Unconstrained Fitting of Non-Central Risk-Neutral Densities Using a Mixture of Normals

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1 Unconstraned Fttng of Non-Central Rsk-Neutral Denstes Usng a Mxture of Normals Rccardo Rebonato and Teresa Cardoso QUARC November 25, 2003 Abstract 1 Statement of the Problem One of the most mportant problems n calbratngopton models (eg, stochastcvolatlty, local-volatlty, jump-d uson, etc) s obtanng a relable smle surface from the (often nosy and non-contemporaneous) market prces of planvanlla optons. For some models t can be debated whether usng undoctored nosy prces mght be better than regularzng the nput, but, for some approaches, such as local-volatlty models, havng a smooth (and d erentable) nput smle surface s a must (see the bref dscusson below). Many technques have been proposed to obtan from the observed market prces of plan-vanlla optons a smooth volatlty surface. A systematc survey would requre a full revew artcle n tself. Wthout attemptng to cover the lterature systematcally, we smply menton Shmko (1994) as a representatve example of a ttng procedure drectly to prces, Jacquer and Jarrow (1995), Madan, Carr and Chang (1998) and Ellot, Lahae and Madan (1995) as examples of ttng to transformed prces, At-Sahala (2002) for a drect ttng of the mpled volatlty curve, Avellaneda (1998) for an applcaton of the mnmumentropy methodology, Mrfenderesk and Rebonato (2001) for drect modellng of the rsk-neutral densty. Most of these models fall n ether of three categores: smooth nterpolaton of (possbly transformed) prces, smooth nterpolaton of mpled volatltes, or smooth ttng of rsk-neutral denstes. We argue that modellng drectly the densty s the most desrable approach, because a smooth and plausble densty wll certanly produce (by ntegraton) smooth and well-behaved prce and mpled volatlty surfaces, but the converse s not true. See, for nstance, the dscusson n Mrfenderesk and Rebonato 1

2 (2001). Indeed, f one models drectly prces one can easly obtan wldly oscllatng denstes because the latter are obtaned, modulo dscountng, form the call prces by double d erentaton: Á(S) 2 (1) It s mportant to observe that no model-dependent assumptons are needed to obtan Equaton 1: one smply requres that the prce of a European opton can be wrtten as a dscounted expectaton of the termnal payo, G, over some rsk-neutral densty. Wllmot (1998) hghlghts that, f the mpled rsk-neutral densty s not well behaved, local-volatlty solutons can become extremely nosy. Ths s easy to see, f one recalls Dupre s (1994) well-known result ¾ 2 K;T = + (r + K;T (0;S) 2 C K;T(0;S) K 2 2 Note, n fact, that the denomnator n the expresson above s smply lnked to the rsk-neutral densty. Rebonato (2003) argues that obtanng smooth and well-behaved densty functons s mportant also n more general modellng contexts. One mght argue that, as long as the mpled volatlty has been smoothly nterpolated, the resultng rsk-neutral denstes wll be smooth as well. Fgs 1 and 2 show that ths s not the case: Fg. 1 dsplays two optcally ndstngushable mpled volatlty curves, and Fg 2 the assocated rsk-neutral denstes. It s clear that relyng on the smoothness of the mpled volatlty curve would not guarantee, for nstance, that the local volatlty extracted usng Equaton 2 would have desrable propertes. Ths paper therefore sets as ts goal to obtan an excellent t to the market prces of plan-vanlla optons by drectly modellng the rsk-neutral densty. We choose to do so by extendng the mxture-of-normals approach presented n Alexander (2001). We present a numercal technque that wll allow us to nd the solutons n an e cent and unconstraned manner, whle automatcally satsfyng all the nancal and mathematcal requrements. 2 Summary of Known Results If one wants to nd the rsk-neutral densty, Á(ln S T ), mpled by a set of market prces for a gven tme horzon T, one can express t as a lnear combnaton of normal denstes, '(¹ ;¾ 2 ): Á(ln S T ) = X w '(¹ ;¾ 2 ) (3) wth the normalzaton condton X w = 1 (4) 2

