Option Pricing Variance Reduction Techniques Under the Levy Process

Transcription

1 Appled and Computatonal Mathematcs 205; 4(3): Publshed onlne June 8, 205 ( do: 0.648/.acm ISS: (Prnt); ISS: (Onlne) Opton Prcng Varance Reducton Technques Under the Levy Process L Zhou, Hong Zhang, Jan Guo, Shucong Mng 2 School of Informaton, Beng Wuz Unversty, Beng, Chna 2 Chnese Academy of Fnance and Development, Central Unversty of Fnance and Economcs, Beng, Chna Emal address: dr.yuwenunxan@gmal.com (Hong Zhang) To cte ths artcle: L Zhou, Hong Zhang, Jan Guo, Shucong Mng. Opton Prcng Varance Reducton Technques Under the Levy Process. Appled and Computatonal Mathematcs. Vol. 4, o. 3, 205, pp do: 0.648/.acm Abstract: After the 2008 fnancal crss, the global dervatves tradng volume n optons proporton s growng, more and more nvestors buld portfolos usng optons to hedge or arbtrage, our futures and stock optons wll soon open. Theoretcal research of optons s also changng, opton prcng models under Levy processes developed rapdly. In ths context, a revew of the Chna's warrants market and the ntroducton of opton prcng models can not only help us to reflect Chnese fnancal dervatves market regulaton, but also to explore the opton prcng theory for Chna`s fnancal market envronment. In the framework of Monte Carlo smulaton prcng, we establshed muft-levy process opton prcng models, the structural model for the gven parameter estmaton and rsk-neutral adustment method are dscussed, the last part of ths chapter s an emprcal analyss of Chna warrants tradng data n order to prove the valdate of Levy models. Key word: Levy stochastc processes, opton prcng models, Chnese warrants market, Amercan opton prcng, rsk-neutral adustment, varance reducton technques. Keywords: Opton Prcng, Varance Reducton Technques, Levy Process. Introducton The theoretcal bass of Monte Carlo smulaton s on the Law of Large umber. Ths law guarantees we can estmate the real value by the sample mean after enough tmes random smulaton. And the errors of estmaton can gradually converge wth the ncrease n smulaton tmes. But the convergence rate s negatvely correlated wth the sample varance, whch mples that estmates by Monte Carlo smulaton can be more accurate f the varances between the smulaton values, wth the smulaton tmes beng equal. Thus, the reducton of samples varance s key to mprovng the estmaton s precson. The ncumbent technques to reduce varances nclude Antthetc Varables Method, Importance Samplng, Control Varates Method, etc. ote that the core of the Antthetc Varables Method s to take the random numbers par by par and to ensure that each par of random numbers s negatvely assocated wth the mean. Ths method appears to be easy to operate. Importance Samplng Method endows each sample pont wth a weght to reduce the varances by transformng the measures. Control Varables Method helps to reduce the varance by selectng and artfcally constructng an nstrumental varable through the correlatons of nstrumental varables and orgnal varables. Ths chapter s focusng on the generatng algorthm of tme-varyng Brown Moton n Levy Process and ntroducng the Control Varables Method to deal wth the reducton of sample ponts varances of Levy random numbers whch to a great extent mproves the effcency of Levy opton prcng model. Furthermore, owng to the fact that the algorthm should be matched up wth mult-dmensonal random numbers, ths algorthm also consders the quas Monte Carlo smulaton technque, whch generates the mult-dmensonal varables n a sngle pass. As a result, ths method at large ncreases the degree of unformty and the randomness of random numbers and optmzes the algorthm. Secton n ths artcle explans the correlatons between the sample varance and the accuracy of Monte Carlo smulaton technque. Secton 2 ntroduces the ratonales and mplementaton steps. The man algorthm n ths chapter, whch ams at the varance reductons of Levy Process of varyng-tme Brownan algorthm, s on the base of the ratonales of ordnary control varables method. Therefore, ths secton lays the theoretcal foundatons to the later algorthm. Secton 3 formulates the generatng algorthm of

