HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACK-MERTON-SCHOLES?

HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACK-MERTON-SCHOLES? WALTER SCHACHERMAYER AND JOSEF TEICHMANN Absrac. We compare he opion pricing formulas of Louis Bachelier and Black-Meron-Scholes and observe heoreically as well as for Bachelier s original daa ha he prices coincide very well. We illusrae Louis Bachelier s effors o obain applicable formulas for opion pricing in pre-compuer ime. Furhermore we explain by simple mehods from chaos expansion why Bachelier s model yields good shor-ime approximaions of prices and volailiies. 1. Inroducion I is he pride of Mahemaical Finance ha L. Bachelier was he firs o analyze Brownian moion mahemaically, and ha he did so in order o develop a heory of opion pricing (see []). In he presen noe we shall review some of he resuls from his hesis as well as from his laer exbook on probabiliy heory (see [3]), and we shall work on he remarkable closeness of prices in he Bachelier and Black- Meron-Scholes model. The fundamenal principle underlying Bachelier s approach o opion pricing is crysallized in his famous dicum (see [], p.34) L ésperance mahemaique du spéculaeur es nul, i.e. he mahemaical expecaion of a speculaor is zero. His argumen in favor of his principle is based on equilibrium consideraions (see [] and [11]), similar o wha in oday s erminology is called he efficien marke hypohesis (see [10]), i.e. he use of maringales o describe sochasic ime evoluions of price movemens in ideal markes. L. Bachelier wries on his opic (see he original french version in [], p. 31). I seems ha he marke, he aggregae of speculaors, can believe in neiher a marke rise nor a marke fall, since, for each quoed price, here are as many buyers as sellers. The reader familiar wih oday s approach o opion pricing migh wonder where he conceps of risk free ineres rae and risk neural measure have disappeared o, which seem crucial in he modern approach of pricing by no arbirage argumens (recall ha he discouned price process should be a maringale under he risk neural measure). As regards he firs issue L. Bachelier applied his fundamenal principle in erms of rue prices (his is erminology from 1900 which corresponds o he concep of forward prices in modern erminology), since all he paymens involved (including he premium of he opion) were done only a mauriy of he conracs. See [] for an explici descripion of he rading rules a he 1

WALTER SCHACHERMAYER AND JOSEF TEICHMANN bourse de Paris in 1900. I is well-known ha he passage o forward prices makes he riskless ineres rae disappear: in he conex of he Black-Meron-Scholes formula, his is wha amouns o he so-called Black s formula (see [4]). As regards he second issue L. Bachelier apparenly believed in he maringale measure as he hisorical measure, i.e. for him he risk neural measure conincides wih he hisorical measure. For a discussion of his issue compare [10]. Summing up: Bachelier s fundamenal principle yields exacly he same recipe for opion pricing as we use oday (for more deails we refer o he firs secion of he S. Flour summer school lecure [11]): using forward prices ( rue prices in he erminology of 1900) one obains he prices of opions (or of more general derivaives of European syle) by aking expecaions. The expecaion perains o a probabiliy measure under which he price process of he underlying securiy (given as forward prices) saisfies he fundamenal principle, i.e. is a maringale in modern erminology. I is imporan o emphasize ha, alhough he recipes for obaining opion prices are he same for Bachelier s as for he modern approach, he argumens in favour of hem are very differen: an equilibrium argumen in Bachelier s case as opposed o he no arbirage argumens in he Black-Meron-Scholes approach. Wih all admiraion for Bachelier s work, he developmen of a heory of hedging and replicaion by dynamic sraegies, which is he crucial ingredien of he Black- Meron-Scholes-approach, was far ou of his reach (compare [11] and secion.1 below). In order o obain opion prices one has o specify he underlying model. We fix a ime horizon T > 0. As is well-known, Bachelier proposed o use (properly scaled) Brownian moion as a model for forward sock prices. In modern erminology his amouns o (1.1) S B := S 0 + σ B W, for 0 T, where (W ) 0 T denoes sandard Brownian moion and he superscrip B sands for Bachelier. The parameer σ B > 0 denoes he volailiy in he Bachelier model. Noice ha in conras o oday s sandard Bachelier measured volailiy in absolue erms. In fac, Bachelier used he normalizaion H = σb and called his quaniy he coefficien of insabiliy or of nervousness of he securiy S. The reason for he normalisaion H = σb is ha H T hen equals he price of an a he money opion in Bachelier s model (see []) The Black-Meron-Scholes model (under he risk-neural measure) for he price process is, of course, given by (1.) S BS = S 0 exp(σ BS W (σbs ) ), for 0 T. Here σ BS denoes he usual volailiy in he Black-Meron-Scholes model. This model was proposed by P. Samuelson in 1965, afer he had led by an inquiry of J. Savage for he reaise [3] personally rediscovered he virually forgoen Bachelier hesis in he library of Harvard Universiy. The difference beween he wo models is somewha analogous o he difference beween linear and compound ineres, as becomes apparen when looking a he associaed Iô sochasic

