A Geeral Closed Form Optio Pricig Formula Cipria NECULA, Gabriel DRIMUS Walter FARKAS,3 Departmet of Bakig ad Fiace, Uiversity of Zurich, Plattestrasse 4, CH-83, Zurich, Switzerlad Email: cipria.ecula@bf.uzh.ch, gabriel.drimus@bf.uzh.ch, walter.farkas@bf.uzh.ch. Departmet of Moey ad Bakig, DOFIN, Bucharest Uiversity of Ecoomic Studies, Bucharest, Romaia 3 Departmet of Mathematics, ETH Zurich, Raemistrasse, CH-89 Zurich, Switzerlad First versio: Jauary 7, 3 This versio: Jauary 5, 3 Abstract A ew method to retrieve the risk-eutral probability measure from observed optio prices is developed ad a closed-form pricig formula for Europea optios is obtaied by employig a modified Gram-Charlier series expasio, kow as the Gauss-Hermite expasio. This ew optio pricig formula is a alterative to the iverse Fourier trasform methodology ad ca be employed i geeral models with probability distributio fuctio or characteristic fuctio kow i closed form. We calibrate the model to both simulated ad market optio prices ad fid that the resultig implied volatility curve provides a good approximatio for a wide rage of strikes. Keywords: Europea optios, Gauss-Hermite series expasio, calibratio. JEL Classificatio: C63, G3 Ackowledgemets: This work was supported by a SCIEX-NMS Fellowship, Project Code.59. Electroic copy available at: http://ssr.com/abstract=359
. Itroductio The risk-eutral valuatio framework is oe of the pillars of moder fiace theory. Uder this framework, the risk-eutral probability measure is a essetial igrediet for asset valuatio, sice the value of a fiacial derivative is give by the expected value uder the risk-eutral measure of the future payoffs geerated by the derivative, discouted at the riskless iterest rate. This approach was pioeered i the cotext of Europea optio pricig by Black ad Scholes (973) ad Merto (973) who postulated a Gaussia risk-eutral measure ad obtaied the celebrated Black-Scholes-Merto (BSM) formula. The stylized fact of volatility smiles has ecouraged the developmet of various extesios that accout for the fact that the risk-eutral desity is egatively skewed ad leptokurtic, such as stochastic volatility models (Hesto, 993), jump-diffusio models (Merto, 976; Kou, ), models based o the Geeralized Hyperbolic process or other pure-jumps Levy processes (Bibby ad Sorese, 997). It is oetheless appealig to obtai the risk-eutral desity from optio prices data by usig o-parametric methods (Ait-Sahalia ad Lo, 998; Ait-Sahalia ad Duarte, 3) or usig a expasio aroud a desity which is easy to compute, such as the ormal or the logormal. Jarrow ad Rudd (98) pioeered the desity expasio approach to optio pricig usig a Edgeworth series expasio of the termial uderlyig asset price risk-eutral desity aroud the log-ormal desity. Corrado ad Su (996, 997) adopted the Jarrow Rudd framework ad derived a optio pricig formula usig a Gram Charlier type A series expasio of the uderlyig asset log-retur risk-eutral desity aroud the Gaussia desity. The pioeerig papers of Jarrow ad Rudd (98) ad Corrado ad Su (996) focused o optio pricig formulae based o a Gram-Charlier type A expasio, geerated a semial brach of optio pricig research. A overview o the Gram-Charlier desity expasio approach to optio valuatio is provided by Jurczeko, Maillet, ad Negrea (). Brow ad Robiso () corrected a typographic error i the iitial Corrado ad Su (996) formula ad poit out that call-put parity is o loger verified whe the risk-eutral log-retur desity fuctio is approximated by a Gram Charlier Type A series expasio, due to the lack of the martigale restrictio (Logstaff, 995). Jurczeko, Maillet, ad Negrea (4) slightly modified the origial formula to provide cosistecy with a martigale restrictio. They also employ CAC 4 idex optios ad show that the differeces betwee the various modificatios of the Corrado Su model are mior, but could be ecoomically sigificat i Electroic copy available at: http://ssr.com/abstract=359
