An introduction to quasi-random numbers

Transcription

1 An ntroducton to quas-random numbers By George Levy, umercal Algorthms Grou Ltd. Introducton Monte-Carlo smulaton and random number generaton are technques that are wdely used n fnancal engneerng as a means of assessng the level of eosure to rsk. ycal alcatons nclude the rcng of fnancal dervatves and senaro generaton n ortfolo management. In fact many of the fnancal alcatons that use Monte-Carlo smulaton nvolve the evaluaton of varous stochastc ntegrals whch are related to the robabltes of artcular events occurrng. A case n ont s the rcng of a smle Euroean oton, where the value of a call oton s c r e r E[ma(( X )),)] whle the value of a ut s e E[ma(( X ),)]. Here X s the strke rce, s the maturty of the oton, r s the rsk free nterest rate, value of the asset at maturty and E[] denotes the eectaton oerator. r he value of a Euroean ut s therefore, e ( ) ma(( X ),) d the robablty that the asset wll have market value at maturty. s the market where ( ) s If t s assumed that the value of the asset follows geometrc Brownan moton and ( ) s the lognormal dstrbuton then the Black-choles formula [] can be used to rce the otons as follows: c r ( d ) e X ( d ) + e where ( d ), X ( d )), r d (log ( / X ) + ( r σ / ) ) /( σ ), d d σ and ( ) π / e d where s the current value of the asset,σ s the volatlty of the asset, and () s the cumulatve standard normal dstrbuton. In many cases however, the assumtons of constant volatlty and a lognormal dstrbuton for are qute restrctve. Real fnancal alcatons may requre a varety of etensons to the standard Black-choles model. Common requrements are for: non-lognormal dstrbutons, tme varyng volatltes, cas, floors, barrers etc. In these crcumstances t s often the case that there s no closed form soluton to the roblem. Monte-Carlo smulaton can then rovde a very useful means of evaluatng the requred ntegrals.

2 Monte-Carlo Integraton When we evaluate the ntegral of a functon, f (), n the s -dmensonal unt cube, I, by the Monte-Carlo method we are n fact calculatng the average of the functon at a set of randomly samled onts. hs means that each ont adds lnearly to the accumulated sum that wll become the ntegral and also lnearly to the accumulated sum of squares that wll become the varance of the ntegral. When there are samle onts the ntegral s: ν f ( ) where ν s used to denote the aromaton to the ntegral and,, K, are the, s -dmensonal, samle onts. If a seudo-random number generator s used the onts wll be (should be) ndeendently and dentcally dstrbuted. From standard statstcal results [] we can then estmate the eected error of the ntegral as follows: If we set χ f ( ) then snce s ndeendently and dentcally dstrbuted χ s also ndeendently and dentcal dstrbuted. he mean of χ s ν and the varance s ar( χ ). It s a well known statstcal roerty that the varance of ν s gven byar( ν ). We can / therefore conclude that the estmated ntegral ν has a standard error of. hs means that the estmated error of the ntegral wll decrease at the rate of /. Is t ossble to acheve a better convergence than ths? If samle onts are chosen that le on a Cartesan grd and we samle each grd ont eactly once then the Monte-Carlo method effectvely becomes a determnstc quadrature scheme, whose fractonal error decreases at the rate of or faster. he trouble wth the grd aroach s that t s necessary to decde n advance how fne t should be, and all the grd onts need to be used. It s therefore not ossble to samle untl some convergence crteron has been met. Quas-random number sequences seek to brdge the ga between the fleblty of seudorandom number generators and the advantages of a regular grd. hey are desgned to have a hgh level of unformty n multdmensonal sace, but unlke seudo-random numbers they are not statstcally ndeendent. Quas-random sequences Quas-random numbers are also called low dscreancy sequences. he dscreancy of a sequence s a measure of ts unformty and s defned as follows: Gven a set of onts,, L, I and a subset G I, defne the countng functon (G) as the number of onts G. For each (,, K, s ) I, let G be the rectangular s -dmensonal regon G [, ) [, ) L [, ) wth volume, K,. hen the dscreancy of the onts,, K, s gven by:

3 D * (,, K, ) su ( G), L, I he dscreancy s therefore comuted by comarng the actual number of samle onts n a gven volume of multdmensonal sace wth the number of samle onts that should be there assumng a unform dstrbuton. It can be shown that the dscreancy of the frst terms of quas-random sequence has the form: * D (,, K, ) C (log ) + O((log ) ), for all he rncal am n the constructon of low-dscreancy sequences s thus to fnd sequences n whch the constant C s as small as ossble. arous sequences have been constructed to acheve ths goal. Here we consder the followng quas-random sequences: ederreter [3] obol [4] Faure [5] he results of usng AG random number generator software [6] wth Gentat grahcs [7] s shown below. Fgures -3 llustrate the vsual unformty of the sequences. hey were created by generatng, 6-dmensonal samle onts, and then lottng the 4 th dmensonal comonent of each ont aganst ts 5 th dmensonal comonent. In Fgure, t can be seen that the seudo-random sequence ehbts clusterng of onts, and there are regons wth no onts at all.. Fgure : Pseudo-random sequence onts.

4 sual nsecton of Fgure and Fgure 3 show that both the obol and ederreter quasrandom sequences aear to cover the area more unformly. It s nterestng to note that the obol sequence aears to be a structured lattce whch stll has some gas. he ederreter sequence on the other hand aears to be more rregular and covers the area better. However, we can't automatcally conclude from ths that the ederreter sequence s the best. hs s because we haven't consdered all the other ossble ars of dmensons. Perhas the easest way to evaluate the random number sequences s to use them to calculate an ntegral. Fgure : obol sequence onts. Fgure 3: ederreter sequence onts. In Fgure 4 Monte-Carlo results are resented for the calculaton of the s dmensonal ntegral:

5 I 6 ( cos( ) d d d3 d4 d5 d6 he eact value of ths ntegral s: 6 I sn( ), whch for 6, gves I. 9 Fgure 4: Monte Carlo ntegraton usng random numbers. It can be seen that the seudo-random sequence gves the worst erformance. But as the number of onts ncreases ts aromaton to the ntegral mroves. Of the quas-random sequences t can be seen that the Faure sequence has the worst erformance, whlst both the obol and ederreter sequences gve rad convergence to the soluton. o conclude t has been shown that quas-random sequences can evaluate ntegrals more effcently than seudo-random sequences. hey thus rovde fnancal engneers wth a very useful technque for rsk assessment. George Levy works at AG Ltd UK, he can be contacted at george@nag.co.uk. References [] Hull., J. C. Otons, Futures and other Dervatves, Prentce Hall Internatonal Inc, 3 rd Edton 997. [] Goldberger, A.. A course n Econometrcs, Havard Unversty Press, 997. [3] ederreter, H. Random umber Generaton and Quas-Monte Carlo Methods, IAM, 99. [4] obol, I. M. he dstrbuton of onts n a cube and the aromate evaluaton of ntegrals, UR Comut. Math. Math. Phys. 7, 4, 86-, 967. [5] Faure, H. Dscreance de sutes assocees a un systeme de numeraton (en dmenson s). Acta Arth. 4, [6] AG Ltd, he Fortran 77 Lbrary Mark, AG Ltd, Oford, UK,. [7] Gentat, Internatonal Ltd, Oford, UK.