An Empirical Comparison of Two Stochastic Volatility Models using Indian Market Data

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1 Asa-Pacfc Fnan Marets (2013) 20: DOI /s An Emprcal Comparson of Two Stochastc Volatlty Models usng Indan Maret Data Sranth K. Iyer Seema Nanda Swapnl Kumar Publshed onlne: 8 March 2013 Sprnger Scence+Busness Meda New Yor 2013 Abstract We conduct an emprcal comparson of hedgng strateges for two dfferent stochastc volatlty models proposed n the lterature. One s an asymptotc expanson approach and the other s the rs-mnmzng approach appled to a Marov-swtched geometrc Brownan moton. We also compare these wth the Blac Scholes delta hedgng strateges usng hstorcal and mpled volatltes. The dervatves we consder are European call optons on the NIFTY ndex of the Indan Natonal Stoc Exchange. We compare a few cases wth proft and loss data from a tradng des. We fnd that for the cases that we analyzed, by far the better results are obtaned for the Marovswtched geometrc Brownan moton. Keywords Opton prcng Stochastc volatlty Mean revertng Regme swtchng Rs mnmzng JEL Classfcaton C02 C90 G13 Sranth K. Iyer: Research Supported n part by UGC SAP -IV. S. K. Iyer Department of Mathematcs, Indan Insttute of Scence, Bangalore, Inda e-mal: syer@math.sc.ernet.n S. Nanda (B) Center for Applcable Mathematcs, Tata Insttute of Fundamental Research, Bangalore, Inda e-mal: nanda@math.tfrbng.res.n S. Kumar Department of Mathematcs, Indan Insttute of Technology, Delh, Inda

2 244 S. K. Iyer et al. 1 Introducton The Blac Scholes model for prcng optons has long been a standard tool n the world of fnance. The model operates under several assumptons, n partcular that of constant volatlty and the assumpton that marets are complete. That volatlty s not constant n the real world s well accepted and evdenced n the non-constant mpled volatlty surface. Stochastc volatlty models have been proposed n lterature to account for the random behavour of maret volatlty, the volatlty smle and to produce dstrbutons of returns wth heaver tals. The two models n ths paper lead to ncompleteness of the maret, that s, there s no unque far prce of an opton. Several prcng models based on the assumpton of stochastc volatlty have appeared n the lterature. See for example Deshpande and Ghosh (2008), Fouque et al. (2000), Heston (1993), Hull and Whte (1987) and Schwezer (2001a,b), for models that explctly model the volatlty process, the case we are nterested n. There are n addton dscrete tme models that fall nto the category of GARCH models Heston and Nand (2000), whch are not what we wsh to consder here. In a complete maret model, the hedgng strategy does not nvolve any addtonal cost. In an ncomplete maret, hedgng strateges wll not be self fnancng, so t s mportant to ascertan the actual cost nvolved n hedgng. In ths paper we compare the performance of two stochastc volatlty models for prcng European call optons wth the Blac Scholes model by loong at the proft and loss (P&L) resultng from delta hedgng, for optons traded on the Indan marets. These are models proposed by Fouque et al. (2000) and by Basa et al. (2009). They represent two dfferent approaches to the problem of prcng contngent clams n an ncomplete maret. We provde two comparsons usng the Blac Scholes model: one wth a fxed hstorcal volatlty, and a second usng half-hourly maret mpled volatltes for hedgng. The latter may be a more realstc scenaro used by traders. In fact the data we compare comes from a tradng des that has used ths last scenaro. In the two models consdered, the underlyng asset evolves accordng to a geometrc Brownan moton, but wth stochastc volatlty. Some studes such as Fouque et al. (2000) have used maret data to llustrate the applcablty of ther models. An emprcal study for a dfferent class of stochastc volatlty models ncludng the models due to Heston (1993), Hull and Whte (1987), and others can be found n Poulsen et al. (2009). The Hull and Whte model (Hull and Whte 1987) and a model by Sten and Sten (1991) deal wth stuatons where the correlaton between volatlty and the spot asset s prce s zero, almost never the case n realty, so we dsregard these models. The Heston model (Heston 1993) does address the case where correlaton between volatlty and the spot asset s prce s not zero, as n the two models consdered n ths paper. A major challenge n the Heston model s the estmaton of several parameters, whch depend on the volatlty of the underlyng. In the end one s left wth a nontrval optmzaton problem to determne the parameters, where the functon to be optmzed s not convex. The results of the Heston model fnally depend upon a good optmzaton technque. Developng a good optmzaton scheme whle a worthwhle exercse, falls outsde the scope of ths paper and has been attempted by many others (Bora et al. 2005; Carr and Madan 1999; Moodley 2005). The Heston model has a much hgher level of computatonal complexty as compared to the models that we have ncluded n our study.

