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# Black Swan

# British Exit

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The Nominal Short Rate Cannot Be Negative.

/* Fischer Black, 1995 */

/* 2008 ~ 2015 */

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2016/06/30

Derivative Pricing Under Negative Interest Rate environment- %

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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

log(F0/K)

�Implied

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dFt = �FtdWt

Ft = F0exp(�Wt � �2t2

)

Vc = P(0,T)[F0�(d1) � K�(d2)]

Vp(0) = P(0,T)[K�(�d2) � F0�(�d1)]

log(F0K

) � �K � 0

d1 =log(F0/K) + (�2

2 )T��T

d2 = d1 � ��T

d1 � �

d2 � �

�(�d1) � 0

�(�d2) � 0 Vp = 0

Vp(0) = P(0,T)[K�(�d2) � F0�(�d1)]

log(F0K

) � �K � 0

d1 =log(F0/K) + (�2

2 )T��T

d2 = d1 � ��T

d1 � �

d2 � �

�(�d1) � 0

�(�d2) � 0 Vp = 0

f(�) = P(0,T)[K�(�d2) � F0�(�d1)]� �� �Vpblack

�Vpmarket = 0

�Implied(K,T)

dFt = �LV(t,Ft)FtdWt

�LV(T,K) =

���� 2�C(T,K)�T

K2 �2C(T,K)�2K

dFt = �LV(t,Ft)FtdWt

�LV(T,K) =

���� 2�C(T,K)�T

K2 �2C(T,K)�2K

�cLV � �VcLV

�F=

�B�F

+�B��B

��B�F

= �Bc + �B ��B

�F

dFt = �LV(t,Ft)FtdWt

�LV(T,K) =

���� 2�C(T,K)�T

K2 �2C(T,K)�2K

�cLV � �VcLV

�F=

�B�F

+�B��B

��B�F

= �Bc + �B ��B

�F

dFt = �F(�)t dWt 0 � � � 1

p(t,f,F0)

(p)t � 12

(F(2�))FF = 0

pA(t,f) =1

1� �

f1�2�

t(ff0

)� 12e� q2+q20

2t I|�|(qq0t

)

pR(t,f) =1

1� �

f1�2�

t(ff0

)� 12e� q2+q20

2t I�(qq0t

)

dFt = �tF�tdW

(1)t

d�t = ��tdW(2)t

dW(1)t dW(2)

t = �dt

F0 = F

�0 = �

0 � � � 1

� �

� �

� �

� �

� �

� �

� �

� �

� �

� �

� �

� �

�H � a(K)b(T,K)(c(K)

g(c(K)))

a(K) = �[(FK)(1��)

2 (1+(1� �)2

24log2

FK

+(1� �)4

1920log4

Fk

)]�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��)/2log

FK

g(x) = log(

�1� 2�x+ x2 + x� �

1� �)

�ATMH � �

F1��[1+ (

(1� �)2

24�2

(F)2�2�

+����

4(F)1��+2� 3�2

24�2)T]

BSABRc = VCSABR(K,F, �,T)

� = �H(K,F; �, �, �, �)

(�, �, �) = argmin�����,�,�

�

i

[�Mi � �H(Fi,Ki; �, �, �)]2

� = �2T � 1

Vc(t) � Vp(t) = P(t,T)(Ft � K)

fTP(t,T)

=�2VcSABR

�2K=

�2VpSABR�2K

q(t,f) =12

(pR(t,f) + pA(t,f))

pR(t,f)

pA(t,f)

dFt = �tF�tdW

(1)t

d�t = ��tdW(2)t

dW(1)t dW(2)

t = 0

F0 = F

�0 = �

dFt = �tF�tdW

(1)t

d�t = ��tdW(2)t

dW(1)t dW(2)

t = 0

F0 = F

�0 = �

dFt = �tF�tdW

(1)t

d�t = ��tdW(2)t

dW(1)t dW(2)

t = �dt

dF̃t = �̃tF̃t�̃dW̃t

(1)

d�̃t = �̃�̃tdW̃t(2)

dW̃t(1)dW̃t

(2)= 0

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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�2T)

� =�

�(F0 � K) �(�) = log(

�1� 2�� + �2 � � + �

1� �)

dF̃t = �tF̃t�dW(1)

t

d�t = ��tdW(2)t

dW(1)1 dW(2)

t = �dt

F0 = F̃

�0 = �

F̃ = F+ s

�atmshift � �

F̃1��[1+ (

(1� �)2

24�2

(F̃)2�2�+

����

4(F̃)1��+2� 3�2

24�2)T]

CN = P(0,T)[(F0 � K)�(d) + ��T�(d)]

CD = P(0,T)[F̃0�(d̃1) � K̃�(d̃2)]

F > 0 �� < F < � F > �s

q(t,f) =12

(pR(t,f) + pA(t,f))

q(t, �f) =12

(pR(t, �f) � pA(t, �f))

f > 0

f < 0

dFt = �t|Ft|�dW(1)t

d�t = ��tdW(2)t

dW(1)t dW(2)

t = �dt

F0 = F

�0 = �

0 � � <12

�FBSABR =�(F0 � K)(1� �)

F0/|F0|� � K/|K|� · �

x· (1+ T(

��(2� �)�2

24|Fav|2�2�+

����sign(Fav)4|Fav|1��

))

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- %Hello world;)

𝜷

log �ATMH � log � � (1� �) log F

𝜷

(�, �, �) = arg min�����,�,�

�

i

[�Mi � �H(Fi,Ki; �, �, �)]2

𝜷

log �ATMH � log � � (1� �) log F

𝜷

(�, �, �) = arg min�����,�,�

�

i

[�Mi � �H(Fi,Ki; �, �, �)]2

𝛃

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𝞀

𝒗

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𝞀

𝒗

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𝒗

- %Hello world;)

- %Hello world;)

- %Hello world;)

𝜷

# Mean Reversion ?