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Price-difference BS-hyperbolic in DM
0.00
0.10
0.20
-0.10
T = 3 days
T = 5 days
T = 10 days
0.85
0.90
€Z;KE
0.95
1.00
1.05
stockprice-strike ratio
1.10
1.15
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e BS-hyp in DM
price differenc0.1
0.2 0.3 0.4
-0.2 -0.1 0
30
25
20
tim
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0.6
0.8
1
rice-s
stockp
1.2
tio
trike ra
1.4
time 50 1
to ma 00 1
turity 50
olatility
plicit v
0.8
BS im
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0
0.4
0 0.2
1
mean= 0.2557 / sd= 0.06902 / #obs= 28325
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-F!]:'/))@A1IC1]!12N!‡Q5P!cTVU)W !XY!=!)2167
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Ud
0.8
0.8
0.9
stock 1
price-s 1.1
trike ra 1.2
tio
time50
100
to m
atu1r 50
ity
time50 1
to m 00 1
aturit50
y
olatility 1
plicit v
BS im0.4 0.6 0.8
0 0.2
Siemens calls
olatility 1
plicit v
BS im0.4 0.6 0.8
.2
0
0
Deutsche Bank calls
mean= 0.1792 / sd= 0.054 / #obs= 78571
DAX future calls 7-12.1994
DAX future puts 7-12.1994
ike ra1ti.05
o
1.1
0.9
atu 150
rity
tim 50
e to
m
0.9
stockp5
rice-s1tr
1.3
tility
licit vola
0.5
BS imp.2
0.3 0.4
0 0.1 0
0.9
tim 50
e to
m
atu 150
rity
it volatility
BS implic 0.3 0.4 0.5
.2
0 0.1 0
0.9
stockp1
rice-s1.1
trike ra1.2
1.3
tio
mean= 0.1879 / sd= 0.065 / #obs= 68803
0.95
stockp1
ri
ce-str1.05
ike rati
1.1
o
mean= 0.2186 / sd= 0.02978 / #obs= 467
mean= 0.1876 / sd= 0.032 / #obs= 830
0.5
Thyssen calls
0.5
Thyssen calls
rho=.59-.85
rho=.85-.95
T=0-20
T=21-40
T=41-80
T=81-120
0.9
1.0
1.1
stockprice-strike ratio
€Z;0vE p0f
q 1.3
0
50
100
time to maturity
150
% ! q+] XpUKQR5Hh9d pHYq`HJI NXpUfDITHpHJITHUKQR5XV
\
RrILXDq XdnITHXPSRV0ZQXh D
s_p^
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o KaZ (x if&P M9
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x ILX
1.2
rho=1.15-1.57
0.2
0.2
T=121-160
T=161-192
0.8
rho=1.05-1.15
BS implicit volatility
0.3
0.4
BS implicit volatility
0.3
0.4
rho=.95-1.05
U!U
TVU)W!W!XYD:!:.)7o!;K!(D:(1;4*,-)G</z':!1nC(*<!@
~I!R=!)Q5ˆ!1xTVU)W!W!XY+!17“K!)1GQ”![$[-TVU)W!W!XY+1O+D,)/()!C1
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/*<:(1C!MNN!"$#&%'()Z!1C$6$:..`*<-)OJN=!‹6
:5(!*,CF=!)yMi* ,=‘(!1/4;=!)181ŠT‡WYF@,D,*,:$6$:..
=!‹6 ´+©$² §P¨>© \/*,:) 1Š2!*,2J>6!n1 yN!"$#&%'()
!!@]Dc)*,:(!6B!=!) ´+©O² §S¨>© RJ1?1I3;o„yT^!:wM^FYDM^!
BS minus hyperbolic implicit volatility
1.3
0.9
stockp 1
rice-str1.1
ike rati1.2
o
1.3
50
to m 100
aturit
y
0.8
time
-0.04
0.9
stockp 1
rice-s 1.1
trike ra 1.2
tio
time
0.8
50
to m 100
aturit
y
0
0.04
0.8
0.4 0.6
0 0.2
1
0.08
Hyperbolic implicit volatility
mean= -0.0041 / sd= 0.00925
BS minus hyperbolic implicit volatility
0.10
0.06
mean= 0.192 / sd= 0.06004
BS minus hyperbolic implicit volatility
ratio= 0.71-0.85
ratio= 0.85-0.95
ratio= 0.95-1.05
ratio= 1.05-1.15
ratio= 1.15-1.36
0.0
0.0
difference
0.05
difference
0.02
0.04
T= 0-20
T= 21-40
0.8
0.9
1.0
1.1
1.2
T= 41-80
-0.02
T= 81-120
T=121-160
T= 161-192
1.3
0
50
stockprice-strike ratio
€Z;Ž„
E
100
150
time to maturity
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Historical volatilities of Bayer 1993
0.25
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