3 An Emprcal Comparson of Two Stochastc Volatlty Models 245 Where the present study dffers from prevous s that we compare the actual performance of the P&L of the hedgng portfolo based on hgh frequency data for the models mentoned above, wth the outcomes of a tradng des. The data set used consttutes optons on the NIFTY ndex of the Indan Natonal Stoc Exchange (NSE). To the best of our nowledge, such a study s not avalable n the lterature. There s a study whch compares the hedgng performance of several portfolo credt rs models (Cont and Kan 2011). They use ndex data from European marets. Complexty of parameter estmaton and data requrements of these models are also dscussed. We begn by descrbng the two stochastc volatlty models brefly. The exact formulas used and the estmaton procedures are gven n the subsequent sectons. We frst consder the approach by Fouque et al. (2000). In ths model, the volatlty process evolves accordng to a fast mean-revertng contnuous dffuson. The mean reverson s characterzed by a parameter ɛ whch s small f the rate of mean reverson s large. The dea s to expand the prcng equaton and gnore terms that are of order smaller than ɛ. The advantage of ths approach s that the resultng formula for the opton prce does not depend on the unobserved current value of the volatlty. However, calbraton of ths model requres nformaton on near-the-money mpled volatltes. Wth ths addtonal nformaton, estmaton n ths case s straghtforward. The prce and hedge ratos are obtaned usng smple expressons that are correctons to the correspondng formulas n the standard Blac Scholes model. The second approach we consder was proposed by Föllmer and Schwezer (1990) to prce contngent clams n ncomplete marets. The contngent clam s assumed to be of the European type and the underlyng s assumed to be a sem-martngale. They ntroduce the noton of a locally rs mnmzng strategy, whch unquely determnes the prce n the class of strateges that match the payoff from the contngent clam at the termnal tme. Such strateges are n general non-self fnancng. Roughly speang, a strategy s locally rs mnmzng f small perturbatons of the strategy over small tme ntervals lead to an ncrease n the expected quadratc cost of the hedge. The ey to fndng such a locally rs mnmzng strategy s the exstence of a Follmer Schwezer decomposton. The contngent clam admts a decomposton as a sum of a measurable (at tme 0) random varable, a stochastc ntegral wth respect to the underlyng asset and a martngale that s strongly orthogonal to the martngale part of the underlyng. The addtonal cost of mantanng the hedge s gven by the thrd term n the decomposton. A good ntroducton to ths approach can be found n Schwezer (2001a). The explct formulas for rs mnmzng strategy usng the Follmer and Schwezer approach for a Marov regme-swtched geometrc Brownan moton was wored out n Deshpande and Ghosh (2008). The maret s assumed to exhbt a fnte number of regmes whch change accordng to a contnuous tme Marov chan (MC). The drft and volatlty parameters of the asset prce process are functons of the regme, as s the nterest rate. There are several ssues to be addressed n mplementng ths model. The frst s that the prce of the opton and the hedge depend on the parameters of the underlyng Marov chan whch s not drectly observed. Estmaton of the parameters of the Marov chan can be qute nvolved. We propose an expectaton maxmzaton (EM) algorthm for the parameters of the Marov chan. The second

