Post on 11-Feb-2017
transcript
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# Black Swan
# British Exit
This picture was .aken by Paul Lloyd.
The Nominal Short Rate Cannot Be Negative.
/* Fischer Black, 1995 */
/* 2008 ~ 2015 */
- %Hello world;)
2016/06/30
Derivative Pricing Under Negative Interest Rate environment- %
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- %Hello world;)
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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
- %Hello world;)
- %Hello world;)
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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- %Hello world;)
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��
))
- %Hello world;)
- %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
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𝞀
𝒗
𝛃
𝝰
𝞀
𝒗
𝛃
𝝰
𝞀
𝒗
- %Hello world;)
- %Hello world;)
- %Hello world;)
𝜷
# Mean Reversion ?