3 (Ths s the approach followed, for nstance, by Alexander (2001)). The mean and the varance of the mxture, ¹ Á and ¾ 2 Á; are related to the means and varances of the orgnal normal dstrbuton by the relatonshps ¹ Á = X w ¹ (5) ¾ 2 Á = X 2 w ¾ X w ¹ 2 Ã X! 3 2 w ¹ 5 (6) Note carefully that, snce one s ttng rsk-neutral denstes, the rst moment s not a free- ttng parameter, but must recover the forward condton, e the expectaton n the rsk-neutral measure of the stock prce must equal ts forward value. Very often ths condton has been enforced n the lterature (see, eg, Alexander (2001)), by requrng that all the ¹ should be equal to the rskneutral drft. Ths s, however, unnecessarly restrctve, because t forces the dstrbuton of log-prces to dsplay no skew. We show below how ths feature can be naturally ncorporated n the method we propose. Snce one of the most common features of emprcal dstrbutons s ther leptokurtoss, t s also useful to gve an expresson (see Alexander (2001)) for the excess kurtoss for a mxture of normals. In the case when all the terms ¹ are be equal to the rsk-neutral drft one obtans: " P Á = 3 w # ¾ 4 ( P w ¾ 1 (7) 2 )2 From Equaton 7 t s clear that the densty of a mxture of normals (wth same means ¹!) wll always have a postve excess kurtoss (e, wll be more leptokurtc than a normal densty). Ths s because, for any non-degenerate case, Ã 2 X X w ¾ 4 > w ¾! 2 (8) If one wanted to use a mxture of normals to t an emprcal prce denstes there are therefore two man routes: 1. one can estmate the rst four moments of the emprcal dstrbuton of the logarthms of the prce densty, select two normals as the bass set and t the four moments exactly. Wth the tted mxture-of-normals dstrbuton the prces for the calls, C T K, can be determned and compared wth the market values. The procedure s very straghtforward, but the t to the market prces s unlkely to be very good. 2. one can determne the optmal weghts fwg by means of a least-square t to the opton prces after convertng the densty nto call prces. Ths procedure s made easy by the fact that the opton prce CK T s smply gven by a lnear combnaton wth the same weghts fwg of Black-and-Scholes formulae. 3

4 As mentoned above, however, f all the normal denstes n the mxture are centered (n log space) around the forward value, one s automatcally guaranteed to recover the no-arbtrage forward prcng condton, but the resultng prcng densty wll dsplay no skewness. Ths s at odds wth emprcal ndngs (see, eg, Madan et al). Skewness can be easly obtaned by allowng the d erent consttuent Gaussan denstes to be centered around d erent locaton coe cents, ¹. By so dong, however, some care must be gven to recoverng the rst moment of the densty exactly, snce ths s lnked to the no-arbtrage cashand-carry forward condton. Furthermore, f the weghts are left unconstraned (apart from Equaton 4), there s no guarantee that the resultng densty wll be everywhere postve. The followng secton shows how both these problems can be overcome. 3 Fttng the Rsk-Neutral Densty Functon: Mxture of Normals 3.1 Ensurng the Normalzaton and Forward Constrants Denote by S the prce of the stock at tme T : S(T ) = S. If we denote by (S ) ts rsk-neutral probablty densty, we want to wrte (S ) = X k w k'(s k) (9) where '(S k) = LN (¹ k ;¾ 2 k;s 0 ) (10) and LN (¹ k ;¾ 2 k ;S 0) denotes a log-normal densty wth E(S k) = S 0 exp(¹ k T ) (11) V ar(s k) = [S 0 exp(¹ k T )] 2 exp ¾ 2 kt 1 By ths expresson, the rsk-neutral 1 densty for the stock prce s expressed as a sum of log-normal denstes, and therefore the stock prce densty s not log-normal. In order to ensure that the densty s everywhere postve, we requre that all the weghts should be postve. Ths can be acheved by mposng: The normalzaton condton, whch requres that (12) w k = k 2 (13) X w k = 1 (14) k 1 To lghten the prose, the qual er rsk-neutral s often omtted n the followng where there s no rsk of ambguty. 4