4 246 S. K. Iyer et al. dffculty wth ths approach s that the prce and the hedge depend on the current regme or state of ths chan whch s not observed. Hence we need to estmate t. Fnally, the system of coupled partal dfferental equatons (PDEs) governng the prce do not have a closed form soluton. Obtanng numercal solutons to these PDEs s computatonally ntensve. One can use more sophstcated technques such as nonlnear flterng etc. to estmate the current state of the MC. We propose an estmaton technque that s fast, and gves satsfactory results. Ths s mportant snce we are comparng ths model wth other models whch are also computatonally effcent. As a frst step we plot the drft and volatlty estmated usng a rollng wndow. Ths plot gves us a farly good dea of the number of regmes that we can consder. Ths s only a prelmnary step followed by parameters estmated usng the EM algorthm. We tested ths procedure on smulated data extensvely, before usng t for the actual data sets. It s a drawbac n any stochastc volatlty model that the prce depends on the current volatlty. In our case we need to estmate the current state of the Marov Chan. To hedge usng the Marov regme-swtched geometrc Brownan moton model, we have to estmate the current state of the Marov chan. That s, we need to now what the current regme s. There s an added rs exposure due to a postve probablty of shft to a dfferent regme from the current one. We neglect ths rs just as we gnore transacton costs etc. In any ncomplete maret model, there s no unque martngale measure, and so we have to pc one accordng to some crteron. The choce n the regme swtched case s the mnmal martngale measure whch s locally rs mnmzng. Gven ths, we do have a hedgng strategy. Also ths mnmal martngale measure s exactly the one gven by the Escher transforms, where the change of measure s carred out usng the Grsanov s theorem and the maret prce of rs. For the Fouque et al. (2000) model the prcng measure s determned by the opton prce surface. Both the models we compare n ths paper can be thought of as approxmatons to contnuous volatlty models. It s a lot harder to estmate the current volatlty n a contnuous volatlty model than to estmate the state of the Marov chan as we demonstrate. Further, contnuous tme models do not capture jumps n volatltes or prces. Large changes such as those that happen durng a fnancal crss can possbly be captured by the Marov swtchng model but not by a contnuous tme volatlty model. So n that sense havng some rare states that can capture possblty of such large changes, even wth some lac of precson, may yeld more realstc prces than those obtaned from contnuous volatlty models. If the Marov swtched model s calbrated based on lqud opton prces, then states representng such large changes, f any wll be determned by the current opton prces. The rest of the paper s organzed as follows. In Sect. 2, we gve the explct formulas used to compute the prce as well as the hedge ratos for the two models descrbe above. Secton 3 descrbes some of the numercs, where explanaton s deemed necessary. It s assumed that the reader has famlarty wth most of the numercs and hence detals are omtted. Ths secton manly descrbes the results of our numercal calculatons, ncludng parameter estmaton procedures. Ths s followed by a dscusson of our conclusons n Sect. 5.

5 An Emprcal Comparson of Two Stochastc Volatlty Models Models, Prcng and Estmaton Throughout ths paper, we assume the maret s restrcted to two tradable assets: a money maret account and a stoc prce. The tme horzon of nterest s [0, T ]. The money maret account {St 0, t 0} satsfes S 0 t = S 0 0 ert (1) where r s the fxed nstantaneous rs free rate of nterest. The contngent clam under study s a European call opton gven by H = (S 1 T K )+, (2) where {St 1, t 0} s the prce process underlyng the contngent clam and K s the stre prce. 2.1 Geometrc Brownan Moton (GBM) wth Fast Mean Revertng Volatlty We frst descrbe the approach by Fouque et al. (2000). Here the underlyng asset process evolves accordng to the stochastc dfferental equaton ds 1 t = μs 1 t dt + σ t S 1 t dw t, (3) wth σ t = f (Y t ), where f s a postve functon and the process Y t drvng the volatlty process s assumed to be a fast mean-revertng dffuson process. That s, Y t evolves accordng to the followng stochastc-dfferental equaton: dy t = α(m Y t )dt + βdz t (4) Here Z t, and W t, t 0, s a standard Wener process. The two drvng processes Z t and W t may be correlated. The long term mean of Y t s m and α s the rate of mean reverson, whch s assumed to be large. It follows that the process Y t s ergodc, that s, Y t has a unque nvarant dstrbuton. Snce the process Y t s fast mean-revertng t wll come close to ts mean several tmes durng the lfe of the opton. Let σ be the hstorcal volatlty, that s, the varance of the random varable f (Y ), where the dstrbuton of Y s the nvarant dstrbuton of the process Y t. Assume that the mpled volatlty surface I, across stres and tmes to maturty, satsfes the followng equaton for near-the-money optons. ( stre prce log stoc prce I = a tme to maturty ) + b. (5) Defne

