Tesng echnques and forecasng ably of FX Opons Impled Rsk Neural Denses Oren Tapero 1
Table of Conens Absrac 3 Inroducon 4 I. The Daa 7 1. Opon Selecon Crerons 7. Use of mpled spo raes nsead of quoed spo rae 8 II. Exracng he mpled rsk neural PDF from opon prces 10 1. The "Two Lognormal mxure" assumpon 10. A volaly smle based approach o exrac mpled PDF 13 III. Tesng Impled Denses 16 1. Goodness of f comparson 16. Blss & Pangrzoglou es for robusness 17 3. Probably Inegral Transform 18 IV. Resuls 1. Goodness of f comparson. Blss & Pangrzoglou es for robusness 3 3. Probably Inegral Transform 5 4. Resuls analyss 6 V. Concluson 7 Bblography 9 Annex 1: Non-Synchroncy Effecs on he mpled densy 31 Annex : Rules for exracng he Impled Exchange Rae 35
Absrac The purpose of hs paper s o es and compare wo mehods o exrac mpled rsk neural densy from Tel-Avv Sock Exchange (TASE) raded opon prces on he exchange rae beween he US dollar and he New Israel Shekel (NIS). The compared mehods n hs paper are he wo lognormal mxure (paramerc approach) and Shmko's mehods (non-paramerc approach). The comparson s done n erms of her ably o provde mpled denses havng a goodness of f of heorecal o observed opon prces, robusness o varous prcng errors and her ably o generae reasonable densy forecas. I has been found ha Shmko's mehod s a preferable mehod for hese asks. However, may be unsable when provdng goodness of f of heorecal o observed opon prces. 3
Inroducon In recen years an enre leraure on mehods of exracng mpled probably densy funcons of fuure reurns of an underlyng asse has been developed. Cenral Banks, Rsk Managers and nsuonal nvesors use denses mpled from opon prces o have a beer undersandng of uncerany regardng fuure reurns. For example: he Bank of Israel uses mpled denses o evaluae he lkelhood of fuure possble flucuaons n he exchange raes beween he Dollar and he New Israel Shekel (NIS) as a par of decson makng regardng neres raes. The Bank of Israel uses hese mpled denses o calculae, on daly bass, he probables of 5% deprecaon and apprecaon n one monh of he NIS agans he dollar. These probables and oher sascs (such as he skewness and kuross) help o have a beer undersandng of he governng rend n he exchange rae beween he dollar and NIS. Informaon regardng hs exchange rae s mporan because Israel s a small open economy where flucuaons n he exchange rae have a srong mpac on he Israel Consumer Prce Index (CPI). Therefore, s mporan for he Bank of Israel o have a deeper undersandng of exchange raes rsk whch s allowed usng he mpled denses. Impled denses are also valuable for forecasng. For example: he mean of an mpled densy can be used as a pon forecas of he asse prce n he fuure wh a beer undersandng and nformaon regardng he uncerany of hs forecas. They can also provde a forecas range whn a gven percenle. For an opon rader hs s mos mporan for choosng he correc radng sraegy. The advanage of usng raded opon prces for undersandng fuure possble reurns s ha hese fnancal asses are forward lookng by her naure. The prce of an opon embodes expecaons abou he fuure. Snce opon and oher dervave markes have become deep and lqud enough for radng n he las hry years, her embedded nformaon regardng he fuure has 4
become more and more relable and relevan. Furhermore, her lnk o her respecve underlyng secury marke (whch s explcly expressed n he Black & Scholes formula) enables hem o absorb news and new nformaon quckly and o embed furher nformaon ha s no conaned n he cash marke. The man problem wh consrucng mpled denses s ha here s no consensus abou how o exrac hem. Anoher problem s ha hey are usually derved under he assumpon ha he marke s rsk neural and s hard o deermne he rsk premum f hs assumpon fals. Therefore, hey fal o ake no accoun he marke's aude owards rsk. Thus, usng hese mpled denses may be problemac from he pon of vew of nerpreng marke uncerany. Inerpreng marke uncerany under he rsk neural measure mgh lead o false conclusons f he unobserved rsk premum s sgnfcan enough. Char A gves an example of he nraday rajecory of he prce of an opon on he TA5 ndex (he Tel-Avv 5 sock ndex) and he ndex 1 self on July 16 h, 006. Ths char llusraes he close lnk beween dervave and he underlyng markes. Noe ha he prce of he call opon changes almos n parallel o he level of he TA5 ndex. Char A: Inraday rajecory of he TA5 ndex and he prce (n NIS) of a European call opon wh a srke prce of 780 and 30 days o expry. 1800 780 1600 1400 100 1000 800 600 Call Opon (Lef Axs) TA5 ndex (Rgh Axs) 770 760 750 740 730 70 710 400 700 00 690 09:30 09:43 09:56 10:09 10: 10:35 10:48 11:01 11:14 11:7 11:41 11:54 1:09 1:3 1:37 1:53 13:07 13:1 13:37 13:53 14:09 14:8 14:43 14:56 15:09 15: 15:35 15:48 16:01 16:14 1 Inraday daa of he TA5 sock ndex are avalable for download a he Tel Avv Sock Exchange nerne se (www.ase.co.l). Unforunaely, he nraday rajecory of he exchange rae beween he U.S dollar and he New Israel shekel (The opons deal n hs paper) are unavalable. 5