specific cases. Corrado (7) developed a martigale restrictio that is hidde behid a reductio i parameter space for the Gram Charlier expasio coefficiets. The resultig restrictio is ivisible i the optio price. Although probability desities give by Gram-Charlier Type A series expasios are as tractable as the Gaussia desity, they have the drawback that they ca yield egative probability values. Jodeau ad Rockiger () developed umerical methods to impose positivity costraits o the Gram Charlier expasio. Rompolis ad Tzavalis (7) employ a method to retrieve the risk eutral probability desity fuctio based o a expoetial form of a Gram Charlier series expasio, kow as type C Gram Charlier expasio. This type of expasio guaratees that the values of the risk eutral desity will be always positive, but there is o closed form formula for optio values. I geeral, from the Gram Charlier type A desity expasio, oly the first two terms, accoutig for skewess ad kurtosis, are kept i empirical studies related to optio pricig. It is quite probable, however, that icludig higher order terms i the expasio will produce a decrease, rather, as oe would expect, a icrease i the performace of the optio pricig formula based o the Gram-Charlier type A expasio. This couterituitive result is likely to occur due to the lack of covergece of the Gram-Charlier type A expasio for heavy tailed distributios that are of iterest i fiace, as it coverges oly if the probability desity fuctio falls off faster tha exp ( x 4) at ifiity (Cramér, 957). The aim of this paper is to develop a ew method to retrieve the risk-eutral probability measure ad to derive a optio pricig formula by employig a modified Gram- Charlier series expasio. Istead of usig the probabilists Hermite polyomials, as i the classical Gram-Charlier type A series expasio, we replace them by the physicists Hermite polyomials. The mai advatages of the ew expasio cosist i its covergece for heavy tailed distributios ad i the possibility of obtaiig optio prices i closed form. The rest of the paper is structured as follows. I sectio we preset the classical ad the modified Gram-Charlier series expasio ad aalyze a example based o a widely used distributio i fiace, amely the Normal Iverse Gaussia (NIG) distributio. I sectio 3 we derive a pricig formula for Europea call optios i the cotext of the modified Gram- Charlier expasio. I sectio 4 we aalyze various methods for obtaiig the expasio coefficiets from observed optio prices ad preset a simulatio study based o the Hesto model. Although the cotributio of the paper is methodological rather tha empirical, we
also preset, i sectio 4, a calibratio exercise based o observed optio data. The fial sectio cocludes.. The Gram-Charlier ad Gauss-Hermite series expasios A probability desity fuctio p ( x ) with mea µ ad stadard deviatio σ ca be represeted as a Gram Charlier Type A series expasio i the followig form: x µ x µ p ( x) = z c He () σ σ σ = where z ( x ) is the stadard Gaussia desity, He ( ) x deotes a th-order probabilists Hermite polyomial. These form a orthogoal basis o (, + ) with respect to the weight fuctio w( x) e x =. The probabilists Hermite polyomials (Abramowitz ad Stegu, 964) are defied recursively by He ( x) = xhe ( x) He ( x) with ( ) + He ( x) = x. The expasio coefficiets are give by ( ) He x = ad x µ c = He p x dx! σ ad are, i fact, a liear combiatio of the momets of the radom variable with probability distributio fuctio p ( x ), a property which is quite coveiet whe it comes to estimate them. Figure. The classical Gram-Charlier approximatio of the NIG distributio 8 7 6 5 GC N=3 GC N=4 GC N=5 Gaussia NIG 8 6 GC N=3 GC N=4 GC N=5 Gaussia NIG pdf 4 3 - pdf 4 - - -4-3 - - stadard deviatios A -4-4 -3 - - stadard deviatios The graphs depict the probability distributio fuctio of the target distributio (NIG), of the Gaussia distributio, ad the Gram-Charlier approximatios trucated after N=3,4,5 terms. I pael a the target distributio has mea -., stadard deviatio., skewess -.6 ad excess kurtosis 5, ad i pael b the target distributio has mea -.5, stadard deviatio., skewess - ad excess kurtosis. B 3
The Gram-Charlier series expasio has poor covergece properties for heavy tailed distributios (Cramer, 957). I order to illustrate the divergece of this expasio we preset a illustrative example based o the Normal Iverse Gaussia (NIG) distributio. Figure presets the Gram-Charlier approximatios with a icreasig umber of terms i the expasio ad the exact NIG distributio for two sets of parameters. Figure a depicts a NIG distributio with mea -., stadard deviatio., skewess -.6 ad excess kurtosis 5, ad figure b the case with mea -.5, stadard deviatio., skewess - ad excess kurtosis. It is obvious that the series quickly becomes iaccurate by icludig a larger umber of terms. The parameters of the NIG distributios were obtaied from the first four cumulats usig the method described i Eriksso, Forsberg ad Ghysels (4). I fact, the Gram-Charlier expasio is a result of the expasio of the fuctio p ( x) z ( x ) i the basis of the probabilists Hermite polyomial. However, oe could choose aother set of orthogoal polyomials as the basis of the expasio. I particular, the physicists Hermite polyomials are employed i astrophysics (va der Marel ad Frax, 993). This yields a modified Gram-Charlier expasio or the so called Gauss-Hermite expasio. A result i Myller-Lebedeff (97) implies that the Gauss-Hermite expasio coverges eve for heavy tailed distributios. More specifically, the covergece of the expasio is assured if x 3 p ( x) lim =. x ± Therefore, a probability desity fuctio p ( x ) with mea µ ad stadard deviatio σ ca be represeted as a Gauss-Hermite (or modified Gram Charlier) series expasio i the followig form: where H ( ) x µ x µ p ( x) = z a H () σ σ σ = x deotes a th-order physicists Hermite polyomial. These form a orthogoal basis o (, ) + with respect to the weight fuctio w( x) = e x. The physicists Hermite polyomials (Abramowitz ad Stegu, 964) are defied recursively by ( ) = ( ) ( ) with H ( x ) = ad ( ) H x xh x H x + H x = x. Usig the orthogoality coditio of the physicists Hermite polyomials it follows π x µ x µ =! σ σ a z H p x dx. that the expasio coefficiets are give by ( ) 4
These coefficiets are o loger a liear combiatio of the momets, but of some weighted momets, i.e. x µ x µ E z, of the radom variable with probability distributio σ σ fuctio p ( x ). Later i the paper we will preset some methods for calibratig these expasios coefficiets. The improved covergece properties for heavy tailed distributios of the Gauss- Hermite series expasio are illustrated i Figure, for the NIG distributios with the same parameter sets as i the previous exercise. Figure. The modified Gram-Charlier (Gauss-Hermite) approximatio of the NIG distributio 6 5 4 Gauss-Hermite approx. Gaussia NIG - pdf 3 log pdf - -3-4 -4-3 - - stadard deviatios A -5 Gauss-Hermite approx. -6 Gaussia NIG -7-4 -3 - - stadard deviatios pdf 3.5 