6 248 S. K. Iyer et al. ( V 2 = σ ( σ b) a (r + 32 )) σ 2, V 3 = a σ 3. (6) For detals of dervaton of V 2 and V 3 the reader may refer to Fouque et al. (2000, pages ). Let P 0 (t, x) be the Blac Scholes prce of a European call opton at tme t and spot prce x, maturng at tme T wth constant volatlty σ and nterest rate r. The corrected prce of a call opton s derved n Chapter 5 of Fouque et al. (2000), as ( ) C(t, x) = P 0 (t, x) (T t) V 2 x 2 2 P 0 x 2 + V 3x 3 3 P 0 x 3. (7) The prcng n ths case nvolves nowledge of dervatve prces across stres and maturtes so that the parameters a and b can be estmated usng (5). The advantages are the ease of mplementaton of ths procedure and the parsmony n the number of parameters. The hedgng rato s gven by [see (7.13) n Fouque et al. 2000] P 0 x (T t) x ( ) 2V 2 x 2 2 P 0 x 2 + (V 2 + 3V 3 )x 3 3 P 0 x 3 + V 3x 4 4 P 0 x 4. (8) Thus both the prce as well as the hedge rato can be seen as a correcton to the respectve Blac Scholes formulas. 2.2 Marov-Swtched Geometrc Brownan Moton (MS-GBM) In ths secton we descrbe the model studed n Deshpande and Ghosh (2008). Ths model assumes that the maret operates n a fnte number of regmes. The maret swtches from one regme to another accordng to a contnuous tme Marov chan. Let {X t, t 0} be an rreducble Marov chan on the state space S ={1, 2,...,M} wth generatng matrx denoted by Q = (q j ) 1, j M. That s, the evoluton of X t s gven by P[X t+h = j X t = ] =q j h + o(h), = j, where q j 0, for = j and q = j = q j. The evoluton of the stoc prce s accordng to a Marov modulated geometrc Brownan moton descrbed by the stochastc dfferental equaton ds 1 t = μ(x t )S 1 t dt + σ(x t)s 1 t dw t, (9) where μ : S R s the drft coeffcent, σ : S (0, ) s the volatlty parameter and {W t : t 0} s a standard Wener process.

7 An Emprcal Comparson of Two Stochastc Volatlty Models 249 Table 1 Partculars of optons used for comparson Opton Stre Expraton Hedgng perod Aprl 24, 2008 March 3 Aprl 23, Aprl 24, 2008 March 5 Aprl 23, June 26, 2008 March 28 May 12, 2008 The opton prce process s gven by the functon h :[0, T ] R + S R +, that s h(t, S t, X t ) s the locally rs mnmzng opton prce at tme t. From Theorem 3.1 n Deshpande and Ghosh (2008), h(t, s, ) s the soluton to the followng system of coupled partal dfferental equatons h(t, s, ) t σ()2 s 2 2 h(t, s, ) h(t, s, ) s 2 + rs + s M q j h(t, s, ) = rh(t, s, ), S, (10) wth the termnal condton h(t, s, ) = (s K ) +, S. Recall that r s the nstantaneous rate of nterest as n (1). The analyss n Deshpande and Ghosh (2008)assumesthatr depends on X, that s, t vares wth regme changes but for our purposes we eep ths fxed across regmes. The above system has a unque soluton n the class C([0, T ] R) C 1,2 ((0, T ) R) of functons havng at most polynomal growth. Observe that the above prce depends on the current state of the maret X t whch s unobserved. Thus one needs to estmate the dstrbuton of the current maret state and use the prce wth the hghest probablty wth respect to ths dstrbuton. The hedgng rato for ths model s gven by ξ t = h s (t, S1 t, X t). (11) In the next secton we perform the numercal calculatons to arrve at the P&L for the two hedge ratos gven by Eqs. (8) and (11), to compare the performance of the two stochastc volatlty models descrbed here. j=1 3 Numercal Results The performance of the two hedgng strateges outlned above s tested on three call opton contracts at varyng stres. For our calculatons, the hedgng perods for each of these optons, along wth ther expraton dates are stated n Table 1. For convenence we refer to these as optons 1, 2 and 3 respectvely, as ndcated n Table 1. The performance comparson of the two stochastc volatlty models s done by carryng out the delta hedgng and computng the P&L. Addtonally, we compare