The am of hs paper s o examne and compare wo mehods for obanng mpled rsk neural denses. The mpled denses wll be obaned from opons raded a he Tel- Avv Sock Exchange (TASE) on he exchange rae beween he U.S dollar and he New Israel Shekel (NIS). Thus, hs paper n some ways serves as a connuaon of R. Sen's (003&004) work on mpled denses from opons on he exchange rae beween he U.S dollar and he NIS. In hs work, R. Sen presened wo mehods for exracng mpled denses from opon prces. The frs mehod s based on a paramerc assumpon of he underlyng exchange raes dynamcs. The second mehod, whch also deals wh he expeced fuure evoluon of exchange raes, s no based on any paramerc assumpon. In hs paper I wll examne and compare hree aspecs of hese wo. Frs I compare how accurae hese mehods are n creang heorecal opon prces ha are close o hose observed n he marke. Noe ha he heorecal prce of an opon depends on some probably densy funcon. Ths wll es how well does he obaned mpled densy (gven he mehod o oban ) s for prcng opons and oher dervave producs. The second aspec s he robusness of mpled denses o varous unobserved msakes n he daa usng a Mone Carlo based procedure proposed by R.Blss and N.Pangrzoglou. Fnally, we compare boh mehods ably o oban a relable forecas of he probably densy funcon of fuure underlyng asse reurns. Ths paper s dvded no fve secons. The frs secon descrbes he daa. The second descrbes he wo mehods for obanng he mpled rsk neural denses from opon prces. The hrd descrbes he hree ess on hese mpled denses. The fourh presens he resuls and he ffh secon concludes hs paper. 6
I. The Daa The daa covers he perod from January 4 h 004 o November 30 h 005, and are aken from he TASE quoe book. The quoes are a snapsho of rade a around 14:00 PM, where mos ransacons ake place and herefore s he me of he day where he marke s mos acve. The choce of hese quoes raher hen closng prce s due o lqudy ssues, snce closng opon prce daa have a hgher average Bd - Ask spread han md day prces. In hs secon I wll dscuss my crera for opon selecon, and my choce of usng mpled spo prces nsead of quoed spo prces. As a proxy for he rskfree rae, I use he yeld on he relevan me o maury Israel zero coupon bond. As a proxy for he foregn rsk free rae I use he yeld on a relevan me o maury LIBOR 3 rae on he US dollar. 1. Crera for opon selecon The frs creron I consder relaes o he quoed Bd Ask spread of raded opons. Ths spread mgh be a major source of error, when exracng nformaon from opon prces. Snce he average of he Bd and Ask prces s used as a proxy for he observed prce of he opon, s desrable o have a narrow spread, whch reduces uncerany. An addonal ssue n opon selecon s moneyness. Takng opons ha are oo much ou of he money can lead o negave probables and oulers when calculang he mpled volaly. A hrd ssue concerns he me o maury, whch I chose o vary from 1 days o 6 days. I has been found ou ha raded opons a hese maures are mos lqud n erms of radng volumes and herefore bearng a prce whch may be more reasonable han shorer erm maures. Opon selecon n hs research paper nvolves hree seps: Sep 1: Choose opons wh quoed Bd-Ask such ha: Bd, Ask ), Bd, Ask, E Ask Ask,, Also known as he MAKAM Israel Shor erm lendng rae. 3 The London Inerbank Offered Rae An neres rae a whch fnancal nsuon can borrow funds from oher banks n he London nerbank marke. I s used as a benchmark for shor erm neres raes on he Dollar, Euro and oher major currences. 7
Where: Bd,, Ask, Quoed Bd and Ask of he h opon a me ) Bd, Ask, - Daly average of relave dfference beween bd and ask. Ths E Ask, sep comes n place n order o om from he sample he mos llqud opons. On average, he daly average of relave dfference beween bd and ask s around 100% Sep : Om opons wh annual mpled volaly hgher han 0% (oo far ou of he money). Sep 3: Choose opons wh me o maury rangng beween 1 o 6 days.. Use of mpled spo raes nsead of quoed spo rae One mporan aspec of FX opon radng n Israel s ha radng akes place on Sundays when here s no rade on he underlyng asse (The US dollar) 4. Ths can cause a problem for exracng nformaon from opon prces. Furhermore, he dollar spo marke has changed consderably whn he sample perod 5. As a resul, nraday movemens of he dollar agans he NIS have been more frequen, causng an amplfcaon of errors due o non-synchroncy beween he opon and s underlyng markes. Ths means ha he prce of an opon may no reflec he laes avalable prce of he underlyng. Char 1 and Table 1, whch show he squared devaon beween he prce mpled from he Pu Call pary equaon and quoed marke spo prce llusraes he problem of non - synchroncy beween he dervave and he underlyng markes. Ths nonsynchroncy has mporan mplcaons for he mpled dsrbuon 6. When usng he non-paramerc mehod, an unusual hck aled densy s obaned whle wh he paramerc mehod, he mean of he dsrbuon s affeced. Throughou he sample 4 On Sundays, he TASE uses he dollar rae whch s deermned on Frday 13:00PM as he underlyng asse prce. 5 Whn ha perod suppor bands for exchange rae beween he NIS and dollar were removed, makng he NIS compleely a floang currency. Also, he "Bachar" reform decenralzng Israel capal markes has conrbued o an ncrease n flows of fund no and ou of Israel. 6 See Annex 1 on non-synchroncy effecs 8