2 Appled and Computatonal Mathematcs 205; 4(3): Levy random numbers va the characterstcs of Levy random process and manly ntroduces the generatng algorthm of pure ump Levy Process. And ths algorthm makes the cushon to man algorthm of ths chapter. Secton 4 elaborates the quas Monte Carlo mult-dmensonal random number generatng algorthm. The algorthm deals wth the generaton of mult-dmensonal random numbers. Snce the man algorthm n ths chapter requres several matched random numbers, thus for the purpose of elevatng the evenness of data dsperson, we use the quas Monte Carlo smulaton technque. On the bass of prevous sectons, Secton 5 combnes the characterstcs of Levy random numbers, the ratonales of control varables method and quas Monte Carlo technque and gves the varance reducton technque for the varyng-tme Brownan algorthm, whch s tested emprcally n Secton 6. The emprcal result llustrates that the two pars of random numbers have a good correlaton and offers a satsfactory effect to reduce the varances. 2. Varance Reducton Technque Prncple Frst, we ntroduce how the varance affects the effcency of estmaton n the Monte Carlo smulaton. Suppose that the random varable X has probablty densty functon f. We consder computng the value of functon g ( X F ) smulaton. In the context of opton prcng, we regard the X as the varable of logarthmc return rate of the underlyng assets, where f s the respectve Levy dstrbutonal functon, and ( ) g X F s the respectve opton s prcng functon. We obtan the sample set by n-tme smulatons. In ths case, we need to evaluate the opton s prce on the nformaton set F: C ( X F ) = E g ( X ) F Let us consder the sample mean: ˆ C = g x t = ( ) by (2.) What we need s that the sample mean C approxmaton to the true values, by the law of large numbers can attest, any ε > 0, there are: 2 2 ( ˆ < εσ ) ( ε ) Prob C C / (2.2) The σ s a random varable, the standard devaton of X, the correspondng confdence level can be obtaned, under the condton of α, the number of smulatons must be determned by the formula: αε 2 (2.3) ow the confdence nterval s wthn Cˆ εσ, Cˆ εσ. + We can nfer from ths result that wth gven estmaton tmes, the confdence ntervals can be shortened by reducng the varance of random varables to mprove the accuracy of estmates. Here note that, wth the standard devaton of the random varable beng unknown, for some real-world calculaton, we can replace t by sample standard devaton. 3. Levy Process Varance Reducton Technques Under Tme-Varyng Brown Algorthm As for the selecton of control varable CY, Dngec, Hormann has creatvely use the smulaton result under the geometrc Brownan moton of opton Monte Carlo as the control varables wth the Levy Process s generatng algorthm. It smplfes the generaton of the control varables and facltates the applcaton to the opton prcng. By ths means, our study has constructed ths varance reducton technque based on the tme-varyng Brownan algorthm and quas-monte Carlo method of Levy process for opton prcng. Contrary to the ordnary control varables method for opton prcng, ths algorthm has the followng characterstcs:. It ntroduces the subordnate Brownan moton Levy random number generatng algorthm and could be convenently expanded to most of Levy processes, whch makes the varance reducton technque for opton prcng more unversal. 2. The control varable CY generated by ths technque has a much hgher correlaton level wth the orgnal varable Y. It to a larger part mproves the effect of reducng varance. 