8 250 S. K. Iyer et al. wth the P&L obtaned by delta hedgng, usng the more famlar Blac Scholes model n two ways. The frst uses a fxed hstorcal volatlty, and the other uses half-hourly mpled volatltes as the optons data we use has half-hourly quotes. We understand that tradng dess often use Blac Scholes wth mpled volatltes for delta hedgng. Thus we are able to comment on how the stochastc volatlty models fare n comparson to a trader usng mpled volatltes for a few optons. Note that we compare the P&L for only three optons as the hgh frequency data we need for ths computaton s avalable only over the shorter tme ntervals gven n Table 1. We utlze a much longer dsjont tme nterval for estmaton of parameters such as the hstorcal volatlty and the parameters of the Marov chan. The contnuously compoundng rate of nterest r s taen to be per year. 3.1 Estmaton of Parameters For estmatng the parameters of the GBM model wth fast mean revertng volatlty, we follow Fouque et al. (2000). Ths requres hstorcal daly returns data and a few days of end-of-the-day optons prce data. For the Marov-swtched GBM, we need to dentfy regme changes that may happen frequently. We use the half hourly asset prce data for the estmaton whch s also the tme gap between consecutve adjustment of the hedgng portfolo Estmaton of Parameters for the GBM wth Fast Mean Revertng Volatlty The standard devaton of the hstorcal log-returns, σ s estmated usng daly returns data. For optons 1 and 2, we use the data from November 30, 2007 to February 29, We refer to ths as data set 1. For opton 3 we use data from December 26, 2007 to March 27, We refer to ths as data set 2. Our data sets are chosen so as to allow for out of sample data for hedgng. See Table 1 for the hedgng perods used for the three optons. To determne estmates â and ˆb for parameters a and b respectvely, we do a lnear ft to mpled volatlty data that we have from a tradng des, as mentoned earler. More precsely, we ft to closng call optons contracts wthn 5 % of at-the-money optons prces on closng days just pror to the start of the hedgng perod. Usng the above crteron for optons 1 and 2, we used data from the consecutve tradng days of February 28 and 29, For opton 3 data from the tradng days March 26 and 27, 2008 s used. A total of 25 data ponts was avalable for optons 1 and 2 whle for opton 3 we had 56 data ponts. These are a large enough set of numbers to estmate the lnear coeffcents â and ˆb. Snce we have the mpled volatlty data and the logmoneyness-to-maturty-rato[see Eq. (5) above and equaton (6.1) n Fouque et al. 2000], t s a straghtforward exercse n lnear regresson to determne estmates of a and b. The estmates that we obtan for the above parameters are gven n Table 2. Calculaton of V 2 and V 3 are now straghtforward usng Eq. 6. In Fg. 1 we plot the estmates of a and b computed, usng daly closng call optons contracts wthn 5 % at-the-money for 13 sets of tradng days between Jan 1, 2008 and March 27, We do ths to get a sense of how a and b may vary over a 3 month

9 An Emprcal Comparson of Two Stochastc Volatlty Models 251 Table 2 Estmated parameters used for hedgng optons 1, 2 and 3 usng the Fouque et al. model Opton ˆ σ â ˆb V 2 V 3 1, ˆ σ s the standard devaton of the hstorcal log-returns, â s the estmate for a, ˆb s the estmate for b Parameter Estmates for Fouque et. al Model Parameter a mean value of a = std dev.= Parameter b mean value of b = std dev.= Jan 1, 2008 March 27, 2008 Fg. 1 Parameter values a and b obtaned by fttng to Impled Volatltes for Nfty Call Optons. Each of the 13 values of a and b have been ft to 2 days of data where there s suffcent lqudty. These fts are for Jan 1 and 2, Jan 9 and 10,Jan 16 and 17, Jan 23 and 24, Jan 30 and 31, Feb 6 and 7, Feb 13 and 14, Feb 21 and 22, Feb 28 and 29, Mar 5 and 7, Mar 13 and 14, Mar 18 and 19, Mar 26 and 27, Note that marets worldwde plunged on January 21, 2008 and ths s reflected n parameter b on Jan 23 and 24 perod wthn our data set. Both parameters exhbt stablty as seen from ther standard devaton and mean values n the fgure, allowng us to conclude that the estmated values are relable for the perod beng used. Next we dscuss parameter estmaton for the MS-GBM model, whch presents several challenges Estmaton of Parameters for the Marov-Swtched GBM Model The frst challenge here s to dentfy the number of regmes and the drft and volatlty parameters assocated wth each of them, for the Marov-Swtched GBM Model. We use hgh frequency (half hourly quotes) NIFTY data from October 16, 2007 to March 2, 2008, for the frst two optons. We call ths data set 3. We use data from October 16, 2007 to March 27, 2008 for opton 3. Ths s called data set 4. Agan, the data sets are chosen so as to allow for out of sample data for hedgng. For almost each tradng day we have 12 quotes from a tradng des. We denote the half-hourly log-returns by x j, j = 1,...,N and we assume that ths data s dstrbuted as

