black swaption modadel
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Black-Swaption copyright THC 2003. All rights reserved by Thom
reference number 32 classific 60204
Version 2.0 level intermediate Instructions
publication date Jun-03 Use "cspline_interp" function of X
author Hanyang Financial Engineering L Unzip xnumber31.zip at local dire
affiliation Hanyang University 3) Once in the Addins Manager, s
email address [email protected] input notional principal, swapti
last revised date Mar-04 output swaption value
references Jamshidian, F., 1989, An Exact Bond Option Formula, Journal of Fin
Ch. 6 of "The Oxford Guide to Financial Modeling" by Thomas S.Y.
Descriptions
financial model class Implied Volatility Surface: Calibrating the Models
issuer/modeler practitioner
model type Price Quotes of Benchmark Securities
risk sources forward swap rate
risk distribution lognormal
economic assumptions perfect capital market
technical assumptions continuous compounding interest rate
key words arbitrage-free interest rate model
Links
data http://www.federalreserve.gov/
financial models 7. Bond Arithmetic, 8. Bond Model
freeware http://digilander.libero.it/foxes/
Inputs
notional principal F 10000
swaption maturity (year) OT 5
tenor of swap (year) ST 3
total maturity (year) T 8 T=OT+ST, 0 < T 10, The maximum total maturity
compounding frequency/year m 2 We can choose the compounding frequency amon
the number of periods n 16 n=Tm, 0 < n 40, The maximum number of period
time increment dt 0.5 dt = T/nstrike K ######
volatility of forward swap rate 0.2
time period (year) 0 1 2 3 4 5 6
yield curve (continuous compounding) 0.05 0.05 0.05 0.05 0.05 0.05 0.05
Outputs
swaption value ######
Interim Calculations
adjusted time period (t-year) t 0 0.5 1 1.5 2 2.5 3interpolated yield curve r(t) ###### ###### ###### ###### ###### ###### ######
discount function P(t) ###### ###### ###### ###### ###### ###### ######
one period forward rate f(t-dt,t) ###### ###### ###### ###### ###### ######
one period forward price F(t-dt,t) ###### ###### ###### ###### ###### ######
zero-coupon bond price with t-maturity B(t) ###### ###### ###### ###### ###### ###### ######
checking swap period 0 0 0 0 0 0 0
mailto:[email protected]://www.federalreserve.gov/http://digilander.libero.it/foxes/http://digilander.libero.it/foxes/http://www.federalreserve.gov/mailto:[email protected] -
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forward discount function ###### 0 0 0 0 0 0 0
the last discount function ###### 0 0 0 0 0 0 0
calculate A ###### 0 0 0 0 0 0 0
forward swap rate ######
d1 ######
d2 ######
1 2
1
1 2
1
2 *
1*
*
2 1
( ) ( ) ( )
( ) ( )
1( ) : the value of a contract that pay
ln 2
mn
i F X
i
F X
mn
i
i
F X
Lswaption P t R N d R N d
m
L A R N d R N d
where
A P tm
R R Td
T
d d T
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as Ho Company, New York, NY. USA www. thomasho.com [email protected]
numbers freeware package to interpolate the yield curve and volatility curve.
tory (D:). 1) Open Excel 2) Select from the menu
earch for and select XNUMBERS.XLA 4) Click OK and have fun!
n maturity, swap tenor, compounding frequency, strike, volatility of forward swap rate, initial yield curv
ance , Vol.44, 205-209.
Ho and Sang Bin Lee, 2003, Oxford University Press
is 10, that is, we can choose both swaption maturity and tenor of swap arbitrarily on condition that the
1, 2, and 4.
s is 40 since the maximum values of total maturity and compounding frequency are 10 and 4 respectiv
7 8 9 10
0.05 0.05 0.05 0.05
3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ######
###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ######
###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ######
###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ######
###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ######
0 0 0 0 1 1 1 1 1 1 0 0
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0 0 0 0 ###### ###### ###### ###### ###### ###### 0 0
0 0 0 0 0 0 0 0 0 ###### 0 0
0 0 0 0 ###### ###### ###### ###### ###### ###### 0 0
s 1/m at times , (1 )i
t i mn
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e
sum of them is less than or equal to 10.
ely.
9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ######
###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ######
###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ######
###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ######
###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ######
0 0 0 0 0 0 0 0 0 0 0 0
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0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
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15.5 16 16.5 17 17.5 18 18.5 19 19.5 20###### ###### ###### ###### ###### ###### ###### ###### ###### ######
###### ###### ###### ###### ###### ###### ###### ###### ###### ######
###### ###### ###### ###### ###### ###### ###### ###### ###### ######
###### ###### ###### ###### ###### ###### ###### ###### ###### ######
###### ###### ###### ###### ###### ###### ###### ###### ###### ######
0 0 0 0 0 0 0 0 0 0
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0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
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