3.5.5 Gauss-Hermite approx. Gaussia NIG log pdf - - -3-4.5-4 -3 - - stadard deviatios B -5-6 Gauss-Hermite approx. -7 Gaussia NIG -8-4 -3 - - stadard deviatios The graphs depict the probability distributio fuctio (pdf) ad the logarithm of the pdf of the target distributio (NIG), of the Gaussia distributio, ad the modified Gram-Charlier (Gauss-Hermite) approximatio trucated after N=5 terms. I pael a the target distributio has mea -., stadard deviatio., skewess -.6 ad excess kurtosis 5, ad i pael b the target distributio has mea -.5, stadard deviatio., skewess - ad excess kurtosis. There is oe drawback of the Gauss-Hermite approximatio relative to the Gram- Charlier approximatio related to coditio that the total mass of the desity should be. Due 5
to the properties of the probabilists Hermite polyomials, i the Gram-Charlier approximatio this coditio is satisfied idepedet of the umber of terms used i the approximatio. O the other had for the Gauss-Hermite series expasio the coditio is valid oly i the limit sice the requiremet of uitary mass is equivalet to the idetity k= a k ( k )! =. However, if the trucatio is doe after a large umber of terms are k! icluded, the mass of the distributio should be very close to. Alteratively oe could ormalize the expasio coefficiets such as the trucated sum adds to. The followig lemma poits out that the characteristic fuctio ca also be easily expaded usig the same expasio coefficiets. Lemma. (Characteristic fuctio represetatio) Cosider a probability desity fuctio p ( x ) with mea µ, stadard deviatio σ ad Gauss-Hermite expasio coefficiets ( a ) N where i =.. The the associated characteristic fuctio ca be writte as: σ ϕ φ µφ φ σφ (3) = ( ) = exp i a i H ( ) Proof: If oe deotes by pɶ ( xɶ ) the stadardized desity, oe has that p ( x) x µ = pɶ. σ σ Usig a well-kow property of the physicists Hermite polyomials, amely x x d ( ), the Gauss-Hermite expasio of p ( x) e H x = x e dx d p ɶ x ɶ z x ɶ ah x ɶ a x z x = = ɶ dxɶ ɶ ( ) = ( ) ( ) = ( ) d dφ ɶ ɶ ca be writte as. Therefore, the Fourier trasform of pɶ is ai φ exp φ = exp φ ai H ( φ ). The Fourier trasform of p = = follows immediately. 3. Optio pricig The Gauss-Hermite series expasio is a attractive alterative for approximatig the risk-eutral measure desity ad, as the followig result poits out, it allows for a closed form formula for pricig Europea optios. 6
Propositio. (Optio pricig formula) Assume that the log-retur risk-eutral measure for time horizo τ is characterized by a aualized mea µ, a aualized stadard deviatio σ ad Gauss-Hermite expasio coefficiets ( ) t a N. The the premium at time t of a Europea call optio with strike price K ad maturity + τ is give by: qτ rτ ( t,,,, µ, σ, τ ) t c S K r q = S e Π Ke Π (4) where S t is the spot price of the uderlyig, r is the risk-free iterest rate, q deotes the divided yield. We have I ad σ Π = exp µ ( r q) + τ a I ad = J satisfy the recursio equatios ( ) ( ) Π = a J where = I = z d H d + σ τ I + I ad + = ( ) ( ) + with I = N ( d ), I z ( d ) σ τ N ( d ) J z d H d J J + = N ( d ), J z ( d ) =, d ( St K ) + ( + ) log µ σ τ =, d d σ τ σ τ cumulative distributio fuctio of the stadard Gaussia distributio. = +, =, ad ( ) N is the Proof: If oe deotes by pt τ ( St τ ) ad by p ( x ) the stadardized log-retur risk-eutral desity the we have that: + + the termial uderlyig asset price risk-eutral desity (,,,,,, ) r τ µ σ τ = max (,) ( ) c S K r q e S K p S ds t t+ τ t+ τ t+ τ t+ τ x ( t ) ( ) rτ µτ + σ τ = e max S e K, p x dx ( t ) ( ) rτ µτ + σ τ x qτ rτ t d = e S e K p x dx = S e Π Ke Π ( µ r+ q) τ σ τ x with Π = e e p ( x) dx ad ( ) d Π = p x dx. d Takig ito accout the Gauss-Hermite expasio of the log-retur risk-eutral desity, oe has that Π = exp ( ) + σ µ r q τ σ τ σ τ x ai with = ( ) ( ) = d I e e z x H x dx = σ τ + d ( x ) ( + σ τ ) ( + σ τ ) = ( + σ τ ) ( ) σ τ σ τ e e z x H x dx H x z x dx ad σ τ d 7