10 252 S. K. Iyer et al Plot usng half hourly log returns to determne Intal Regmes std dev σ mean μ x 10 3 Fg. 2 Each pont on ths graph s calculated usng a rollng 30 day wndow. Each wndow conssts of 15 days of hstorcal data, and 15 days of subsequent data. Data s hgh frequency Nfty, from October 16, 2007 to Mar 2, 2008, (data set 3) M φ(x j ) = p φ(μ,σ, x j ). =1 Here φ(μ,σ) s the normal densty functon wth mean μ and standard devaton σ. The probablty of beng n regme s denoted by p. We estmate the parameters usng the EM algorthm (Blmes and Gentle 1998) whch s descrbed below. For our data, M s chosen to be 4 as explaned below. Step 1: We frst dentfy the number of regmes n whch the maret operates and obtan an ntal estmate of the drft and volatlty n each of these regmes. We estmate the mean and standard devaton of the half hourly log-returns, usng a rollng wndow of 30 days around each data pont. That s, ths set of 30 days conssts of 15 days precedng and 15 days startng from each data pont. We show the plot of standard devaton versus the mean for these data sets, n Fg. 2 for data set 1. Recall that data set 1 s used for estmatng parameters for hedgng of optons 1 and 2. The statstcs of data set 4 are smlar to data set 1. Based on ths fgure, we select four clusters wth ntal values (μ (0),σ (0) ), = 1,...,4, roughly at the center of each cluster. Then we compute the lelhoods φ(μ (0),σ (0), x j ). Based on the computed lelhoods, the data pont x j s assgned to aregme f ( φ μ (0),σ(0) ) (, x j = max φ 1 4 μ (0),σ (0) ), x j, j = 1,...,N.

11 An Emprcal Comparson of Two Stochastc Volatlty Models 253 Table 3 Estmated drft, volatlty and probabltes of each regme usng data set 3 and EM algorthm Regme ˆμ ˆσ ˆp Once we classfy the data ponts nto regmes we estmate the ntal proportons of data ponts p (0), for = 1...4, n each of the regmes based on the above classfcaton. We note that regmes obtaned for data set 2 (used for estmatng parameters for hedgng of Opton 3) are smlar, probably snce the dfference n data set 3 and 4 s from March 2 to March 27, 2008, whch had almost constant volatlty. For each = 1, 2, 3, 4, the lelhood of the data pont x j beng n regme,gven by p (0) (, j) = p (0) ( φ 4=1 p (0) φ μ (0),σ(0) ), x j ( ). (12) μ (0),σ (0), x j For m 1, repeat the followng step untl a convergence crteron s met. Step 2: The estmate of the mean of the asset prce n regme s Nj=1 μ (m) p (m 1) (, j)x j = Nj=1 p (m 1) (, j), (13) the estmate of the volatlty s σ (m) = Nj=1 p (m 1) (, j)(x j μ (1) Nj=1 p (m 1) (, j) )2, (14) and the estmate for the average tme spent n regme s p (m) = 1 N N p (m 1) (, j). (15) j=1 Compute p (m) (, j) by substtutng the updated values from Eqs. (13) (15) n(12). Estmates of the drft, volatlty and the regme probabltes are gven n Tables 3 and 4. It s nterestng to note from Tables 3 and 4 that, n the regme wth large volatlty, the drft s negatve. Thus the MS-GBM model captures the mportant emprcal observaton that asset prces declne durng perods of hgh volatlty.

12 254 S. K. Iyer et al. Table 4 Estmated regme mean, volatlty and probabltes for each regme usng data set 4 and EM algorthm Regme ˆμ ˆσ ˆp For each data pont x j, j = 1,...,N, we compute the lelhood φ (x j ), where φ s the normal densty functon wth estmate of the mean μ and standard devaton σ obtaned usng the EM algorthm. Based on these lelhoods the data pont x j s reassgned to regme for whch φ (x j ) = max 1 M φ (x j ), j = 1,...,N. Note that the estmaton of regmes s not done usng eye-ballng alone. The plots gve us a farly good dea of the number of regmes that we can consder. We found no mprovement n performance upon ncreasng the number of regmes. Usng the prelmnary estmates, the parameters are then estmated usng the EM algorthm. We tested our procedure on smulated data extensvely before usng t for the actual data sets. One can use more sophstcated technques such as non-lnear flterng etc. to estmate the current state of the MC. But the method we propose wors, s fast, and gves satsfactory results. Ths s mportant snce we are comparng ths model wth other models whch are also computatonally effcent. Once the data ponts are classfed nto varous regmes, the state of the Marov chan s nown. So, we now need to calculate the generator matrx whch specfes the evoluton of the Marov chan. We estmate the transton probabltes p j,for all = j by ˆp j = n j /n, where n s the number of vsts to regme and n j s the number of to j transtons. Estmate ˆv to be the recprocal of the average tme spent n regme. Then ˆq j =ˆv ˆp j for = j gves the transton rates. Lastly, ˆq = = j q j. The generator matrces Q = (q j ) are gven below for the two dfferent data sets 3 and Q = Estmated generator matrx usng data set Q = Estmated generator matrx usng data set 4