BACHELIER VERSUS BLACK-SCHOLES 3 differenial equaion, ds B = σ B dw, ds BS = S BS σ BS dw. This analogy makes us expec ha, in he shor run, boh models should yield similar resuls while, in he long run, he difference should be specacular. Forunaely, opions usually have a relaively shor ime o mauriy (he opions considered by Bachelier had a ime o mauriy of less han monhs).. Bachelier versus Black-Meron-Scholes We now have assembled all he ingrediens o recall he derivaion of he price of an opion in Bachelier s framework. Fix a srike price K, a horizon T and consider he European call C, whose pay-off a ime T is modeled by he random variable C B T = (S B T K) +. Applying Bachelier s fundamenal principle and using ha ST B is normally disribued wih mean S 0 and variance (σ B ) T, we obain for he price of he opion a ime = 0 (.1) (.) C B 0 = E[(SB T K) +] = 1 (S 0 + x K) K S 0 σ B T exp( x (σ B ) T )dx = (S 0 K)Φ( S 0 K σ B T ) + σb Tφ( S 0 K σ B T ), where φ(x) = 1 exp( x ) denoes he densiy of he sandard normal disribuion. We applied he relaion φ (x) = xφ(x) o pass from (.1) o (.). For deails see, e.g., [5]. For furher use we shall need he very well-known Black-Meron-Scholes price, oo, C BS (.3) (.4) (.5) 0 = E[(ST BS K) + ] = = (S 0 exp( (σbs ) T log K S 0 + (σbs ) T σ BS T + σ BS Tx) K) + 1 exp( x )dx (S 0 exp( (σbs ) T + σ BS 1 Tx) K) exp( x )dx S0 log K = S 0 Φ( + 1 (σbs ) S0 T log σ BS K ) KΦ( 1 (σbs ) T T σ BS ). T Ineresingly, Bachelier explicily wroe down formula (.1), bu did no boher o spell ou formula (.), see [, p. 50]. The main reason seems o be ha a his ime opion prices a leas in Paris were quoed he oher way around: while oday he srike prices K is fixed and he opion price flucuaes according o supply and demand, a Bachelier s imes he opion prices were fixed (a 10, 0 and 50 Cenimes for a rene, i.e., a perpeual bond wih par value of 100 Francs) and herefore he srike prices K flucuaed. Wha Bachelier really needed was he inverse version of he above relaion beween he opion price C0 B and he srike price K.

4 WALTER SCHACHERMAYER AND JOSEF TEICHMANN Apparenly here is no simple formula o express his inverse relaionship. This is somewha analogous o he siuaion in he Black-Meron-Scholes model, where here is also no formula for he inverse problem of calculaing he implied volailiy as a funcion of he given opion price. We shall see below ha L. Bachelier designed a clever series expansion for C B 0 as a funcion of he srike price K in order o derive (very) easy formulae which approximae his inverse relaion and which were well suied o pre-compuer echnology..1. A he money opions. Bachelier firs specializes o he case of a he money opions (called simple opions in he erminology of 1900), when S 0 = K. In his case (.) reduces o he simple and beauiful relaion C B 0 = σ B T. As explicily noiced by Bachelier, his formula can also be used, for a given price C = C0 B of an a he money opion wih mauriy T, o deermine he coefficien of nervousness of he securiy H = σb, i.e., o deermine he implied volailiy in modern language. Indeed, i suffices o normalize he price C B 0 by T o obain H = CB 0 T. We summarize his fac in he subsequen proposiion. For convenience we phrase i raher in erms of σ B han of H. Proposiion 1. The volailiy σ B in he Bachelier model is deermined by he price C0 B of an a he money opion wih mauriy T by he relaion (.6) σ B = C0 B T. In he subsequen Proposiion, we compare he price of an a he money call opion as obained from he Black-Meron-Scholes and Bachelier s formula respecively. In order o relae opimally (see he las secion) C0 B and CBS 0 we choose σ B = S 0 σ and σ BS = σ for some consan σ > 0. We also compare he implied volailiies, for given price C 0 of an a he money call wih mauriy T, in he Bachelier and Black-Meron-Scholes model. We denoe he respecive implied volailiies by σ B and σ BS and discover ha he implied Bachelier volailiy esimaes he Black-Scholes implied volailiy quie well a he money. Proposiion. Fix σ > 0, T > 0 and S 0 = K (a he money), le σ BS = σ and σ B = S 0 σ and denoe by C B and C BS he corresponding prices for a European call opion in he Bachelier (1.1) and Black-Meron-Scholes model (1.) respecively. Then (.7) 0 C0 B CBS 0 S 0 1 σ3 T 3 = O((σ T) 3 ). The relaive error can be esimaed by C0 B (.8) CBS 0 C0 B T 1 σ. Conversely, fix he price 0 < C 0 < S 0 of an a he money opion and denoe by σ B he implied Bachelier volailiy and by σ BS he implied Black-Meron-Scholes