Π = a J with J = H ( x) z ( x) dx. Usig the properties of Gauss-Hermite = d ' polyomials amely H ( x) = x H ( x) H ( x), ( ) + by parts, oe ca obtai the recursio equatios for I ad H x = H ( x) ad itegratio J as follows: ( σ τ ) ( ) ( σ τ ) ( σ τ ) ( σ τ ) ( ) I = H x + z x dx = x + H x + H x + z x dx + + d d ( σ τ ) ( ) σ τ ( σ τ ) ( ) ( σ τ ) ( ) = H x + z x dx + H x + z x dx H x + z x dx d d d ( ) ( ) = H d + σ τ z d + σ τ I + I where we have used z '( x) = xz( x). ( ) ( ) ( ) ( ) ( ) J = H x z x dx = xh x H x z x dx + + d d ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = H d z d + H x z x dx = H d z d + H x z x dx d d ( ) ( ) = H d z d + J. Fially, we have I = z ( x) dx = N ( d) ad similarly J = z ( x) dx = N ( d ) d d. Remark. If oe assumes that the risk eutral measure desity is ormal the the Gauss-Hermite expasio coefficiets are give by a = ad a =, ad, therefore, equatio (4) reduces to the Black-Scholes-Merto formula. Whe usig series expasios to approximate risk-eutral desities oe has to determie the martigale restrictio associated to that expasio. Lemma allows us to easily derive the martigale restrictio i the case of the Gauss-Hermite expasio. Corollary. (Martigale restrictio) Assume that the log-retur risk-eutral measure for time horizo τ is characterized by a aualized mea µ, a aualized stadard deviatio σ, ad Gauss-Hermite expasio coefficiets ( a ) N. Give that r is the risk-free iterest rate ad q deotes the divided yield the the martigale restrictio associated to the Gauss-Hermite expasio is: 8
σ exp ( ) Proof: The martiagle coditio requires ( ) µ r q + τ ai H i σ τ = (5) = = rτ qτ E e St+ τ e St. Deotig by ϕ the Fourier trasform of the log-retur risk-eutral measure desity, the result follows immediately from { } rτ qτ ( r q) τ Lemma sice E e S = e S e ϕ ( i) t + τ t 4. Calibratig the expasio coefficiets to optio data Before presetig the result relatig the expasio coefficiets to observed optio data, it is iterestig to poit out that the Gauss-Hermite expasio coefficiets could be computed from the associated characteristic fuctio. More specifically, from equatio (3) ad from the orthogoality coditio of the physicists Hermite polyomials it follows that the expasio coefficiets of a desity with mea µ, stadard deviatio σ ad characteristic fuctio ϕ ca be represeted as: a φ exp ( ) exp! i H φ i µ = φ ϕ φ d φ π σ σ (6) Therefore, give a optio pricig model for which the characteristic fuctio is kow, such as the Hesto model, equatio (6) together with equatio (4) provide a alterative method of obtaiig optio prices. We ow show how to obtai the Gauss-Hermite expasio coefficiets from optio market prices. The followig propositio states the result. Propositio. (Calibratio to optio data) Give optio prices for time horizo τ ad give that the log-retur risk-eutral measure desity for time horizo τ is characterized by a aualized mea µ ad a aualized stadard deviatio σ the the Gauss-Hermite expasio coefficiets ca be computed as: ( r q) τ ( r q) τ ( ( )) ( ( )) π a = z d S te H d Ste +! ( r q) τ Ste rτ ( ) ( t,, ) ( ) ( t,, ) + e F K c S K τ dk + F K p S K τ dk ( r q S ) τ te (7) 9