perod, I wll use he mpled spo rae nsead of quoed marke spo rae 7. As he char and able llusrae, he opon marke s somewha close o beng a complee marke. If he squared devaons beween he mpled and quoed exchange rae were sgnfcan han would sugges ha he opons on he dollar marke was ncomplee or had some marke falure. Char 1: The Squared devaon beween mpled from Pu-Call pary spo rae and he quoed marke spo rae: 4/1/004 30/11/005 8 0.0035 0.003 0.005 0.00 0.0015 0.001 0.0005 0 Table 1: The Squared devaon beween mpled from Pu-Call pary spo rae and he quoed marke spo rae: 4/1/004 30/11/005 9 All days excludng All days ncluded n he Seres Sundays Sample daa 0.1 0.18 Mean 0.1 0.036 Medan 0 0 Mn 1.51 1.51 Max 0.19 0. Sandard Devaon 7 See Annex on opon selecon crera for obanng he mpled spo 8 Whn ha me he dollar vared from 4.5NIS/1$ on Aprl 1 s 004 o 4.661NIS/1$ on Ocober 11 h 005, wh esmaed 6% hsorcal volaly. 9 For purpose of convenence, sascal ndces are mulpled by 10000. 9
II. Exracng he mpled rsk neural PDF from opon prces Ths secon descrbes wo dfferen mehods for exracng mpled rsk neural denses from opon prces. Generally speakng, here are wo man schools of hough regardng he exracon of hese denses: he paramerc school and he non-paramerc school. In he paramerc school, he underlyng secury prces are assumed o follow some parcular dsrbuon (e.g. he log-normal). In he second, non-paramerc school, no such assumpons are made. The able below summarzes he man dfferences regardng hese wo approaches. In hs paper, I use a mxure of wo lognormals (LN) for he paramerc mehod. For he non-paramerc mehod, we shall exrac he mpled densy usng an approach devsed by Shmko (1993). Table : Man characerscs of paramerc and non-paramerc mehods Paramerc Non Paramerc Opon prcng formula Exss. I s based on he Black No opon prcng formula. & Scholes (BS) opon prcng formulae. Dsrbuonal assumpons oher han he lognormal assumpon can be made and adaped whn he framework of he BS formula. Esmaed parameers Depends on he dsrbuonal There are no parameers o assumpon. For example, n he esmae. lognormal assumpon here are wo parameers o esmae he mean and varance. Smoohng splne There s none. Usually, he mpled volaly smle s nerpolaed usng some smoohng procedure. Impled Densy exracon mehod Usually mnmzng some loss funcon. Usng Breeden and Leezenberg numerc approxmaon 1. The wo lognormal mxure assumpon The LN assumpon has been wdely used o exrac nformaon from opon prces. Bahra (1997) apples hs mehod whle sudyng he mpled nformaon from 3-monh Serlng neres rae opons and LIFFE equy ndex opons. Gemmll and Saflekos (1999) use hs mehod o examne he usefulness of mpled probables exraced from opons on he FTSE100. Sen and Hech (003,004) appled hs mehod on opons on he exchange rae beween he US dollar and he New Israel Shekel (as I wll do here). 10
A mxure of wo-lognormals has some advanages when appled o currency opons. I s applcable when here s no oo wde a range of srke prces avalable (when here are no opons raded far away from he money). I s compuaonally easy. Esmaon s relavely smple snce here are only fve parameers o esmae. Moreover, a mxure of wo lognormal dsrbuons s reasonably emprcally relable. However, here are some drawbacks. For example, he mpled dsrbuon may exhb spkes due o oulers n observed opon prces or msspecfcaon of he mxure dsrbuon. Also, may be oo resrcve as an assumpon for he dynamcs of exchange raes snce he governng law of exchange rae flucuaons s unknown and perhaps a more general dsrbuon such as he Generalzed Bea of he nd ype (GB) s more adequae. Under he above assumpon, he prce of a European pu or call opon s bascally a lnear combnaon of wo Black and Scholes 10 opon prces relaed o wo dfferen saes of he world. BS BS C K, τ ) = θc ( K, τ ) + (1 θ ) C ( K, ) (1) ( 1 τ where BS BS C 1 denoes he Black and Scholes prce of a European call n sae 1, C s he prce n sae and θ s he probably of he frs sae. More explcly, he prces a European pu and call opons wll be: [ θl( α1, β1; ST ) + (1 θ ) L( α, β ; ST )]( ST X dst C τ = + ) (a) r ( K, ) e K [ θl( α, β ; ST ) + (1 θ ) L( α, β ; ST )]( X ST dst K r P ( K, ) = e 1 1 ) 0 τ, (b) where L(α,β;S T ) = (ln S T α ) 1 S T β π e β 10 Snce I sudy opons on exchange raes I use he Garman-Kohlhagen formula: r τ τ C = S e f rd T N( d1) Ke N( d1 σ τ ) Where: r f and r d are foregn and domesc neres raes, respecvely. 11
s he lognormal densy and 1 α = ln S + μ σ τ and β = σ τ. (3) Elemenary calculaons, usng he Black and Scholes prcng formula yeld he followng heorecal opon prces for a European pu or call: { [ ]} (4a) C(K,τ) = θ e r d e α 1 +0.5β 1 N(d 1 ) e r f [ KN(d )]+ (1 θ) e r d e α +0.5β N(d ) e rf KN(d ) 3 4 { [ ]} (4b) P(K,τ) = θ e r d e α 1 +0.5β 1 N( d 1 ) + e r f [ KN( d )]+ (1 θ) e r d e α +0.5β N( d ) + e rf KN( d ) 3 4 Where: ln K + α + β 1 1 d 1 =, = d1 β1 β1 d, ln K + α + β d 3 = and 4 = d 3 β β d (5) Noe ha he above prces are closed-form soluons of equaons (a) and (b), as proved by Bahra (1997). In order o oban an mpled PDF based on an emprcal observaon of he opon prces, we may mnmze her relave squared devaon from he heorecal opon prces mpled by he -lognormals mxure dsrbuon, or Mn α 1,α,β 1,β,θ n =1 ^ C C(K,τ) + C(K,τ) m j=1 ^ P P(K,τ) P(K,τ), (6) where C ( K, τ ), P( K, τ ) are he observed marke prces for a gven srke prce (K ) and me o maury (τ) C ), P ) are he mpled heorecal prces and { α, α, β, β, θ } are he 1 1 parameers of he LN dsrbuon o be esmaed. 1
