(SPOT) LIBOR (RATE), denoted L ( t, strictly denote the continuouslycompounding. SHORT (INSTANTANEOUS SPOT) RATE, denoted r (t)

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1 he Bak o halad Facal Isuos Polcy Group Quaave Models & Facal Egeerg eam Mahemacal Face Models Noe 4 ดร.พ มใจ นาคสก ล INERES RAE MODELS SPO VS. FORWARD INSANANEOUS RAES SPO INERES RAE deoed r u rae agreed upo a he prese me or borrowg/ledg some ooal amou or he durao u. SPO LIBOR RAE deoed L geercally a smplycompoudg spo rae.e. leg r srcly deoe he cououslycompoudg spo: exp L r 26 SHOR INSANANEOUS SPO RAE deoed r rae agreed upo a he prese me or borrowg/ledg some ooal amou or a esmal durao oher words: r r lm r u lm r u 27 FORWARD RAE deoed u v rae agreed upo a he prese me or borrowg/ledg some ooal amou begg a some uure me u or he durao v u oao-wse v r v. [Poomja Nacaskul Ph.D.][Mahemacal Face Models][p.4][Ju.255B.E.]

2 u LIBOR FORWARD RAE deoed F geercally a smply-compoudg orward rae.e. leg srcly deoe he couously-compoudg orward: exp F 28 INSANANEOUS FORWARD RAE deoed u rae agreed upo a he prese me or borrowg/ledg some ooal amou begg a some uure me u or a esmal duraooao-wse r oher words: u u lm u v lm u u v u 29 INERES RAE MODELS FUNDAMENAL NUMERAIRE QUANIIES ERO-COUPON BOND deoed prce o a promse made/vald a he prese me o pay s bearer u o local currecy a some uure me hece a a deermsc quay. Wh a Lbor orward F ad wo zeros ad u o local currecy ca be made avalable a he uure me he ollowg wo ways: eher holdg u o he zero or holdg F u o he zero whle lockg he Lbor orward L over he remag erval or he amou F avalable o reves a ; hece ay Lbor orward ca be wre erms o he zeros: [Poomja Nacaskul Ph.D.][Mahemacal Face Models][p.5][Ju.255B.E.]

3 [Poomja Nacaskul Ph.D.][Mahemacal Face Models][p.6][Ju.255B.E.] F F a revesg bod shor he holdg bod log holdg he 3 As wh Lbor orward ay spo Lbor ca also be expressed erms o zeros.e. by rewrg F as L F : L 3 Separaely ay saaeous orward rae ca also be expressed erms o a zero: l lm lm lm F 32 So ha coversely a zero ca also be expressed erms o he saaeous orward rae: dy y exp 33 MONEY-MARE/BAN ACCOUN NUMERAIRE deoed M value a ow or uure me o M u o local currecy omally

4 M deposed a me couously compoudg a he possbly r hus: sochasc shor rae process M M exp r s ds dm r M d dereal eq. M M r acal reasog SOCHASIC DISCOUN FACOR deoed D value a ow or uure me o u o local currecy promsed avalable a some uure me dscoued couously a he possbly sochasc shor rae process r hus: D exp r y dy As such uder srcly deermsc r : More geerally: D 36 r deermsc D 37 uder some prob. measure Q INERES RAE PAYER SWAP agreeme o pay xed eres raes exchage or loag eres raes.e. radom varables a he me o he agreeme over he le o a ooal loa. SWAP ENURE SRUCURE se o evely-spaced daes w.r.. whch beg ed [Poomja Nacaskul Ph.D.][Mahemacal Face Models][p.7][Ju.255B.E.]

5 [Poomja Nacaskul Ph.D.][Mahemacal Face Models][p.8][Ju.255B.E.] he loag raes are xed a or he rasers a o eed dereces he amous o eress owed by boh pares pracce dere ervals are oe used or xed vs. loag legs a complcao whch here s doe away wh. SWAP RAE deoed S xed rae se a he alsao o he swap corac.e. a he prese me beg such ha he prese value o xed ad loag legs are equalsed.e. wh he orward raes used o sad or he uure realsaos o he loag raes:.. raes orward he usg e legs loag o PV legs xed o PV S 38 Solve or S whch ca also be see as a weghed average o he orward raes: w w also S 39 INERES RAE MODELS VANILLA INERES RAE IR OPIONS

6 [Poomja Nacaskul Ph.D.][Mahemacal Face Models][p.9][Ju.255B.E.] INERES RAE CAPLE call opo o he orward rae payg a : s max 4 INERES RAE CAP a se o caples. INERES RAE FLOORE pu opo o he orward rae payg a : s max 4 INERES RAE FLOOR a se o loores. FORWARD MEASURE deoed Equvale Margale Measure assocaed wh he zero coupo bod as he umerare. Apply he margale prcg heory a.e. vs-à-vs he ormao se : Floore Caple max max 42 Assume or he orward rae he logormal dsrbuo parameersed by a scalar volaly : ~ l 43 Oe ca he apply he Black-76 ormula or he prce o a caple: d d d Caple 2 2 l 44

7 SWAPION opo o eer a payer swap payg a.e. pror o he rs rae-xg dae: max S max 45 Recallg S suggess ha he deomaor be used as he umerare: 46 Le Equvale Margale Measure assocaed wh as he umerare. Apply he margale prcg heory a s.e. vs-à-vs he ormao se Swapo s s max S 47 Assume or he swap rae S he logormal dsrbuo parameersed by a scalar volaly : d swap Swapo s s S s 2 l 2 s swap S s swap INERES RAE MODELS HJM & BGM d d s Assume or he sochasc dyamcs o he saaeous orward rae: d d dw dw ~ d s hs ca also be geeralsed o a mul-dmesoal model. s : [Poomja Nacaskul Ph.D.][Mahemacal Face Models][p.2][Ju.255B.E.]

8 d d dw ~ d j m j dw j j dwj dwk jkd correlaed sadard Weer processes j k m Heah Jarrow Moro HJM 99 deermes ha order o preve arbrage cao be arbrarly speced bu mus be deed erms o hus: 5 s ds 5 Seg he volaly uco o cosa or sace smply 2 yelds ds ad recovers he shor-rae based Ho-Lee model. Brace Gaarek Musela BGM 997 deermes how o model he sochasc dyamcs o he Lodo Ier-Bak Oer Rae LIBOR raes dreclyhereby reders he model calbrao redly ad a way ha s cosse wh he HJM ramework. [Poomja Nacaskul Ph.D.][Mahemacal Face Models][p.2][Ju.255B.E.]

Market Models. Practitioner Course: Interest Rate Models. John Dodson. March 29, 2009

Market Models. Practitioner Course: Interest Rate Models. John Dodson. March 29, 2009 s Praciioner Course: Ineres Rae Models March 29, 2009 In order o value European-syle opions, we need o evaluae risk-neural expecaions of he form V (, T ) = E [D(, T ) H(T )] where T is he exercise dae,