S t log + µτ K where d ( K) =, S t is the spot price of the uderlyig, r is the risk-free σ τ iterest rate, q is the divided yield, c ( S, K, τ ) ad (,, ) t p S K τ deotes the premium at time t of a Europea call ad put, respectively, with strike price K ad maturity t fuctio F ( ) is give by t + τ ad the z( d) ( ) = (( σ τ ) ( ) + ( 4 σ τ ) ( ) + 4 ( ) ( ) ) F K d d H d d H d H d K σ τ π x µτ x µτ! σ τ σ τ Proof: Oe has that a = z H p ( x) dx where ( ) p x log-retur risk-eutral measure desity for time horizo τ. By a chage of variable we obtai that = ( ) ( ) where p ( S ) a G S p S ds π t+ τ t+ τ t+ τ t+ τ! risk-eutral desity ad ( ) t τ t τ + + is the termial uderlyig asset price ( S S ) log ( S S ) log t µτ t µτ G S = z H. The propositio σ τ σ τ follows i a straightforward way from a geeral result i Bakshi, Kapadia ad Mada (3) based o the fact that ay fuctio with bouded expectatios ca be spaed by a cotiuum of OTM Europea call ad put optios (Bakshi ad Mada, ). More specifically, oe has to apply equatio (3) i Bakshi, Kapadia ad Mada (3) to the fuctio G ( ) S. Remark. I order to implemet equatio (7) oe eeds the mea ad the stadard deviatio of the log-retur risk-eutral measure desity. These ca be computed usig the method described i Bakshi, Kapadia ad Mada (3). I what follows we coduct a series of calibratio exercises i order to ivestigate the performace of the optio pricig model based o the Gauss-Hermite approximatio to adequately reflect the observed volatility smile. 4.. Calibratio to simulated optio data I order to simulate the optio prices we employ the Hesto stochastic volatility model with the followig parameter values: k = 4.5, θ =.8, σ =.8, ρ =.85 ad v =.3. These values are similar to those obtaied i Gourier, Bardgett ad Leippold () by calibratig the Hesto model to all S&P 5 idex optios traded o October. We
assume that the curret uderlyig price is ad we geerate a set of 3 moths Europea call ad put optio prices with strike prices betwee 35 ad 55 with a step of.5 uits. The risk-free iterest rate is set to r =.5 ad the divided yield to q =.3. Sice the characteristic fuctio of the Hesto model is kow i closed form we ca compute the expasio coefficiets either by usig equatio (6) or directly from the geerated optio pricig usig (7). I all the computatio we trucated the Gauss-Hermite expasio after N=5 terms. Figure 3 depicts the approximated desity ad i the case the expasio coefficiets are computed usig the characteristic fuctio (pael a) or are iferred from optio prices (pael b). Figure 3. The Gauss-Hermite approximatio of the Hesto model distributio 6 5 4 Gauss-Hermite approx. Hesto Gaussia - pdf 3 log pdf - -3-4 -4-3 - - stadard deviatios a -5 Gauss-Hermite approx. -6 Hesto Gaussia -7-4 -3 - - stadard deviatios 6 5 4 Gauss-Hermite approx. Hesto Gaussia - pdf 3 log pdf - -3-4 -4-3 - - stadard deviatios b -5 Gauss-Hermite approx. -6 Hesto Gaussia -7-4 -3 - - stadard deviatios The graphs depict the probability distributio fuctio (pdf) ad the logarithm of the pdf of the target distributio (Hesto model), of the Gaussia distributio, ad the modified Gram-Charlier (Gauss-Hermite) approximatio trucated after N=5 terms. I pael a the expasio coefficiets are computed usig the characteristic fuctio, ad i pael b are iferred from the simulated optio prices. Next we employed the optio pricig formula (4) trucated after N=5 terms ad derived the implied volatility curve as depicted i Figure 4. I pael a the expasio