BACHELIER VERSUS BLACK-SCHOLES 5 volailiy, hen (.9) 0 σ BS σb S 0 T 1 (σbs ) 3. Proof. (compare [5] and [11]). For S 0 = K, we obain in he Bachelier and Black- Meron-Scholes model he following prices, respecively, Hence C B 0 = S 0σ T C BS 0 = S 0 (Φ( 1 σ T) Φ( 1 σ T)). C0 B CBS 0 = ( S 0 x S 0 (Φ( x ) Φ( x ))) x=σ T ( = S x 0 S x 0 x x (1 exp( y ))dy y dy x=σ T ) x=σ T = S 0 x 3 1 x=σ T = S 0 4 σ3 T 3 = O((σ T) 3 ), since e y 1 + y for all y, so ha y 1 e y for all y. Clearly we obain C0 B C0 BS 0 again from he firs line. For he second asserion noe ha solving equaion C 0 = σb T = S0 (Φ( 1 σbs T) Φ( 1 σbs T)) for given σ B > 0 yields he Black-Meron-Scholes implied volailiy σ BS. We obain similarly as above 0 σ BS σb = σ BS (Φ( 1 S 0 T σbs T) + Φ( 1 σbs T)) 1 = ( x (Φ( x T ) Φ( x ))) x=σ BS T 1 T 1 (σbs ) 3 T 3 (σ BS ) 3 T =. 1 Proposiion 1 and yield in paricular he well-known asympoic behaviour of an a he money call price in he Black-Meron-Scholes model for T as described in [1]. Proposiion ells us ha for he case when (σ T) 1 (which ypically holds rue in applicaions), formula (.6) gives a saisfacory approximaion of he implied Black-Meron-Scholes volailiy, and is very easy o calculae. We noe ha for he daa repored by Bachelier (see [] and [11]), he yearly relaive volailiy was of he order of.4% p.a. and T in he order of one monh, i.e T = 1 1 years so ha

6 WALTER SCHACHERMAYER AND JOSEF TEICHMANN T 0.3. Consequenly we ge (σ T) 3 (0.008) 3 5 10 7. The esimae in S Proposiion yields a righ hand side of 0 1 5 10 7 1.6 10 8 S 0, i.e. he difference of he Bachelier and Black-Meron-Scholes price (when using he same volailiy σ =.4% p.a.) is of he order 10 8 of he price S 0 of he underlying securiy. The above discussion perains o he limiing behaviour, for σ T 0, of he Bachelier and Black-Meron-Scholes prices of a he money opions, i.e., where S 0 = K. One migh ask he same quesion for he case S 0 K. Unforunaely, his quesion urns ou no o be meaningful from a financial poin of view. Indeed, if we fix S 0 K and le σ T end o zero, hen he Bachelier price as well as he Black-Scholes price end o he pay-off funcion (S 0 K) + of order higher han (σ T) n, for every n 1. Hence, in paricular, heir difference ends o zero faser han any power of (σ T). This does no seem o us an ineresing resul and corresponds in financial erms o he fac ha, for fixed S 0 K and leing σ T end o zero, he hinked pay-off funcion (S 0 K) + essenially amouns o he same as he linear pay-off funcions S 0 K, in he case S 0 > K, and 0, in he case S 0 K < 0. In paricular he funcional dependence of he prices on σ T is no analyical. This behaviour is essenially differen from he behaviour a he money, S 0 = K, since here he prices depend analyically on σ T and he difference ends o zero of order 3. 3. Furher resuls of L. Bachelier We now proceed o a more deailed analysis of he opion pricing formula (.) for general srike prices K. Le C = C0 B denoe he opion price from (.). We shall inroduce some noaion used by L. Bachelier for he following wo reasons: firsly, i should make he ask easier for he ineresed reader o look up he original exs by Bachelier; secondly, and more imporanly, we shall see ha his noaion has is own meris and allows for inuiive and economically meaningful inerpreaions (as we have already seen for he normalizaion H = σb of he volailiy, which equals he ime-sandardized price of an a he money opion). L. Bachelier found i convenien o use a parallel shif of he coordinae sysem moving S 0 o 0, so ha he Gaussian disribuion will be cenered a 0. We wrie (3.1) a = σb T, m := K S 0, P := m + C. The parameer a equals, up o he normalizing facor S0, he ime sandardized absolue volailiy σ B T a mauriy T. Readers familiar, e.g. wih he Hull-Whie model of sochasic volailiy, will realize ha his is a very naural paramerizaion for an opion wih mauriy T. In any case, he quaniy a was a naural paramerizaion for L. Bachelier, as i is he price of he a he money opion wih he mauriy T (see formula.6), so ha i can be direcly observed from marke daa. The quaniy m is he difference beween he srike price K and S 0 and needs no furher explanaion. P has a naural inerpreaion (in Bachelier s imes i was called écar, i.e. he spread of an opion): i is he price P of a european pu wih mauriy T and srike price K, as was explicily noed by Bachelier (using,