The opmzaon problem above s he same as n Sen and Hech's work. I have red oher opmzaon problems, noably he one used by Bahra and oher researchers, whch has he followng form Mn α 1,α,β 1,β,θ n =1 ^ m ^ C(K,τ) C + P(K,τ) P + θe α 1 1 + β 1 + (1 θ)e α + 1 β e rτ S j=1 (7) The hrd erm of he opmzaon equaon consrans he parameers such ha he hrd erm s equal o he heorecal forward spo rae. However, I found ha opmzng over he above equaon gves unsasfacory parameer esmaes. The funconal form of equaon (6) s more sable 11 and also has he advanage of beng somewha less sensve o sarng values.. A volaly smle based approach o exrac mpled PDF Unlke he paramerc approach whch assumes dsrbuon, he nonparamerc approach derves he mpled PDF drecly from he second dervave of he heorecal 1 opon prce wh respec o s srke prce. The dsrbuon s obaned by nerpolang he volaly smle of he call opon prce drecly by fng a splne and expressng mpled volaly as a funcon of he srke prce. Then, usng Breeden and Leezenberg approxmaon for he second dervave of opon prce wh respec o srke prce, we oban he mpled rsk neural densy. Baes (1991) for example, fs a cubc splne on S&P 500 opons when sudyng he relave predcve capables of mpled dsrbuons before and afer he 1987 Wall Sree crash. Shmko (1993) fs mpled volaly o srke prces and grafs he als of a lognormal dsrbuon o he mpled dsrbuon. Malz (1997) follows Shmko s 11 More sable n he sense ha f he opmzaon s repeaed 'n' mes (wh same values and sample daa), resuls wll no vary by much. 1 r For a call: C ( S, K, r, ) = e ( S K) p( ST ) ds. + K 13
approach, bu nerpolaes volaly smle across opons delas 13 whle Blss, Pangrzoglou and Syrdal (00) use a naural splne echnque o f mpled volales o opon delas. A Sahala and Lo (1998) use kernel regressons o express he relaonshp beween he opon prce and he srke prce. Many oher smoohng echnques are used o mprove and o confron essenal ssues rased n he applcaon of he non-paramerc approach. The frs ssue relaes o mos accuraely fng mpled volaly gven relavely small number of observaons. The second ssue s he presence of oulers n mpled volaly a opons whch are far ou or n he money. These oulers usually appears for reasons due o lack of lqudy (measured n erms of low radng volumes and relavely large bd and ask spreads) and opon mss-prcng. The Breeden and Leezenberger approach sars by usng a buerfly spread. Ths spread replcaes a sae conngen clam (or Arrow-Debreu secury). I consss of a shor poson n wo calls wh exercse prces of K (A The Money Opons) and a long poson n wo calls, one wh a srke prce of K + Δ K and he oher wh a srke prce of K - Δ K. The payoff of hs porfolo wll be 1 for X = K and 0 oherwse. [ C( K + ΔK, τ ) C( K, τ )] [ C( K ΔK, τ ) C( K ΔK, τ )] X = K = 1 (7) As ΔK approaches o 0, he lm of he buerfly spread becomes a replcaon of an Arrow Debreu secury. If we le P(K,τ;ΔK ) be he prce of such a clam cenered a X = K and dvded by ΔK, we oban a second order dfference quoen: P( K, τ; ΔK ΔK ) [ C( K = + ΔK, τ ) C( K, τ )] [ C( K ( ΔK ), τ ) C( K ΔK, τ )] (8) And For X = K : P( K, τ; ΔK ) C( X, τ ) lm = ΔK K 0 Δ X X = K (9) 13 The dervave of he opon prce wh respec o he underlyng asse prce. 14
Now, snce he prce of an Arrow-Debreu secury s an expresson of he presen value of $1 mulpled by he rsk neural probably of a sae occurrng a X = K we have he followng esmae of he rsk neural densy f(s T ): C( K, τ ) e K rτ f ( S T C ) = + 1 + C ( ΔK 1 ) C, C = C( K ) (10) Pror o applyng he Breeden and Leezenberger resul however, we need o f a volaly smle n order o oban a se of synhec opon prces on a connuum whn a gven range of srke prces. Shmko's mehod for fng he mpled volaly s consdered below. Shmko's approach In order o oban call 14 prces as a funcon of srke prces, Shmko fs a quadrac equaon o mpled-volaly 15 wh parameer coeffcens esmaed usng a smple OLS procedure. σ IV = β 0 + β 1 K + β K + ε (11) Thus we oban a fed funconal form of mpled volaly wh respec o he srke prce and he prce of a synhec call opon s represened by equaon 1. Usng hs equaon, we can apply equaon 10 n order o esmae he mpled rsk neural dsrbuon. C synhec = C BS (S 0,K,τ,r f,r d,σ IV (K )) (1) In hs work, Shmko, assumes (as a maer of convenence) ha he als of hs nonparamerc densy are smlar o he als of he lognormal dsrbuon and herefore grafs ono he non-paramerc dsrbuon. Alernavely, o oban he als, synhec call 14 Pu prces are ranslaed o call prces usng he Pu-Call Pary equaon. 15 Impled volales are exraced from observed opon prces usng Newon-Raphson algorhm. Ths algorhm s mplemened usng VBA Excel. 15
prces can be calculaed ousde he srke prce range. In hs paper, he al of he nonparamerc dsrbuon are no assumed nor calculaed from synhec call prces. The reason for hs s ha calculang he als of dsrbuon usually generaed negave or unreasonable probables. Furhermore, assumng log normal als n he case of TASE raded opon s no sound due o low radng volumes n far ou and n he money opons. III. Tesng mpled denses In hs paper, we consder ess for: Sably/Robusness, Goodness of f of synhec opon prces o observed opon prces and he forecasng ably of mpled denses. For Goodness of f I compare observed and synhec opon prces usng he Mean Squared Error (MSE). For sably (or robusness o errors n he daa), I apply he algorhm by Blss and Pangrzoglou. Fnally, o es forecasng ably I use he Probably Inegral Transform (PIT) es on he esmaed mpled denses. These ess can be used o evaluae and compare paramerc and non-paramerc mehods for exracng mpled rsk neural denses. 1. Goodness of f comparson Goodness of f over he sample perod s compared beween Shmko s and LN approach. For each day he mean squared error (MSE) beween marke and heorecal/synhec prces were calculaed by: MSE = 1 N 1 K N k observed ( C Cˆ ) (14) I sudy he me evoluon of hs sascal ndce. Noe ha on a gven day, one mehod mgh do a beer job n fng synhec prces o real prces han he oher. Thus, o rank and compare he wo approaches used, we shall also use "me" scores, whch coun: he number of mes (or days) where one approach s a beer f han he oher. In addon, I calculae he me seres mean and varance. These comparsons wll allow a beer undersandng of hese mehods. 16