5 therefore becomes X 2 k = 1 (15) k Ths condton can always be sats ed by requrng that the coe cents k should be the polar co-ordnates of a unt-radus hyper-sphere. Therefore we can wrte For nstance, for n = 2 one smply has k = f (µ 1;µ 2;:::;µ n 1) (16) Ths s certanly acceptable, because, for any angle µ 1, 1 = sn(µ 1) (17) 2 = cos(µ 1) (18) sn(µ 1) 2 + cos(µ 1) 2 = = 1 (19) and equaton 15 s sats ed. In the more general case the coe cents k are gven by: k = cos µ k k 1 j=1 sn µ j k = 1; 2;:::;m 1 (20) k = k 1 j=1 sn µ j k = m (21) The reason for expressng the coe cents k n terms of angles s that we wll want to optmze the model densty over the weghts wk n an unconstraned manner, whle automatcally restng assured that the resultng lnear combnaton s a possble densty. Ths wll be the case only f 15 s always sats ed. In general, e, f one tred to optmze drectly over the weghts wk, one would have to carry out a heavly constraned numercal search: not only every weght w k would have to be greater than zero but smaller than one, but also every partal sum over the weghts would have to be strctly postve 2 and less than one. The procedure suggested above automatcally ensures that ths wll always be the case, and therefore allows one to carry out an unconstraned optmzaton over the angle(s) µ: Apart form the requrements n Equaton 15, there s at least one more constrants. The no-arbtrage forward condton Z E [S ] = (S = es js 0 = bs) es d es = S 0 exp(rt ) (22) must n fact always be sats ed exactly under penalty of arbtrage 3. Ths could be trvally acheved by mposng ¹ k = r for any k (23) 2 If, n the real-world measure, we assume that all postve values of the underlyng are possble, the densty cannot go to zero under any equvalent measure. 3 We denote byr ether the short rate or the d erence between the short rate and the dvdend yeld or the d erence between the domestc and the foregn short rates, accordng to whether one s dealng wth the case of a non-dvdend payng asset, of a dvdend stock or of an FX rate, respectvely. 5

6 Ths, however, would gve rse to denstes wth kurtoss but no skew. (Indeed, ths s the approach suggested by Alexander (2001)). To obtan skew, we want to allow the varous bass functons to be centred around d erent locatons n logs-space, but we want to do so whle retanng the forward-prcng condton. Ths can be acheved as follows. From the relatonshps above we can wrte " # X E [S ] = E w ks k = X w ke S k = S0 exp(rt ) (24) k k Recallng that one nds that E S k = S0 exp(¹ k T ) (25) exp(rt ) = X k w k exp(¹ k T ) (26) The summaton over the number of bass functons, k, can be splt nto the rst term, and the sum, P0 ; over the remanng terms: whch can be solved for ¹ 1 : exp(rt ) = w 1 exp(¹ 1 T ) + 0X w k exp(¹ k T ) k exp(¹ 1 T ) = exp(rt ) P 0 k w k exp(¹ k T ) ln ¹ 1 = w 1 h exp(rt) P 0 k w k exp(¹ k T ) w 1! T (27) In other words, f, for any maturty T, we choose the rst locaton coe cent accordng to the expresson above, we can always rest assured that the forward condton wll be automatcally sats ed. Note, however, that, a pror, there s no guarantee that the argument of the logarthm wll always be postve. In practce, we have never found ths to be a problem. So, for any set of angles fµg, for any set of ¾ k ;k = 1; 2;:::;n and for any set of ¹ k ;k = 2; 3;:::;n the forward and the normalzaton condtons wll always be sats ed f ¹ 1 s chosen to be gven by Equaton 27. In the followng we wll always assume that ths choce has been made. 3.2 The Fttng Procedure Let Call T (mod) be the model value of the call exprng at tme T for strke, and Call T (mkt) the correspondng market prces. The quantty Call T (mod) s gven by Call T (mod) = X Z k(µ) 2 '(S k )G(S k ; )ds k (28) k=1;n 6