3. By quas Monte Carlo smulaton technque, the mult-dmensonal random varables are more densely-deployed and the overall computatonal effcency s mproved. The followngs are the detals of mplementaton: For the underlyng asset wth the expraton date T, the openng prce So and the prcng prce, we defne the opton Y. The varance reducton algorthm of Levy Monte Carlo on subordnate Brownan moton algorthm has the followng steps:. for = : where s the smulaton tmes. 2. usng the Ermontecalo smulaton algorthm, generatng the tme of length T K dmensonal unform dstrbuton of [0,] number; U = U,..., U,..., U, k ( k,) ( k,2) ( k, T ) [ ] U U,..., U,..., U, k K, t T, U U 0, = ( ) ( ) 3. We use the frst two dmensons of the unformly-dstrbuted random numbers to generate the one-dmenson normal random number B wth the length of T by BOX-Muller algorthm. 4. Usng the normal random number B, accordng to the model and the hstorcal data of opton Y, we can generate the random number of return rate subect to geometrc Brownan moton. Wth the remanng normal random numbers, takng B as the basc subordnate Brownan moton, we also generate the return rate seres LR by subordnate Brownan moton Levy generatng k k, t K

3 76 L Zhou et al.: Opton Prcng Varance Reducton Technques Under the Levy Process algorthm. 5. Measure transformaton of two groups of random numbers, converted nto a rsk neutral measure value. 6. Smulaton of path constructon of basc assets LR LS = LS,..., LS,..., LS T, LS = e LS by S0 and LR ; at the same tme, Smulaton of path constructon of basc assets CS = CS,..., CS,..., CS T, CR CS = e CS by S and CR 0 7. By Levy process under the asset path, geometrc Brown moton path and asset opton prcng formula, the frst generaton of smulated prce optons: rt rt LY e ψ LS and CY = e ψ CS t = ( ) ( ) 8. Constructon of new varables: 9. End 0. Take the mean of the opton prce Y by -tme smulaton. And take the confdence nterval accordng to the related dstrbutons. 4. The Emprcal Results of Varance Reducton To test the effect of the varance reducton technque of tme-varyng Brownan moton quas-monte Carlo smulaton, ths secton uses European style opton S wth the underlyng IG parameter estmaton FG parameter estmaton ( [ ]) Y = LY α CY E CY * t t asset ted to HSI tradng n the Hong Kong Stock Exchange. By adoptng IG, VG as two types of Levy Process and Halton, Sobol as two low-dscrepancy sequences, we smulate the opton prcng. IG process s a random dstrbuton functon whch s modfed out from IG (Inverse Gaussan) process by Barndvrff(998). Because ths type of functon has good propertes of nfnty Dversble, etc, t s easy to take the model transformaton and effcent to generate random numbers. It wde appled to the dervatve prcng models under the Levy process(stentoft,2008 ). VG process s the generalzed hyperbolc random functon proposed by Madan and Seneta(990). Snce VG process s greatly characterzed by the hgh order moment of fnancal data and can be generated by two ndependent Gamma processes, t s one of the most popular pure ump-levy processes. The selecton of low-dscrepancy sequences s owng to the hgh generatng effcency of Halton, smultaneously the evenness of the hgh-dmensonal dsperson. 4.. The Prcng Results Table. Levy parameter estmaton results. Frst, we use the moment estmaton method to estmate the logarthm return rate of HSI. Table reports the estmaton results. As these parameters are the values under the true measurement, the rsk-neutral measure transformaton s needed before the dervatve prcng. α β δ µ 9.45E E σ ν θ µ Ths model has nothng to do wth the condtonal heteroskedastcty. When measurng the rsk neutral level, we can base on the dentty functon of random process to transform by rsk-free return rate. The calculaton of the new drft term s: ϕ µ = r t ( ) * (4.) Where W s the exponent part of the dentty functon. respectvely. The dentty functons of IG, VG process are: ψ ( ; µ, α, β, δ ) u E e ux e u µ + δ α 2 + β 2 δ α 2 ( β + u) 2 IG = = (4.2) ux t uµ t 2 2 ψ VG ( u; Xt, µ, σ, ν, θ ) = E e = e uθν + σ νu 2 t ν (4.3) Based on the prevous transformaton, we can smulate the logarthm return rate under the rsk neutral measurement. Fgure -2 reports regresson results of the smulaton dstrbuton and the real-data dstrbuton by IG and VG models. As the graph shows, the two models for return rate can demonstrate the dstrbuton characterstcs of fnancal data, especally the rght-skewed and fat-taled.