. Blss & Pangrzoglou es for robusness R. Blss and N. Pangrzoglou (henceforh, BP) analyzed he robusness o prce errors of varous mehods for exracng mpled denses. Such prcng errors can be deeced usng pu/call pary equaon and he convexy of opon prces wh respec o srkes. However, he prcng error's source canno be deermned. BP menoned he followng possble sources of errors ha mgh arse wh raded opons daa: (1) Errors occurrng whle recordng opon prces (human errors), () Non synchroncy beween opons and her underlyng prces, (3) Dfferenal (undeeced) lqudy prema beween opon prces and her respecve srkes (ou and n of he money opons end o be less lqud han a he money opons). A Robusness es s performed usng a Mone Carlo procedure. The algorhm, adaped o TASE raded dollar currency opons, consss of he followng: 1. Opon prces across all srkes are dsurbed by a unformly dsrbued error on an nerval cenered around zero and wh lengh equal o half a ck. In he conex of he TASE raded opons, ck sze s deermned accordng o he followng manner: 1 NIS for an opon raded a a prce up o 0 NIS 5 NIS for an opon raded a a prce beween 0 00 NIS 10 NIS for an opon raded a a prce beween 00-000 NIS 0 NIS for an opon raded a more han 000 NIS. For each mehod, an mpled densy s exraced and he medan, he mode and ner quarle range are calculaed. 3. The above s repeaed a leas 100 mes. 17
Such smulaon provdes a large number of seres (100 for each mehod) whch can be used o compare he esmaon mehods. The sandard devaon, varance, range, mean and he medan of he squared dfference beween he rue and dsurbed ndces was calculaed. On he bass of hese resuls, he mehod ha provdes he seres wh he lowes devaon ndcaors wll be consdered he mos robus. In her paper, BP, perform he es on raded opons on he Shor Serlng neres rae and he FTSE100 n order o compare he LN and her own non-paramerc mehod. They conclude ha her non-paramerc mehod s more robus o errors n prcng. In addon, hey observe neher mehod s robus n he als of he dsrbuon (he 1 s and 99 h percenles of he mpled densy). Ther resuls are explaned by he fac ha he LN approach has assumpons lmng he shape of he mpled rsk neural densy and hus s more sensve o prcng errors. In hs paper, I es robusness for each mehod usng opon prces observed on May 6 h, 005. Ths s due o numercal lmaons of he hardware ha s used: whou such lmaons, I would have conduced hs es over he whole sample daa, gvng a beer undersandng of how robus hese wo mehods are. Furhermore, I seleced only hree emprcal sascal ndces due o he fac ha he non paramerc densy esmaed s wh no als and hus only emprcal ndces can be obaned from (such as he medan, he mode and nerquarle range, The mean, varance and hgher momens are no calculaed). 3. Probably Inegral Transformaon The PIT drecly relaes he rue PDF o he mpled PDF exraced from opon prces, and mgh be more nformave n comparng beween paramercally and nonparamercally obaned dsrbuons. The PIT evaluaes he forecasng ably of he mpled dsrbuon. I can be used o es boh he als of he underlyng dsrbuon as well as he whole dsrbuon. In hs paper, snce he als of he dsrbuon are no esmaed n he non-paramerc derved mpled densy, I wll focus on he whole mpled dsrbuon. 18
Tess based on he PIT for evaluang esmaed densy forecas daes back as early as 195 (Rosenbla). However, only recenly has been appled as a es for he accuracy of mpled rsk neural denses. Debold, Gunher and Tay (DGT, 1998) gve a dealed examnaon of hs mehodology whle applyng o a smulaed GARCH process. They also ndcae s poenal use on a wde range of fnancal models, such as value a rsk. Anagnou e al. (ABHT, 00) appled hs mehod on densy forecass of he S&P 500. In her research, hey evaluae hree paramerc approaches (GB, Negave Inverse Gaussan and he wo-lognormals mxure) and a sngle non-paramerc approach based on he B Splne. They conclude ha he mpled rsk neural PDF s a poor forecas of fuure prces. Crag, Glazer, Keller and Schecher (CGKS, 003) esed mpled denses obaned from opons on he DAX ndces (exraced usng he wo lognormal approach). Ther resuls pon o evdence of srong negave skewness as well as a sgnfcan dfference beween he acual densy and he rsk neural densy". They conclude: marke parcpans were surprsed by he exen of boh he rse and fall of he DAX. Alonzo, Blanco and Rubo (005) derve mpled denses from opons on he IBEX35, also usng a paramerc and a nonparamerc procedure. Usng daa from 1996 unl 003, hey canno rejec he hypohess ha mpled denses successfully forecas fuure realzaons. Neverheless, hey found ha hs resul s no robus whn sub perods. Gurkaynak and Wolfers (GW, 005) apply he PIT on mpled denses derved from Macroeconomc Dervaves. They conclude afer a seres of graphcal ess, ha hese denses are accurae forecass. They menoned ha hs resul s raher surprsng, snce asse prces usually end o nclude a rsk premum. When here are unobserved rsk prema, opons prced n a rsk neural world end o be sysemacally based. The PIT es consss of deermnng wheher he densy forecas s equal o he realzed fuure densy. A frs, hs sounds mpraccal and unfeasble snce he densy canno be 19