7 where G(S ; ) s the pay-o functon: G(S k ; ) = S k + (29) and '(S k ) s the log-normal densty: A23 '(S k 1 ) = p 6 S k exp4 ln( k S S 0 exp(¹ k T ) ) ¾2 kt p 7 5 (30) ¾ k 2¼T 2 ¾ k T Note that each term under the ntegral sgn s smply equal to the value of a Black-and-Scholes call when the rskless rate s equal to ¹ k and the volatlty s equal to ¾ k. Therefore one can wrte: Call T (mod) = X k(µ) 2 Call BS (¹ k ; ;T ;¾ k ) (31) k=1;n where ¹ 1 s xed from the prevous forward relatonshp. Equaton 31 lends tself to a smple nterpretaton: for any gven strke,, the model prce s expressed as a lnear combnaton of Black-and-Scholes prces, wth the same strke, tme to maturty and volatlty, but wth rskless rate equal to ¹ k and volatlty equal to ¾ k. Now de ne  2 as  2 = X hcall T (mod) Call T (mkt) 2 (32) Then, n order to obtan the optmal t to the observed set of market prces we smply have to carry out an unconstraned mnmzaton of  2 over the (n 1) angles fµg; the n volatltes ¾ k ;k = 1; 2;:::;n, the (n 1) locaton coe cents ¹ k ;k = 2; 3;:::;n and wth ¹ 1 gven by (27). Therefore, for each expry we have at our dsposal 3n 2 coe cents. (For n = 1, we smply have one coe cent, e one volatlty. For n = 2 we have four coe cents, e, two volatltes, one weght (e one angle µ), and one locaton coe cent; etc). 4 Numercal Results 4.1 Descrpton of the Numercal Tests In ths secton we explore how well the mxture-of-normals method works n practce. We do so by lookng both at theoretcal denstes and at market prces. The theoretcal denstes are obtaned from three mportant models that wll be dscussed n the followng, e the jump-d uson, the stochastc-volatlty and the varance-gamma model. We have sometmes used rather extreme choces of parameters n order to test the robustness and exblty of the approach. Fnally, for smplcty we assumed a non-dvdend-payng stock (nterest rate at 5%) wth spot prce of $100, and we looked at maturtes of 0.5, 1, 2 and 4 7

8 years. Longer maturtes, because of the Central Lmt Theorem, would actually produce an easer test. All the optmzed coe cents are reported n Tabs I to III. As for the market prces, they were obtaned from the GBP$ caplet market n March Tab I: The means, standard devatons and weghts obtaned for the ts to the jump-d uson, stochastc-volatlty and varance gamma models dscussed n the text usng a mxture of three log-normals. The ch squared statstcs s also dsplayed. Tab II: Same as Tab. I for the mxture of four log-normals n the jumpd uson case Tab III: Same as Tab II for the market data used (GBP$ caplet mpled volatltes, March 2003) 4.2 Fttng to Theoretcal Prces: Stochastc-Volatlty Densty The smplest test s probably that of a stochastc-volatlty process for the underlyng, because we know that, n ths case, the process for the logarthm of the prce drectly generates a termnal rsk-neutral densty whch s made up of a mxture of normals 4. A mean-revertng process was chosen for the volatlty, wth an ntal value of the volatlty equal to the reverson level (12.13%). The volatlty of the volatlty was gven a very hgh value (100%) to stress the test. We assumed no correlaton between the Brownan shocks a ectng the underlyng and the volatlty. The smles produced by these parameters are shown n Fg3. The theoretcal densty, the tted densty and the log-normal densty matched to the rst two moments are shown n Fg. 5. The match s excellent everywhere even wth just three bass functons. The resultng t to the smle s shown n Fg. 4. Fg 3: The smles produced by the parameters dscussed n the text n the stochastc-volatlty case. Fg 4: The t to the 1-year stochastc-volatlty smle obtaned usng three log-normals. Fg 5: The theoretcal densty, the tted densty and the moment-matched log-normal densty n the stochastc-volatlty case It s nterestng to pont out that we found that, after optmzaton, the three bass dstrbutons n the mxture turned out to have the same (rsk-neutral) mean even f they were not requred to be so centered. Ths s consstent wth 4 Each normal densty component would have as varance the square of the root-meansqaured volatlty encountered along each volatlty path. 8