13 An Emprcal Comparson of Two Stochastc Volatlty Models 255 Table 5 P&L for the three optons usng hedgng perod from Table 1 Model Opton 1 Opton 2 Opton 3 BSHV 72, ,115 67,068 BSIV 89, ,497 56,406 GBM-FMR 104, ,570 66,870 MS-GBM 22,407 54,665 32,807 BSHV Blac Scholes wth hstorcal volatlty, BSIV Blac Scholes wth mpled volatlty, GBM-FMR GBM wth fast mean revertng volatlty, MS-GBM Marov-swtched GBM Table 6 The fgures are P&L changes for opton 1 usng spatal step sze dz = n MS-GBM Calculaton method 4 Regmes 3 Regmes 2 Regmes Most lely maret states 21,404 20,911 35,856 Average over maret states 26,896 27,695 37,823 Agan the two generator matrces have very smlar entres due to consderable overlap of data sets 3 and 4. The entry ˆq 12 = 0.24 for both generator matrces, mples a 67 % chance of movng from regme 1 to regme 2 whether we use data from March 2, 2008 to March 27, 2008 or not; and ˆp = 0.01 n Table 4 mples that tme spent n regme 1 s 1 % of the tme. On the other hand usng data set 3 there s a 75 % chance of movng from regme 2 to regme 4 when usng data set 3 but a slghtly ncreased chance at 76 % f we use the larger data set 4. 4 Results Usng the numercal procedures descrbed thus far, we have hedged for each of the three optons, for dentcal tme perods for all four models beng compared here. Ths s summarzed n Table 5 below. Note that opton 3 s not hedged tll expraton and hence the values n the last column of Table 5 s the net poston as of May 12, We further compare the performance of the MS-GBM for the most lely maret states, and usng coarser demarcaton of regmes. That s, nstead of the 4 regmes descrbed earler we collapse to 3 and 2 regmes to see f there s any loss of nformaton. Clearly collapsng the number of regmes to 2 worsens the hedgng outcome, ndcatng that too much loss of nformaton would result n ncreased costs. Collapsng to 3 regmes on the other hand does not create a bg dfference to the P&L. Also we loo at the P&L obtaned by averagng wth respect to the estmated dstrbuton of the regme,.e averagng over maret states. Note that the calculaton of the dstrbuton s done at each hedge here. Averagng seems to result n a margnal degradaton of performance. Ths comparson s shown n Table 6. We noted that n the estmaton perod, roughly 13 % of the tme, the estmated maret state were the hgh volatlty regmes (volatltes of 321 and 84 %, see Table 3). In the hedgng perod the volatlty fluctuatons were small and close to the hstorcal volatlty most of the tme. The computatonal complexty of mplementng the

14 256 S. K. Iyer et al. Fg. 3 The standard devaton s calculated usng daly returns for past 15 days MS-GBM model s substantal. Better computatonal capacty than we used may mprove these results further. It would be useful to carry out ths comparson when the hedge perod s longer and has more fluctuatons n prces. From Fg. 3 we observe that the volatlty s mean revertng. However the occasonal sharp spes n the volatltes seen n the graph are not captured by GBM-FMR model. On the other hand, the MS-GBM model captures the frequency, magntude and duraton of such volatlty spes. The frequency of the spes s reflected n the statonary dstrbuton of the Marov chan. The magntude of the spe s gven by the value of the volatlty n the correspondng hgh volatlty state and the duraton can be obtaned from the generator matrx. For example, n data set 3, state 1 has the hghest volatlty and ts value s ˆσ = 3.21 (see Table 3). The long run proporton of tme the Marov chan spends n ths state s 1 % whch mples that one can expect to see such a spe once every 100 days. Each tme the chan enters ths state, t spends a random amount of tme that s exponentally dstrbuted wth mean (0.36) 1 = 2.78 days. The best performng model s the one for whch the P&L stays closest to zero throughout the hedgng perod. Clearly the MS-GBM model shows a superor performance over other models. As an llustraton, n Fg. 4 we show a plot of the daly P&L for three models, namely the GBM-FMR, the MS-GBM and the Blac Scholes wth mpled volatlty for opton 1, over the hedgng perod consdered n ths paper. As can be seen from Fg. 4, the daly P&L for the MS-GBM model stays closest to zero for much of the hedgng perod as compared to other models. In Table 7 the mean value and standard devaton of the daly P&L s ndcated. Another measure of performance, the P&L dvded by the call opton prce at the start of the hedgng perod, s also ncluded n Table 7. Ths s smlar to the prce-earnngs rato and s