observed, even ex pos. There are a few mporan noons ha should be kep n mnd n such a case. Frs, a densy forecas s bascally a dsrbuon of many possble pon forecass. Therefore, momens of mpled dsrbuon gve a descrpon of poenal fuure pon realzaons snce only one realzaon s possble. DGT pon ou ha s possble o esablsh a relaonshp beween he daa generang process and he sequence of densy forecass hrough he probably negral ransform of realzed reurns (n hs paper I refer o he realzed exchange rae beween he U.S Dollar and NIS) wh respec o he forecass obaned. z y = p ( u) du = P ( y ) (17) Where: p (u) s he esmaed densy forecas, P y ) s he correspondng esmaed cumulave densy funcon, y s he realzaon self and z s he probably negral ( ransform. DGT prove ha z follows he followng sascal law: z..d ~ Unform(0,1) (18) Snce he non-paramerc mpled densy s resrced o a range of srke prces (and hereby runcaed), calculaon of he PIT for boh denses can be done n a manner smlar o ha of ABHT. Ths wll resrc he PIT es o he body of he dsrbuon and allows a comparson beween he wo mpled denses. z * = P (y ) P (K mn, ) P (K max ) P (K mn, ) In order o es he (null) hypohess ha he rsk neural densy s an accurae forecas (he PIT, z, follows a Unform dsrbuon), I used he procedure of Chrsoffersen and 0
Mazoa (CM, 004). In her paper, hey es uncondonal and condonal normal dsrbuons of he followng ransformaon of he PIT: x =Φ 1 (z )~..d N(0,1) (19) In hs paper, esng uncondonal normaly wll be done because he als of he nonparamerc densy are unknown and herefore he mean, varance and hgher momens canno be calculaed. To es he uncondonal normaly of x, we use he followng jon hypohess: E(x ) = 0 E(x ) =1 E(x 3 ) = 0 E(x 4 ) 3 = 0 Usng GMM 16 n order o allow auocorrelaon from overlappng observaons, we esmae a sysem of four equaons and es for he sgnfcance of he coeffcens a 1, a, a 3 and a 4 : (1) x = a 1 + e x () 1= a + e x 3 (3) = a 3 + e x 4 (4) 3 = a 4 + e Ths es s a slgh modfcaon of he Berkowz es and s performed o allow overlap n he daa and as a resul an ncrease n he sample sze. 16 The GMM procedure s run by usng EVIEWS sofware. There s a resrcon relang o auocorrelaon as defned by he EVIEWS sofware. 1
IV. Resuls Ths secon wll presen he resuls obaned from he ess on he obaned mpled denses. I s dvded no four sub secons. The frs par wll presen he resuls of he goodness of f comparson; he second par presens resuls for he Blss & Pangrzoglou robusness es; he hrd par presens resuls for he runcaed verson of he probably negral ransform ess and he forh par wll conclude. 1. Goodness of f comparson The char and able below show he daly evoluon of MSE (as defned n equaon 14) of boh synhec derved opon prces and her respecve summary sascs. Char : MSE of boh synhecally derved opon prces Table 3: Summary sascs of he MSE of boh models 17 Paramerc (LN) Non Paramerc (Shmko) Mean 0.48 1.06 Medan 0.3 0.14 Sandard Devaon 0.58 4.5 Mn 0.004 0.001 Max 4.84 65.6 Range 4.84 65.6 17 For convenence, summary sascs are mulpled by 10000.
Char and Table 3 boh mply a ceran advanage for he paramerc mehod n fng opon prces over me. However, whn he sample (445 observaons) s lkely for he non-paramerc mehod o score beer han he paramerc one. Only 8% of he me does he paramerc mehod score beer han he non - paramerc. These resuls are no surprsng due o he naure of hese synhec prces. The calculaon of non-paramercally derved synhec prces s done va he fed value of mpled volaly a he seleced srke prce. Ths calculaon akes no accoun he volaly smle, whle paramercally derved synhec prces do no. However, he resuls above show some of he shorcomngs of Shmko s approach. The huge jumps n MSE of non-paramerc synhec prces reveal ha on ceran days, he mpled volaly smle was no well fed. Thus, hs mehod may gve ncorrec mpled denses. The MSE of paramerc synhec prces s more sable han he non paramerc synhec prces n he sense ha large devaons from observed prces are less lkely. A closer look a hese resuls reveals ha Shmko s non-paramerc mehod s weak n exracng mpled denses n days n whch he range of srkes s narrow. Ths s due o he fac ha hs densy does no have any als. Furhermore, mpled volales are fed o srke prces, makng he goodness of f exremely sensve o number of srkes and her range.. Blss & Pangrzoglou es for robusness The able and chars llusrae he resuls obaned usng he robusness es. Ths enables a comparson of he sably of hese wo mehods for obanng mpled denses from opon prces. Table 4: Summary sascs of he squared dfference beween he undsurbed and dsurbed calculaed sascal ndces 18 18 For convenence, summary sascs are mulpled by 100. 3
Iner-quarle range Medan Mode Two lognormal Shmko Two lognormal Shmko Two lognormal Shmko Mean 0.3 0.14 0.008 0.13 1.56 1.15 Medan 0.13 0.05 0.001 0.015 0.68 0.56 Varance 0.0013 0.00 0.0000001 0.0013 0.034 0.11 S. Dev 0.36 0.36 0.0033 0.36 1.86 3.8 Char 3: Squared dfference beween undsurbed and dsurbed emprcal sascal ndces (Two - lognormal mehod) 4