9 the assumpton of the underlyng followng a stochastc-volatlty process, whch automatcally produces a mxture of dentcally-centered dstrbutons. Ths results n a non-skewed densty (see Fgure 5). On the other hand, the resultng dstrbuton has postve kurtoss, correctly reproduced by the ttng procedure. 4.3 Fttng to Theoretcal Prces: Varance-Gamma Densty For ths model, we consder the whole set of parameters estmated by Madan, Carr and Chang (1998) for the rsk-neutral densty of the S&P: ¾=12.13%, º = 16.86%, µ = Madan, Carr and Chang (1998) show that, n the rsk-neutral world, the hypothess of zero skewness can be rejected. Ther rskneutral densty wll therefore provde the rst test for our method when the underlyng dstrbuton s skewed. Fg 8 dsplays the t to the 2-year densty obtaned wth just three bass functons, together wth a moment-matched lognormal t. Fg 7 dsplays the t to the two-year smle. It s clear from the gure that the model prces are everywhere recovered well wthn bd-o er spread 5. Fg 6: The smles produced by the parameters dscussed n the text n the varance-gamma case Fg 7: The t to the 2-year varance-gamma smle obtaned usng three lognormals. Fg 8: The theoretcal densty, the tted densty and the moment-matched log-normal densty n the varance-gamma case 4.4 Fttng to Theoretcal Prces: Jump-D uson Densty The last theoretcal smle we consder s the stress case of a log-normal jumpd uson process wth parameters chosen so as to produce a mult-modal rskneutral densty for some maturtes. Under what crcumstances a jump-d uson process can gve rse to mult-modal denstes, and what ths mples for the assocated smles s dscussed n Rebonato (2003). The model parameters used were 1 jump/year for the jump frequency, and an expectaton and volatlty of the jump ampltude ratos of 0.7 and 1%, respectvely. The volatlty of the d usve part was taken to be constant at 12.13%. The theoretcal densty (expry 0.5 years) s shown n Fg 11 wth a thn contnuous lne. The same gure also shows for comparson a moment-matched log-normal densty. The lne wth markers then shows the rsk-neutral densty obtaned wth a mxture of three log-normals. It s clear that, even for such a d cult-to-match theoretcal rsk neutral densty, a very good agreement s obtaned everywhere (wth the excepton of the very-low-strke regon) wth as few as three log-normals. Fg 12 shows that the t to the rsk-neutral densty 5 The bd-o er spread was assumed to be half a vega (50 bass ponts n volatlty). 9

10 becomes vrtually perfect everywhere wth 5 bass functons. The resultng theoretcal and tted smles for expry 0.5 years (the most challengng one) are shown n Fg 10. One can observe that everywhere the target and tted mpled volatltes concde to well wthn bd-o er spread. The largest dscrepancy was found to be 0.7 bass ponts n volatlty (n these unts one vega would be 100 bass ponts). Fg 9: The smles produced by the parameters dscussed n the text n the jump-d uson case Fg 10: The t to the 0.5-year jump-d uson smle obtaned usng three log-normals. Fg 11: The theoretcal densty, the tted densty and the moment-matched log-normal densty n the jump-d uson case (three lognormals). Note the relatvely poor recovery of the theoretcal densty n the far left tal. Fg 12: The theoretcal densty, the tted densty and the moment-matched log-normal densty n the jump-d uson case (four lognormals). The densty s now well recovered everywhere. 4.5 Fttng to Market Prces In order to test the method wth real market prces we looked at the smle curves for caplet prces (GBP, March 2003) for d erent expres. See Fg. 13. Agan, the shortest maturty provded the most challengng test. We show the ts obtaned for all the maturtes (0.5, 1, 2, 4 and 8 years). Also n ths case, the t s vrtually perfect everywhere wth four log-normal bass functons. On the bass of these results, one can conclude that the mxture-of-normals approach provdes a smple and robust method to t even very complex prce patterns. The fact that the model prce s expressed as a lnear combnaton of Black-and-Scholes prces makes t very easy to calculate It also makes t very temptng to nterpret ths dervatve as the delta statstc, e, as the amount of stock that wll allow to hedge (nstantaneously) aganst movements n the underlyng, and to replcate a payo by expry. Ths nterpretaton s however unwarranted, as we dscuss below. Fg 13: The market caplet smles for several maturtes (GBP, March 2003) Fg 14 to 18: The t to the market data usng four log-normals for expres of 0.5, 1, 2, 4, and 8 years 10