coefficiets are computed usig the characteristic fuctio, ad i pael b they are estimated from optio prices. We do ot costrai the coefficiets i order to observe the martigale restrictio. However, the trucated sum i equatio (5) is quite close to beig equal to.. Figure 4. Implied volatility curves iferred from simulated optio prices.45.4 IV Hesto IV Gauss-Hermite approx..45.4 IV Hesto IV Gauss-Hermite approx..35.35 implied volatility.3.5 implied volatility.3.5...5.5. -.5 -.4 -.3 -. -... log (K/F) a. -.5 -.4 -.3 -. -... log (K/F) The graphs depict the implied volatility curves of the Hesto model ad of the optio pricig model based o the Gauss- Hermite approximatio trucated after N=5 terms. I pael a the expasio coefficiets are computed usig the characteristic fuctio, ad i pael b are iferred from the simulated optio prices b The Gauss-Hermite implied volatility curve is a good approximatio of the Hesto model oe for a rage of strike prices spaig five stadard deviatios of the log-retur riskeutral desity, four stadard deviatios to the left ad oe to the right of the mea retur or, equivaletly, for log( K F ) i the iterval (-.4,.), where F is the forward price. 4.. Calibratio to market optio data I this sectio we employ market data about Europea optios i order to ifer the implied volatility curve of optio pricig model based o the Gauss-Hermite approximatio. More specifically, we focus o SPX optios quotes o 5 September, the day before triple-witchig, ad compute implied volatilities for maturities of moth ad 3 moths. The implied volatility surface o this specific date was also aalysed by Gatheral ad Jacquier () with a SVI parameterizatio. Data o optio prices, the term structure of iterest rate ad the divided yield o this specific date are obtaied from OptioMetrics. I order to estimate the expasio coefficiets, we determied the implied volatilities associated to mid prices of the put optios ad the the call ad put prices that appear i the itegrals i equatio (7) are computed by iterpolatio. We employed the optio pricig formula (4)
trucated after N=5 terms. We do ot costrai the coefficiets i order to impose the martigale restrictio ad the trucated sum i equatio (5) is aroud.6. The resultig implied volatility curves of the Gauss-Hermite approximatio are depicted i Figure 5. Figure 5. Implied volatility curves iferred from market optio prices.8.7 IV Gauss-Hermite approx. IV bid IV ask.7.65.6 IV Gauss-Hermite approx. IV bid IV ask implied volatility.6.5.4 implied volatility.55.5.45.4.35.3.3.5. -.4 -.3 -. -... log (K/F). - -.8 -.6 -.4 -...4 log (K/F) moth 3 moths The graphs depict the implied volatility curves for moth ad 3 moths of the optio pricig model based o the Gauss- Hermite approximatio trucated after N=5 terms. The Gauss-Hermite approximatio seems to perform quite well for a iterval of strike prices that spas more tha five stadard deviatios of the log-retur risk-eutral desity, four stadard deviatios to the left ad oe ad a half to the right of the mea retur or for a log( K F ) i the iterval (-.4,.5) for moth maturity ad i the iterval (-.8,.3) for 3 moths maturity. 5. Cocludig remarks I this paper we developed a ew method to retrieve the risk-eutral measure desity from optio prices usig the Gauss-Hermite series expasio aroud the Gaussia desity ad poited out its better covergece properties compared to the Gram-Charlier expasio. We also derived several methods for obtaiig the expasio coefficiets. More specifically, oe ca obtai the coefficiets of the Gauss-Hermite expasio from the probability distributio fuctio, from the characteristic fuctio, or directly from market optio prices. Approximatig the risk-eutral desity usig the Gauss-Hermite expasio is quite appealig because it allows for a closed form optio pricig model that embeds the classical Black-Scholes-Merto formula. This optio pricig formula based o the Gauss-Hermite expasio is a alterative to the iverse Fourier trasform methodology ad is quite geeral 3