Char 4: Squared dfference beween undsurbed and dsurbed emprcal sascal ndces (Shmko s mehod) As we can see from Table 3, Char 3 and Char 4, Shmko s mehod provdes an mpled densy whch s less sensve o small prcng errors. There agan, seems ha he resul above come from he robusness of a lnear regresson of mpled volaly on srke prces. I seems ha he opmzaon procedure n he wo-lognormal mehod s somewha sensve o small prcng errors. Thus, as may seem, Shmko s nonparamerc mehod s more robus n he case of TASE dollar opons. Ths resul s smlar o ha obaned by Blss & Pangrzoglou. 3. Probably Inegral Transform Table 5 presens he resuls obaned from he GMM procedure ha was run for boh PIT ransformaons. 5
Table 5: GMM resuls for he Probably Inegral Transform Two lognormal Shmko Esmaed Sandard P-Value Esmaed Sandard P-Value Coeffcen Error Coeffcen Error a 1-0.9 0.054 0.0000 0.13 0.18 0.4881 a -0.6 0.14 0.0677 0.083 0.17 0.6308 a 3-1.83 0.64 0.0043 0.9 0.44 0.507 a 4 5.13 3.13 0.1019-0.46 0.59 0.4349 The resuls for he Probably Inegral Transform es sugges ha he non-paramerc mehod gves an emprcal mpled densy ha s a reasonable forecas of he rue densy. Ths mgh be evdence ha he wo-lognormal assumpon s no adequae n modelng he prce process of he underlyng asse. 4. Resuls analyss The obaned resuls gve a sgnfcan advanage o Shmko s mehod for exracng he mpled rsk neural densy from opon prces. Ths non-paramerc mehod gves a densy for whch synhec opon prces beer f observed marke prces. I s more robus o prcng errors and n erms of forecas ably does sgnfcanly beer job han he wo - lognormal mehod. However, Shmko s mehod has a major weakness n comparson o he wo-lognormal mehod. In days where opons are raded n a narrow srke range, Shmko s mehod generaes an mpled densy ha has a worse f relave o observed opon prces. Ths mgh also have consequences on s robusness and forecas ably on a gven day. Ths weakness s no surprsng, snce he srengh of Shmko s mehod s a funcon of he goodness of f of he regresson process of mpled volales on observed srke prces. Thus, gven a narrow range of srke prces, he regresson of mpled volales on srkes does no capure he whole smle. Ths naurally has an effec on he shape of Shmko s mpled densy. Under a narrow range of srkes, he shape of he mpled densy s such ha even a graphcal 6
nerpreaon of he mpled densy s no possble. Ths s where he paramerc approach does a beer job han Shmko s mehod. Of course, s possble o fully follow Shmko s fooseps and graf he als of a heorecal lognormal densy. However, I do no beleve ha hs s adequae gven he fac ha here s srong evdence agans he lognormal assumpon n modelng he US dollar/new Israel Shekel exchange rae. V. Concluson In hs research paper, wo mehods for obanng mpled rsk neural denses from exchange raded FX opons were revewed and compared n erm of her ably o: f beween heorecal and observed opon prces, o be more robus o prcng errors and o be a reasonable forecas of expecaons on fuure changes n he underlyng asse prces. In general, Shmko's mehod for obanng he mpled rsk neural denses does a beer job n for exchange rae beween he U.S dollar and he New Israel Shekel. The srengh of hs mehod comes from beng n many ways more emprcally sound han he wo log normal. As menoned prevously, mehods usng he Impled Volaly Smle o oban a probably densy of expeced fuure changes n exchange rae capures beer he nformaon embedded n raded opon prces. However, hs mehod has some major weaknesses. The frs and mos srkng weakness s ha he daa s unnformave abou he als of he dsrbuon, whch prevens he calculaon of some key sascs such as he mean and sandard devaon of he mpled densy. The second weakness, whch relaes drecly o he frs, s ha hs mehod performs poorly on days where he range beween he mnmal and he maxmal srke prce s small. The narrow range of srke prces (n he conex of small number of observaons) also nfluences he degrees of freedom for he curve ha fs he mpled volaly smle. Neverheless, Shmko's mpled densy usually gves us a more relable densy for forecasng fuure changes n he exchange rae. Overall, s also more robus o errors n prces. However, perhaps he robusness of he wo lognormal mehod can be 7
sgnfcanly mproved by adapng a beer opmzaon mehod ha s used by MATLAB opmzaon oolbox. I have red usng he adapve smulaed annealng opmzaon algorhm 19. However, hs algorhm underperformed and ook more me o gve resuls 0. Noe also ha he MATLAB opmzaon oolbox usually performs poorly n comparson o oher sascal and mahemacal sofware. Anoher drecon n mprovng he paramerc mehod s o assume a dfferen probably law for reurns on he exchange rae. Perhaps a more suable sascal law exss for hese reurns. Examples of possble canddaes are he Generalzed Bea of Second Order (GB) dsrbuon, he g-h dsrbuon, he Webull, he Generalzed Exreme Value (GEV) dsrbuon, and oher possble canddaes. 19 For more nformaon: Mons S., 'Implemenaon of a smulaed annealng algorhm for MATLAB ', Techncal Repor, 00, Lnkopng Insue of Technology 0 The average me for he lsqnonln roune n MATLAB o gve esmaed parameers for each densy was 100 seconds. 8