11 5 Is Really a Delta? When one uses a ttng procedure lke the mxture-of-normals approach one expresses the margnal (uncondtonal) prce denstes n terms of one or more bass functons. The termnal denstes therefore dsplay a parametrc dependence on the value of the stock prce today, S 0. Snce the plan-vanlla opton prces (denoted by C for brevty n ths secton) are obtanable as ntegrals over these probablty denstes, also these prces wll dsplay a parametrc dependence on the ntal value of the stock. It s therefore possble to evaluate S. Furthermore, n the mxture-of-normal case one can do so analytcally. If one wanted to, one could also etc. Ths has led to the statement, often found n the lterature, that these closed-form expressons for the prce, the delta, the gamma etc of the plan-vanlla opton consttute an alternatve prcng model, d erent from, but on a conceptual par wth, say, the Black-and-Scholes model. Some authors speak of prcng and hedgng wth the mxture-of-normals approach, and refer to the prcng of optons as the mxture models. Ths however should be qual ed very carefully. S because S plays n the Black-and-Scholes world not just the role of the dervatve of the call prce wth respect to the stock prce today, but also of the amount of stock we have to hold n order to be rst-order neutral to the stochastc stock prce movements. So, n the Black-and-Scholes world we post a process for the underlyng, and we obtan that, f ths process s correct, we can create a rsk-less portfolo by S amount of stock. By followng ths delta-neutral strategy to expry we can replcate the termnal payo. It s the replcaton property that allows us to dentfy the far prce of the opton wth the prce of the replcatng portfolo. The delta quantty s however only correct only because we have assumed that the process was known (a geometrc d uson n the Black-and-Scholes case). Ths s n general not true for the when seen n the context of the ttng procedures presented above. Ths s because the market prces, whch our recover could recover almost perfectly, mght have been produced by a more complex process (perhaps by a process that does not allow perfect replcaton by dealng n the underlyng). If we want, we can stll call delta, gamma, theta etc, but t nancal nterpretaton n terms of a self- nancng dynamc tradng strategy s not warranted. The problem (see, eg, Pterbarg (2003)) s that the process for the underlyng s not unquely spec ed by the margnal denstes. So, even f these were perfectly recovered by the ttng procedure, the trader would stll not know what process has generated them. (Note that Brgo and Mercuro (2000) crcumvent ths problem by carryng out the mxture-of-normals ttng n conjuncton wth a local-volatlty approach, that s desgned to recover exactly any exogenous set of opton prces). 11

12 6 Conclusons We have shown the e ectveness of mxture of normals as bass functons n terms of whch a great varety of rsk-neutral denstes can be expanded. The contrbuton of ths paper s to show how the postvty and forward-prcng condtons can be automatcally and smply sats ed wthout requrng that all the bass Gaussan functons should have the same mean. Ths would mply (n log space) absence of skew, whch, at least n the rsk-neutral measure, s a well-establshed and essental feature of the market smle. References Avellaneda M (1998) Mnmum-Realtve-Entropy Calbraton of Asset-Prcng Models, Internatonal Journal of Theoretcal and Appled Fnance, Vol 1, No. 4, Alexander C (2001) Market Models - A Gude to Fnancal Data Analyss, John Wley & Sons, Chcester, New York, Wenhem, Brsbane, Sngapore, Toronto Bahra, B, (1997), Impled Rsk-Neutral Probablty Densty Functons from Opton Prces: Theory and Applcatons, Bank of England Workng Paper Seres, No. 66 Brgo D, Mercuro F, A Mxed Up Smle, Rsk, September 2000 Jacquer E, Jarrow R A (1995) Dynamc Evaluaton of Contnegent Clam Models: An Analyss of Model Error, Workng Paper, Johnson Graduate School of Management, Cornell Unverty, Ithaca, NY Ellot R J, Lahae C H, Madan D b (1995) Flterng Dervatve Securty Valuatons from Market Prces Proceedngs of the Isaac Newton Workshop n Fnancal Mathematcs, Cambdrdge Unversty Press, Cambrdge Madan D B, Carr P P, Chang E C, (1998) The Varance Gamma Process and Opton Prcng, European Fnance Revew, 2, Madan D B, Seneta E, (1990), The Varance Gamma (VG) Model for Share Market Returns, Journal of Busness, Vol. 63, no. 4, Pterbarg V V, (2003), Mxture of Models: A Smple Recpe for a... Hangover?, Bank of Amerca Workng Paper Rebonato R, (2004), Volatlty and Correlaton: The Perfect Hedger and the Fox, Second Edton, Wley, Chchester 12