15 An Emprcal Comparson of Two Stochastc Volatlty Models 257 Fg. 4 Ths fgure shows a comparson of P&L acheved over the same hedgng perod for Opton 1 (stre 4900) by 3 models. The daly hedge rato graph can be obtaned from ths graph by dvdng each data pont by 366,000 Table 7 The fgures are mean and standard devaton for end of day P&L for the 3 models as well as for the hedge rato of these models GBM-FMR MS-GBM BSIV P&L Mean 48,205 11,338 33,591 Standard devaton 33,966 14,573 28,993 Hedge rato Mean Standard devaton sometmes referred to as the hedge rato. We do not separately plot the graph of the daly hedge ratos as t exactly mmcs the P&L graph n Fg. 4 and can be obtaned by dvdng each data pont n ths graph by INR 366,000, the begnnng opton prce. 5 Concluson We compare the performance as measured by the P&L of two models proposed recently, that have low computatonal complexty. These are the Marov-swtched GBM and the GBM wth fast mean-revertng volatlty. The preferred model would be the one for whch the P&L s as close to zero as possble. It s seen from Table 5 that the performance of the frst three models s comparable, whle the Marov-Swtched

16 258 S. K. Iyer et al. GBM gves a substantally better performance for the data we are loong at. That s, the P&L s closest to zero compared to the other three methods of hedgng. Impled volatltes are consdered to reflect the maret s percepton of the prce. Therefore t s of partcular nterest that sgnfcantly better results are obtaned usng the MS-GBM as compared to the BSIV, whch s what many tradng dess use. Admttedly the data used for llustraton s for only a few optons. There s dffculty n obtanng prces from actual trades made by a trader as ths s typcally propretary. We were fortunate to obtan ths data over a sgnfcant perod of tme, so that we were able to offer a reasonable comparson study here. Ths wor serves as an llustraton of the dfference n performance of models, a worthwhle exercse especally snce the MS-GBM model has not been examned, unle the other models dscussed here. To our nowledge ths nd of a model comparson has not been done on any data, that s, ths comparson of P&L as would be seen at a tradng des does not seem to appear anywhere n the lterature. It would be nterestng and worthwhle to conduct our comparson over a larger data set, whch admttedly s hard to obtan, as well as over a longer tme perod to see f our conclusons hold. Acnowledgments We wsh to than Kota Securtes for provdng us wth optons traded data for ths paper. We also wsh to acnowledge Abhnav Srvastava for some of the prelmnary computatons done durng hs summer project. Thans to the referee for several useful suggestons for mprovements to ths paper. References Basa, G. K., Ghosh, M. K., & Goswam, A. (2009). Rs mnmzng opton prcng for a class of exotc optons n a marov modulated maret. Worng paper. Blmes, J. A., & Gentle, A. (1998, Aprl). Tutoral of the EM algorthm and ts applcaton to parameter estmaton for Gaussan mxture and hdden Marov models optons prcng, nterest rates and rs management. Internatonal Computer Scence Insttute Bereley, CA. Retreved June, 2008 from Bora, S., Detlefsen, K., & Härdle, W. (2005). FFT based opton prcng. Dscusson paper: Center For Applcable Statstcs and Economcs, Humboldt Unverstät du Berln, Germany. Carr, P.,& Madan, D. B. (1999). Opton valuaton the usng fast fourer transform. Journal of Computatonal, Fnance 2(4), Cont, R., & Kan, Y. (2011). Dynamc hedgng of portfolo credt dervatves. SIAM Journal on Fnancal Mathematcs, 2(1), Deshpande, A., & Ghosh, M. K. (2008). Rs mnmzng opton prcng n a regme swtchng maret. Stochastc Analyss and Applcatons, 26, Fouque, J. P., Papancolaou, G., & Srcar, K. R. (2000). Stochastc volatlty correcton to Blac Scholes dervatves n fnancal marets wth stochastc volatlty. Cambrdge: Cambrdge Unversty Press. Föllmer, H., & Schwezer, M. (1990). Hedgng of contngent clams under ncomplete nformaton. In M. H. A. Davs & R. J. Ellott (Eds.), Appled stochastc analyss, stochastc monographs (Vol. 5, pp ). New Yor: Gordon and Breach. Heston, S. L. (1993). A closed form soluton for optons wth stochastc volatlty wth applcatons to bond and currency optons. The Revew of Fnancal Studes, 6, Heston, S. L., & Nand, S. (2000). A closed-form GARCH opton valuaton model. Revew of Fnancal Studes, 13(3), Hull, J., & Whte, A. (1987). The prcng of optons wth stochastc volatltes. The Journal of Fnance, 42(2),

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