Fugit (options) The terminology of fugit refers to the risk neutral expected time to exercise an

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1 Fug (opons) INTRODUCTION The ermnology of fug refers o he rsk neural expeced me o exercse an Amercan opon. Invened by Mark Garman whle professor a Berkeley n he conex of a bnomal ree for Amercan opon hs concep has been snce exended o oher numercal mehods for Amercan opons and n parcular o Amercan Mone Carlo. The concep of fug s que relevan for converble bonds equy lnked converble noes Bermudan opon and any. puable or callable exoc coupon noes. I provdes an esmae of when he opon would be exercsed. Ths s a useful ndcaon for he deermnaon of he maury o use o hedge Amercan or Bermudan producs wh European opons. In he case of a ree or a pde he fug s easly compued when dscounng he cashflow backward. We nally sar n he las slce of he ree or pde wh an esmaed me equal o he fnal dae. When compung backward he cashflow one checks wheher he opon s exercsed or no. If so he me of exercse a hs node becomes precsely he curren dae. Oherwse he esmaed exercse me s compued as he probably weghed average of he exercse me of he prevous slce. The esmaed exercse me or fug s gven when reachng he las slce.

2 In he case of an Amercan Mone Carlo he compuaon of he fug mples o frs deermne he exercse boundary and hen usng prevous or new sampled pahs (accordng o he mehod) compue he expeced exercse me. EXERCIE BOUNDARY PROBLEM More mporanly he concep of fug for Amercan and Bermudan (also referred o as Md-Alanc opons) provdes valuable nformaon abou he exercse boundary. The prcng of Amercan or Bermudan opons can be n fac reformulaed n erms of dynamc programmng and he Bellman prncple. mply speakng he Bellman equaon saes ha he opmal exercse sraegy for an Amercan opon s o compare he early exercse value wh he presen value of all fuures cash flow (wa for a furher exercse) and ake he sraegy wh greaer value. where ( ) ( ) = max{ F( ) P [ ( d d )]} (.) s he underlyng asse ( ) s he value of he Amercan opon F he value n case of an early exercse and P [ ( d d )] s he presen value of he sraegy conssng n wang for he nex exercse perod. The dynamc programmng formulaon s generally used by Amercan Mone Carlo echnques ogeher wh he fac ha one can generally approxmae he early exercse decson by a funcon of he sae varable A sraghforward Amercan Mone mehod consss n maxmzng he exercse boundary for a gven smulaon. Obvously hs leads o a lower bound for he Amercan opon value..

3 The oher formulaon of he problem used by rees and fne dfference uses he dual formulaon n erm of Paral Dfferenal Inequaly. The relaonshp beween he wo approaches les n he Feyman Kac formulaons ha ranslae hs probablsc problem no a Paral Dfferenal Inequaly problem. In he case for nsance of local volaly models hs leads o ( ) r r 0 (.) wh he early exercse condon ( ) F( ) (.3) and he fnal boundary condon ( T ) F( T ) (.4). T T Compared wh sandard European Opon prcng he boundary condon are no well known hence he name free boundary problem for. The usual dscresaon based on general hea scheme leads o use a Crank Ncolson mehod ( = / ) on he logarhm of he underlyng asse (guaraees ha he equaon s a real hyperbollc one). Denong by he dscresed funcon where he frs varable sands for me whereas he second one for he space varable one obans he followng resuls: Alhough a Douglas scheme s more accurae and very smlar o he Crank Ncolson scheme he bankng ndusry seems o prefer Crank Ncolson algorhm. Recen developmens n erms of Alernave Drecon Mehods have been popularsed by local volaly and umps and sochasc volaly and umps models.

4 ( ) ( ) ( ) () ( ) ( ) ( ) ( ) r r r T = wch can be regroup o lead o he radonnal rband dffuson marx solved wh he lnear LU decomposon. The boundary condons can ake varous forms (see boundary condon for opons) lke Drchle an Neuman and second order dervaves. CAE TUDY: BERMUDAN WAPTION Consder he case of a 0 no 0 year Bermudan swapon wh exercse dae every year sarng a year 5 and onwards. Lke for any oher Bermudan or Amercan srucure he queson of he opmal exercse s que crucal boh for he seller of he opon (for her hedgng sraegy) and he opon holder (monese he opon value). Alhough he pary long he opon ofen would lke o sell he opon o cash n he opon value s rarely easy o do so as hese producs are OTC and barely rade cheaply. The pary long he opon would have no choce bu o exercse he opon f she hnks ha hs s deal mng for he opon. In conras o exercsng he opon sellng he opon could have provded more value as he opon s value s he combned resul of he early exercse sraegy and he oher wa and see sraegy.

5 Unforunaely hs heorecal value s very hard o cash n. The Bermudan swapon marke s a bg busness marke snce many corporae are neresed n flexble swapons. On he lably sde recever Bermudan swapons have been massvely sold o ssurance companes for her lables problems as hey offer hem proecon agans neres rae rse. Bermudan swapon can be seen as an opon on he mos expensve European one plus he rgh o swch. Usually (n common examples) when compung he fugone fnds a maury que close o he one of he mos expensve European opon. However for Bermudan opons ha do no have a clear mos expensve European opon he fug can sgnfcanly dffer from he maury of he mos expensve European opon. Hedgng only wh he mos expensve European opon can lead o serous mng problem as he hedge may be nacvaed before he exercse of he Bermudan opon or vce versa. Enry caegory: opons cope: Opon prcng Amercan opon. Relaed arcles: Amercan-syle opon prcng Greeks

6 Erc Benhamou waps raegy London FICC Goldman achs Inernaonal The vews and opnons expressed heren are he ones of he auhor s and do no necessarly reflec hose of Goldman achs