4 Appled and Computatonal Mathematcs 205; 4(3): Fg.. IG model random dstrbuton fttng. Reduce the data of return rate by smulaton to the prce path. Them, by the prcng formula of European opton, use the quas Monte Carlo varance reducton technque prcng Fg. 2. VG model random dstrbuton fttng. the 25 put and call optons wth dstnct expraton prces respectvely. Fg. 3. Call opton prcng results under the IG model. Fg. 4. Put opton prcng results under the IG model. Fg. 5. Call opton prcng results under the VG model. In order to compare the IG, VG models and test the effect of Sobol, Halton seres on the prcng result, our study contrasted the prcng methods. Table 2 presents the Fg. 6. Put opton prcng results under the VG model. dfferences. Frst, usng IG, VG models, then prce the 25 call and put optons by Halton and Sobol low-dscrepancy sequences respectvely. Second, use RMSE (root mean square

5 78 L Zhou et al.: Opton Prcng Varance Reducton Technques Under the Levy Process error) and AAE(average absolute error) to account the prcng result. The two ndcators are desgned to measure the dfference between the prcng result and market prce. A smaller error shows a greater precson. The formulas of these two ndcators are: RMSE = AAE = = Model Maket ( C ) 2 C = C Model C Maket (4.4) Table 2. Levy prcng model results. Opton type Call optons Put optons ndcators IG process IG process VG process VG process Halton sequence Sobol sequence Halton sequence Sobol sequence RMSE AAE RMSE AAE As s llustrated n the results, the precson s slghtly hgher than IG. These two types of Levy process can both help offer a greatly precse prcng. As for the VG process, the prcng results of call optons appear to be better than put optons. For call optons, Halton sequences 4.2. Random Dstrbuton Characterstcs Comparng to ordnary Monte Carlo smulaton, the maor advantage of quas Monte Carlo varance reducton technque s to accelerate the speed of convergence by applyng the quas Monte Carlo technque and varance reducton technque. Thus, ths Secton presents the dynamc process of convergence n ths smulaton by ths method. Fg. 7. The IG opton prcng process smulaton convergence process. Fg. 8. The VG opton prcng process smulaton convergence process.

6 Appled and Computatonal Mathematcs 205; 4(3): Table 2 shows the rates of convergence to the mean wth the ncrease n the smulaton tmes of the prcng results, under the IG and VC process, respectvely by the quas Monte Carlo varance reducton technque and ordnary Monte Carlo smulaton. In case of Monte Carlo smulaton, the prcng of opton requres at least 500 tmes smulaton to acheve the convergence to the mean. Whle takng ths varance reducton technque, only 50 tmes smulaton can the prcng result be controlled wthn a relable range. Ths shows that, for dervatves prcng models, quas Monte Carlo varance reducton technque can effectvely reduce the varance and the necessary smulaton tmes, whch helps mprove the effcency of prcng. 5. Summary The applcaton of Levy process can promote the overall prcng performance. At the expense of effectveness, the Levy process seems more complcated and less effcent than the tradtonal normal random numbers. Consderng ensurng the effectveness of the prcng result after the use of Monte Carlo smulaton technque, a large number of smulatons should be taken. Thus how to mprove the prcng performance under the crcumstance of a low-effcent smulaton s crtcal to all the Levy process Monte Carlo smulaton. In years of research, varance reducton technques are capable of mprove the effcency of Monte Carlo smulaton. Moreover, the algorthm s relatvely smple. It lays the techncal foundaton for Levy process varance reducton technque on the tme-varyng Brownan algorthm n ths chapter. In the framework of ths algorthm, we combne the followng algorthm. It corporates the subordnate Brownan moton Levy random number generatng algorthm. Because ths algorthm s based on the Brownan moton random number, we use the Subordnators to compress and transform the Brownan moton to Levy process. The Brownan motons before and after the compresson have a strong assocaton. Thus we use the correspondng Brownan moton as the bass of generatng two groups of smulaton data. One group s Levy process and the other s the related geometrc Brown moton. Then followng the framework of control varables method leads to the reducton of varance. 