Bblography A Sahala, Y. and Lo, A., 'Nonparamerc esmaon of sae-prce denses mplc n fnancal asse prces', Journal of Fnance, Vol. 53, 1998, pp. 499-548 Alonzo F. e al., 'Tesng he Forecasng Performance of IBEX 35 Opon Impled Rsk Neural Denses', Workng Paper, 005, Bank of Span Anagnou I. e al., 'The Relaon beween Impled and Realsed Probably Densy Funcons', Workng Paper, 003, Unversy of Warwck Baes, D.S, 'The crash of '87: Was expeced? The evdence from opons markes', Journal of Fnance, Vol. 46, 1991, pp. 1009-1044 Bahra, B., 'Impled Rsk-Neural Probably Densy Funcons from Opon Prces: Theory and Applcaon', Workng Paper, 1997, Bank of England Black, F. and Scholes, M., 'The prcng of opons and corporae lables', Journal of Polcal Economy, Vol. 81, 1973, pp. 637-654 Blss, R. and Pangrzoglou, N., 'Tesng he Sably of Impled Probably Densy Funcons', Journal of Bankng and Fnance, Vol. 6, 00, pp. 381-4 Breeden, D.T., and Lzenberger, R.H, 'Prces of Sae-Conngen Clams Implc n Opon Prces', Journal of Busness, Vol. 55, 1978, pp. 61-651 Crag B. e al, 'The forecasng performance of German Sock Opon Denses', Dscusson Paper, 003, Deusche Bundesbank Chrsoffersen P., and Mazzoa S., 'The Informaonal Conen of Over-he-Couner Currency opons', Workng Paper, 004, European Cenral Bank Debold F. X. e al., 'Evaluang Densy Forecass wh applcaons o Fnancal Rsk Managemen', Inernaonal Economc Revew, Vol. 39, 1998, pp. 863 883 Garman M.B., and Kohlhagen S.W., 'Foregn Currency Opon Values', Journal of Inernaonal Money and Fnance, Vol., 1983, pp. 31-37 Gemmll G., and Saflekos A., 'How useful are Impled Dsrbuons? Evdence from Sock Index Opons', Journal of Dervaves, Vol. 7, 000, pp. 83-98 Gurkaynak R., and Wolfers J., 'Macroeconomc Dervaves: An nal analyss Of Marke Based Macro Forecass, Uncerany, and Rsk', Workng Paper, 005, NBER Malz A.M., 'Usng Opon Prces o Esmae Realgnmen Probables n The European Moneary Sysem', Workng Paper, 1995, Federal Reserve Bank of New-York 9
Mons S., 'Implemenaon of a smulaed annealng algorhm for MATLAB ', Techncal Repor, 00, Lnkopng Insue of Technology Rosenbla M. (195), 'Remarks on a Mulvarae Transformaon', Annals of Mahemacal Sascs, 3, 470-47. Shmko D., 'Bounds of Probably', Rsk, Vol. 6, 1993, pp. 33-37 Sen R. and Hech Y., 'Dsrbuon of he Exchange Rae Implc n Opon Prces: Applcaon o TASE', Workng Paper, 003, Bank of Israel Sen R., 'The Expeced Dsrbuon of Exchange Rae: A Non Paramerc mpled Densy n FX Opons (In Hebrew)', Workng Paper, 005, Bank of Israel Syrdal S.A., 'A sudy of Impled Rsk-Neural Densy Funcons n he Norwegan Opon Marke', Workng Paper, 00, Bank of Norway 30
Annex 1: Sunday effecs on he Impled Densy For he purpose of demonsrang he effecs of non synchronous radng beween he dervave and he underlyng marke, he Impled Rsk Neural Densy wll be derved once usng he quoed exchange rae and once wh he mpled from he pu call pary equaon exchange rae. I chose Sundays, where non synchronous radng s mos apparen (here s no rade n he underlyng marke). The daes chosen were he followng: Sunday February 0 h, 005 (quoed exchange rae: 4.361NIS per 1$, mpled exchange rae: 4.3167NIS per 1$) Sunday January 15 h, 006 (quoed exchange rae: 4.606NIS per 1$, mpled exchange rae: 4.5761NIS per 1$) Sunday May 7 h, 006 (quoed exchange rae: 4.4736NIS per 1$, mpled exchange rae: 4.461NIS per 1$) An examnaon of he devaon beween he mpled and quoed exchange raes on oher days of he week suggess ha he opon on he dollar marke s no neffcen. Effecs apparen n he Non Paramerc Mehod The chars below of mpled denses sugges ha he devaon beween he mpled and quoed spo exchange rae has an effec on he lef and rgh sdes of he mpled denses. Ths mgh brng o unreasonable probables for exreme flucuaon n he exchange rae. There s also an effec on he ner quarle range (whch s a proxy of volaly) Char A: Impled non-paramerc densy on Sunday February 0 h, 005: 31
Char B: Impled non-paramerc densy on Sunday January 15 h, 006: Char C: Impled non-paramerc densy on Sunday May 7 h, 006: 3
Effecs apparen n he Paramerc Mehod As he followng able and chars llusrae, he devaon beween he mpled and quoed exchange raes has an effec on he locaon of he obaned mpled denses. To be more precse, hs devaon affecs he mean of he mpled densy and slghly affecs he sandard devaon (whch s an esmae of mpled volaly n hs case). Table A: Parameer esmaes of he mpled densy 0//005 15/1/006 7/5/006 Quoed FX Impled FX Quoed FX Impled FX Quoed FX Impled FX Θ 1 1 1 1 1 1 μ 1 0.0055 0.1019-0.0043 0.0899-0.0803 0.0177 μ - - - - - - σ 1 0.0635 0.0635 0.0768 0.0768 0.0757 0.076 σ - - - - - - Char D: Impled paramerc densy on Sunday February 0 h, 005: 33
Char E: Impled paramerc densy on Sunday January 15 h, 006: Char F: Impled paramerc densy on Sunday May 7 h, 006: 34
Annex : Rules for exracng he Impled Exchange Rae In hs annex, I wll presen he rules for exracng he prevous mpled exchange rae. The rules are he followng: Bd-Ask Spread: Whn a gven Bd and Ask quoes we om opons where: Bd, Ask Ask,, 1, where o Bd, - s he bd quoe of he 'h opon a me o Ask - s he ask quoe of he 'h opon a me Moneyness: We om opons ha are far from he money such ha: S K K,, 0.1, where o S,K The exchange rae and srke prce respecvely The mpled exchange rae s hen calculaed as he mean of he mos a he money opons a each me o maury 1. 1 There are hree seres arranged by me o maury raded n he marke for dollar opons a he TASE 35