
c + B B
F

30. dFt = LV(t, Ft)FtdWt
LV(T, K) =
2 C(T,K)
T
K2
2C(T,K)
2K
c
LV
Vc
LV
F
=
B
F
+
B
B
B
F
= B
c + B B
F

31. dFt = F
( )
t dWt 0 1
p(t, f, F0)
(p)t
1
2
(F(2 )
)FF = 0

33. pA(t, f) =
1
1
f1 2
t
(
f
f0
)
1
2 e
q2+q2
0
2t I| |(
qq0
t
)
pR(t, f) =
1
1
f1 2
t
(
f
f0
)
1
2 e
q2+q2
0
2t I (
qq0
t
)

34. dFt = tFt dW
(1)
t
d t = tdW
(2)
t
dW
(1)
t dW
(2)
t = dt
F0 = F
0 =
0 1

38. H a(K)b(T, K)(
c(K)
g(c(K))
)
a(K) = [(FK)
(1 )
2 (1 +
(1 )2
24
log2 F
K
+
(1 )4
1920
log4 F
k
)] 1
b(T, K) = [1 + (
(1 )2
24
2
(FK)1
+
4(FK)(1 )/2
+
2 3 2
24
2
)T]
c(K) = (FK)(1 )/2
log
F
K
g(x) = log(
1 2 x + x2 + x
1
)

39. ATM
H
F1
[1 + (
(1 )2
24
2
(F)2 2
+
4(F)1
+
2 3 2
24
2
)T]

40. BSABR
c = VC
SABR(K, F, , T)
= H(K, F; , , , )
( , , ) = arg min
, , i
[ M
i H(Fi, Ki; , , )]2

41. = 2
T 1

43. Vc
(t) Vp
(t) = P(t, T)(Ft K)
fT
P(t, T)
=
2
Vc
SABR
2K
=
2
Vp
SABR
2K

46. q(t, f) =
1
2
(pR(t, f) + pA(t, f))
pR(t, f)
pA(t, f)

47. dFt = tFt dW
(1)
t
d t = tdW
(2)
t
dW
(1)
t dW
(2)
t = 0
F0 = F
0 =

48. dFt = tFt dW
(1)
t
d t = tdW
(2)
t
dW
(1)
t dW
(2)
t = 0
F0 = F
0 =
dFt = tFt dW
(1)
t
d t = tdW
(2)
t
dW
(1)
t dW
(2)
t = dt
d ˜Ft = ˜t ˜Ft
˜
d ˜Wt
(1)
d˜t = ˜˜td ˜Wt
(2)
d ˜Wt
(1)
d ˜Wt
(2)
= 0

50. - %Hello world;)
- %Hello world;)

56. dFt = tdW
(1)
t F0 = F
d t = tdW
(2)
t 0 =
dW
(1)
t dW
(2)
t = dt
N(K) =
( )
(1 +
2 3 2
24
2
T)
= (F0 K) ( ) = log(
1 2 + 2 +
1
)

59. d ˜Ft = t ˜Ft dW
(1)
t
d t = tdW
(2)
t
dW
(1)
1 dW
(2)
t = dt
F0 = ˜F
0 =
˜F = F + s
atm
shift ˜F1
[1 + (
(1 )2
24
2
(˜F)2 2
+
4(˜F)1
+
2 3 2
24
2
)T]

63. CN = P(0, T)[(F0 K) (d) + T (d)]
CD = P(0, T)[ ˜F0 ( ˜d1) ˜K ( ˜d2)]

64. F > 0 < F < F > s

67. q(t, f) =
1
2
(pR(t, f) + pA(t, f))
q(t, f) =
1
2
(pR(t, f) pA(t, f))
f > 0
f < 0

69. dFt = t|Ft| dW
(1)
t
d t = tdW
(2)
t
dW
(1)
t dW
(2)
t = dt
F0 = F
0 =
0 <
1
2
FBSABR =
(F0 K)(1 )
F0/|F0| K/|K|
·
x
· (1 + T(
(2 ) 2
24|Fav|2 2
+
sign(Fav)
4|Fav|1
))

71. - %Hello world;)
- %Hello world;)

76. 𝜷
log ATM
H log (1 ) log F
𝜷
( , , ) = arg min
, , i
[ M
i H(Fi, Ki; , , )]2

77. 𝜷
log ATM
H log (1 ) log F
𝜷
( , , ) = arg min
, , i
[ M
i H(Fi, Ki; , , )]2

78. 𝛃
𝝰
𝞀
𝒗

79. 𝛃
𝝰
𝞀
𝒗

80. 𝛃
𝝰
𝞀
𝒗

81. 𝛃
𝝰
𝞀
𝒗

88. - %Hello world;)
- %Hello world;)

89. - %Hello world;)

92. 𝜷

94. # Mean Reversion ?