sice it ca be employed for models with the probability distributio fuctio kow i closed form, for models with the characteristic fuctio kow i closed form, ad ca also be calibrated to market optio prices. We calibrated the ew optio pricig model to optio prices simulated usig Hesto stochastic volatility model ad to market optio prices. These calibratio exercises have revealed that the resultig implied volatility curves are quite accurate for a rage of strike prices that spas five stadard deviatios of the log-retur risk-eutral desity, four stadard deviatios to the left ad oe to the right of the mea retur. Refereces Abramowitz, M., ad I.A. Stegu, (Eds.) (964). Hadbook of Mathematical Fuctios, Natioal Bureau of Stadards. Ait-Sahalia, Y. ad A.W. Lo. (998). Noparametric estimatio of state-price desities implicit i fiacial asset prices. Joural of Fiace 53 (998) 499 548 Ait-Sahalia, Y., ad J. Duarte. (3). Noparametric optio pricig uder shape restrictios. Joural of Ecoometrics, 6, 9 47. Bakshi, G. ad D. Mada. (). Spaig ad derivative-security valuatio. Joural of Fiacial Ecoomics, 55 5 38. Bakshi, G., N. Kapadia, ad D. Mada. (3). Stock Retur Characteristics, Skew Laws, ad the Differetial Pricig of Idividual Equity Optios, Review of Fiacial Studies, 6(): -43 Bibby, B. M. ad M. Sorese. (997). A hyperbolic diffusio model for stock prices. Fiace & Stochastics, 5-4 Black, F., ad M. Scholes. (973). The pricig of optios ad corporate liabilities. Joural of Political Ecoomy, 8, 637 654. Brow, C. ad D. Robiso. (). Skewess ad kurtosis implied by optio prices: A correctio. Joural Fiacial Research 5 79 8. Corrado, C. J., ad T. Su. (996). S&P 5 idex optio tests of Jarrow ad Rudd s approximate optio valuatio formula. Joural of Futures Markets, 6, 6 69. 4
Corrado, C. J., ad T. Su. (997). Implied volatility skews ad stock idex skewess ad kurtosis implied by S&P 5 idex optio prices. Joural of Derivatives, 4, 8 9. Corrado, C. J. (7) The hidde martigale restrictio i Gram-Charlier optio prices, Joural of Futures Markets, vol. 7, o. 6, pp. 57-534. Cramer. H. (957). Mathematical Methods of Statistics. Priceto Uiversity Press, Priceto. Eriksso, A., L. Forsberg, ad E. Ghysels. (4). Approximatig the Probability Distributio of Fuctios of Radom Variables: A New Approach, Discussio paper, CIRANO. Gatheral, J., ad A. Jacquier. (). Arbitrage-Free SVI Volatility Surfaces. available at SSRN: http://dx.doi.org/.39/ssr.3333 Gourier, E., C. Bardgett, ad M. Leippold. (). Iferrig volatility dyamics from the S&P5 ad VIX markets. Presetatio, Uiversity of Zurich Hesto, S. L., (993). A closed-form solutio for optios with stochastic volatility with applicatios to bod ad currecy optios. Review of Fiacial Studies 6 37 343. Jarrow, R., ad A. Rudd. (98). Approximate optio valuatio for arbitrary stochastic processes. Joural of Fiacial Ecoomics,, 347 369. Jodeau, E., ad M. Rockiger. (). Gram Charlier desities. Joural of Ecoomic Dyamics ad Cotrol, 5, 457 483. Jurczeko, E., B. Maillet, ad B. Negrea. (). Revisited multi-momet approximate optio pricig models: A geeral compariso (Discussio Paper of the LSE-FMG, No. 43). Jurczeko, E., B. Maillet, ad B. Negrea. (4). A ote o skewess ad kurtosis adjusted optio pricig models uder the martigale restrictio. Quatitative Fiace, 4, 479 488. Kou, S. G. (). A Jump-Diffusio Model for Optio Pricig. Maagemet Sciece, 48, 8, 86- Logstaff, F. A. (995). Optio pricig ad the martigale restrictio. Review of Fiacial Studies, 8, 9 4. Merto, R. (973). Ratioal theory of optio pricig. Bell Joural of Ecoomics ad Maagemet Sciece, 4, 4 83. Merto, R. C. (976). Optio pricig whe uderlyig stock returs are discotiuous. Joural of Fiacial Ecoomics 3, 5-44. 5
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