2. The algorthms above nvolve many a random number. So for the reason of ncreasng the dsperson evenness, we use the quas Monte Carlo smulaton and match wth the Box-Buller algorthm to generate normal random numbers. Ths offers the basc data support for the Levy process varance reducton technques on tme-varyng Brownan algorthm. Fnally, we use warranty dataset n Hong Kong to further the smulaton theory. The emprcal result shows that ths method can help to fasten the rate of convergence of smulaton and promote the effcency of prcng. The precson of prcng can be mproved. The specfc effects of varance reductons also rest wth the styles of optons, the type of Levy process and the selecton of low-dscrepancy sequences. Acknowledgements Ths paper s funded by the proect of atonal atural Scence Fund, Logstcs dstrbuton of artfcal order pckng random process model analyss and research(proect number: ); and funded by ntellgent logstcs system Beng Key Laboratory (o.bz02); and funded by scentfc-research bases--- Scence & Technology Innovaton Platform---Modern logstcs nformaton and control technology research (Proect number: PXM205_0424_00000); and funded by school year, Beng Wuz Unversty, College students' scentfc research and entrepreneural acton plan proect (o.68); and funded by Beng Wuz Unversty, Yunhe scholars program ( /007); and funded by Beng Wuz Unversty, Management scence and engneerng Professonal group of constructon proects. (o. PXM205_0424_000039). Unversty Cultvaton Fund Proect of 204-Research on Congeston Model and algorthm of pckng system n dstrbuton center ( ) References [] Koponen, I. Analytc approach to the problem of convergence of truncated Levy flghts towards the Gaussan stochastc process[j]. Physcal Revew E, 995,52: [2] Lays Stentoft. Amercan opton prcng usng smulaton: an ntroducton wth to the GARCH opton prcng model[c]. CREATES workng paper, 202. [3] Lehar A, Schecher M, Schttenkopf C. GARCH vs. stochastc volatlty: opton prcng and rsk management[j]. Journal of Bankng & Fnance, 2002,60(): [4] Longstaff F A, Schwartz E S. Valung Amercan optons by smulaton: a smple least-squares approach[j]. The Revew of Fnancal Studes, 200, 4():3-47. [5] Lyda W. Amercan Monte Carlo opton prcng under pure ump Levy models[d]. Stellenbosch Unversty, 203. [6] Km J, Jang B G, Km K T. A smple teratve method for the valuaton of Amercan optons[j). Quanttatve Fnance, 203, 3(6): [7] Chorro C, Guegan D, hyperbolc Lelpo F. Opton prcng for GARCH-type models wth nnovaton[j]. Fnance, 202, 2(7): [8] Chrstoffersen P, Jacobs K, Ornthanala C. GARCH opton valuaton: and evdence[z]. Aarhus Unversty, Workng Paper, 202.theory [9] Byun SJ, Mn B. Condtonal volatlty and the GARCH opton prcng model wth non-normal nnovatons[j]. 3ournal of Futures Market, 243, 33(): -28.

7 80 L Zhou et al.: Opton Prcng Varance Reducton Technques Under the Levy Process [0] Carr P, Madan D B. Opton valuaton usng the fast Fourer transform[j].journal of Computatonal Fnance, 999, 2(4): [] Carr P, Geman H, Madan D H and Yor M. The fne structure of asset returns: an emprcal nvestgaton[j]. Journal of Busness, 2002, 75(2): [2] Carr P and Wu L R. The fnte moment log stable process and opton prcng[j]. Journal of Fnance, 2003, 58(2): [3] Carrere J F. Valuaton of the early exercse prce for optons usng smulatons and nonparametrc regresson[j]. Insurance: Mathematcs and Economcs, 996, 9(): 9-30;