&  sW.X@Bookman Old Style- A2 Z$`gInterest Rate Curve Calibration with/P5D6C=6(ZD6C(WL6FD'WD(*E6D6)HP'^*5P 2 `g ],2 `gMonte Carlo SimulationnHP6C(VD6'I(M)tM(D6)IP 2  `g ],@Bookman Old Style--&2 X`gProblem raised by I3AI">n%3D#<=I$H?%- 3f12 X`gIndizen Technologies S.L.r/PK)BCP(QDCPOI'ID*C=(M'K( 2 X`g ] ,@Bookman Old Style-- D2 %x&`gCoordinating teachers of the problem: WAB3I#LD,#M?%,>C<M>3=$B%%,M>%I2BI"<n%%- 2 % `g < - 2 `gGerardo Oleaga]>3D3HB%^"=D?D2 $`g (%"=2 k!`gUniversidad Complutense de Madridl[M#<>3=#HDI$WAmI#P+>L<>%H>$lCI2#I 2 `g)# 2 `g E,2 3`gJorge Valdehita PrietoFB3>>%RD"H>M#,D%H3#>,B2  `g (%"12 c `gIndizen Technologies S.L.r)LI#7>M$H><MMB#B?"==%M%F% 2  `g)# 2 -`g F 2 [`g FX@Bookman Old Style-2 K `gExpositionH=@>4$.$>D#2 K`g of the problem:">&".D:"?.><":d" 2 K3`g P@"Arial Narrow-2 O`gThe value of an interest rate curve at a future time can be known today and can2..).........)..).)-....-D.(.-..(..;--..).-.)-. 2 %`g &2 >`ghelp us to obtain -...(-.-.-g`'(2 k `gtoday's values for ...()).-.).2 k g`gfiU2 k 1`gxed income securities, futures, derivatives, etc. )-..(.D.).(..)-..)..).).).) 2 k M `g 52 k i `gHowever, if we move forward :.;.).;-D.)..;.-g`'J2 *`gin time and wish to know the value of thos.D..-.;)..(..;-.)......)2  `ge securities-).)..) 2 `g O2 -`gin the future, then the interest rate curves ..-.......-..-)..).).'g`',2 P `gought to be simulated...-..-.)D..-. 2 P `g ,g`'2 S`gA Monte Carlo simulation of a stochastic process is a procedure for sampling random7D...;..)D....-.).)..))..).)))...).-...).D....-..D 2 `g 2 9 `goutcomes ..).D.'g`'w2 6 H`gfor that process. Interest rate curves can therefore be simulated once a.....).))...(-.).).))-........)D....-.).. 2 6  `g 2 6 , `gspecim(..)2 6 `gfi2 6 `gc )#2 6 Y`gevolution model -)....D-..g`'_2 8`gfor the curve has been assumed. 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For instance,.-D.).2..)..(. 2  `g >2  "`ga simple simulation will consider .)D.-)D..-.;)..)--g`'\2 6`gshort and long term interest rate structures as equal.)..-..-...D...(..).)..).(.-.. 2 3 `g ,g`'@"Arial Narrow- -2 `gThe 2..+- 2 ~`gNelson;.).. 2 ~`g-2  `gSiegel Model7...+D.-.-^2 7`g is a parametric estimation technique for yield curves.+)+.+...D.)+.)D.-.*.).....+.+)..+).).) 2 `g +2  `gIt is a non+)+.+... 2 `g-g`'X2 g3`gpolynomial model to prevent abrupt changes in the tv..)..D.D.-....)--....(..-..).-.2 g< `germ struct.D).)2 g `gure of int....72 g `gterest rates, mostly for the i..(..(D.))..-g`'2  `glong term.....D 2 2`g -g`' 2 L`g -g`'- -g`,--.  @Times New Roman-  .8Symbol- 2 (8Symbol-  2 U)-  [  e   Symbol- 2 " % 2 r % 2  % 2 " % 2 r % 2  % 2 N -5 2  -5 2 < $ 2  $ 2 z $ 2  $ 2 <i % 2 i % 2 zi % 2 i % 2 "q % 2 rq % 2 q % 2 " % 2 r % 2  % 2  -5 2  -5 2 '+5 2 H+5 2 =5 2  QHSymbol-  2 f t* 2 ! b8 2 f t* 2  t* 2 sb8 2 b8 2 b8Times New Roman- 2  mF 2  mF 2 f mF 2 mF 2 HR;Times New Roman-  2  exp+21 2  exp+21 2  11 2 ) 2 ( 2 ,Times New Roman- 2 2_ 2 2 22 2 21 2 2 0"System- ---  --'-  '''- 2 )`g - g`'- 2  `g -g`'- -g`,--.  @Times New Roman-  .8Symbol- 2 cv(8Symbol-  2 cb)Symbol-  2 d&t* 2 db8 2 d!b8 2 db8Times New Roman-  2 d , 2 d~, 2 d,Times New Roman-  2 |2 2 |Y1 2 |0Symbol-  2 d.=5 2 dQH"System-  --- --'-  '''-2 `g, !- 2 `gwheren;... 2 `g  -g`,--.  @Times New Roman-  .Times New Roman- 2 | 0Symbol-  2 db8"System-  --- --'-  '''- 2 5`g,  -g`,--.  @Times New Roman-  .Times New Roman- 2 1Symbol-  2 l|b8"System-  --- --'-  '''- 2 `g and n... -g`,--.  @Times New Roman-  .Times New Roman- 2 2Symbol-  2 lb8"System-  --- --'-  '''- O2  -`g indicate the contribution of long, short and..).---)......-.-.)-.... 2 Z`g 2 y`gmedium :D-..Dg`'-2  `gterm compon.D).D..-(2 `gents respectively; ..(.)..().) -g`,--.  @Times New Roman-  .Symbol- 2 9t)"System-  --- --'-  '''- 12 `g is the decay factor and )..-.).).)...-- 2  `gmD-"2 . `g is the time to)..D.. 2  `g 2 . `gmaturity. 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