BACHELIER VERSUS BLACK-SCHOLES 7 of course, differen erminology). In oday s erminology his amouns o he pucall pariy. Bachelier inerpreed P as he premium of an insurance agains prices falling below K. This is nicely explained in []: a speculaor á la hausse, i.e. hoping for a rise of S T may buy a forward conrac wih mauriy T. Using he fundamenal principle, which in his case boils down o elemenary no arbirage argumens, one concludes ha he forward price mus equal S 0, so ha he oal gain or loss of his operaion is given by he random variable S T S 0 a ime T. On he oher hand, a more pruden speculaor migh wan o limi he maximal loss by a quaniy K > 0. She hus would buy a call opion wih price C, which would correspond o a srike price K (here we see very nicely he above menioned fac ha in Bachelier s imes he srike price was considered as a funcion of he opion price C la prime in french and no vice versa as oday). Her oal gain or loss would hen be given by he random variable (S T K) + C. If a ime T i indeed urns ou ha S T K, hen he buyer of he forward conrac is, of course, beer off han he opion buyer. The difference equals (S T S 0 ) [(S T K) C] = K S 0 + C = P, which herefore may be inerpreed as a cos of insurance. If S T K, we obain (S T S 0 ) [0 C] = (S T K) + K S 0 + C = (S T K) + P. By he Bachelier s fundamenal principle we obain P = E[(S T K) ]. Hence Bachelier was led by no-arbirage consideraions o he pu-call pariy. For furher consideraions we denoe he pu price in he Bachelier model a ime = 0 by P0 B or P(m) respecively. Clearly, he higher he poenial loss C is, which he opion buyer is ready o accep, he lower he coss of insurance P should be and vice versa, so ha we expec a monoone dependence of hese wo quaniies. In fac, Bachelier observed ha he following prey resul holds rue in his model (see [3], p.95): Proposiion 3 (Theorem of reciprociy). For fixed σ B > 0 and T > 0 he quaniies C and P are reciprocal in Bachelier s model, i.e. here is a monoone, sricly decreasing and self-inverse (ha is I = I 1 ) funcion I : R >0 R >0 such ha P = I(C). Proof. Denoe by ψ he densiy of S T S 0, hen C(m) = P(m) = m m (x m)ψ(x)dx, (m x)ψ(x)dx. Hence we obain ha C( m) = P(m). We noe in passing ha his is only due o he symmery of he densiy ψ wih respec o reflecion a 0. Since C (m) < 0 (see he proof of Proposiion 1) we obain P = P(m(C)) := I(C), where C m(c) invers he funcion m C(m). C maps R in a sricly decreasing way o R >0 and

8 WALTER SCHACHERMAYER AND JOSEF TEICHMANN P maps R in a sricly increasing way o R >0. The resuling map I is herefore sricly decreasing, and due o symmery we obain I(P) = P(m(P)) = P( m(c)) = C, so I is self-inverse. Using he above noaions (3.1), equaion (.) for he opion price C0 B (which we now wrie as C(m) o sress he dependence on he srike price) obained from he fundamenal principle becomes (3.) C(m) = m (x m)µ(dx), where µ denoes he disribuion of S T S 0, which has he Gaussian densiy µ(dx) = ψ(x)dx, 1 ψ(x) = σ B T exp( x (σ B ) T ) = 1 a exp( x 4πa ). As menioned above, Bachelier does no simply calculae he inegral (3.). He raher does somehing more ineresing (see [3, p. 94]): Si l on développe l inegrale en série, on obien, i.e. if one develops he inegral ino a series one obains... (3.3) C(m) = a m + m 4πa m4 96π a 3 + m 6 190π 3 a 5 +.... In he subsequen heorem we jusify his sep. I is worh noing ha he mehod for developing his series expansion is no resriced o Bachelier s model, bu holds rue in general (provided ha µ, he probabiliy disribuion of S T S 0, admis a densiy funcion ψ, which is analyic in a neighborhood of 0). Theorem 1. Suppose ha he law µ of he random variable S T admis a densiy µ(dx) = ψ(x)dx, such ha ψ is analyic in a ball of radius r > 0 around 0, and ha Then he funcion C(m) = xψ(x)dx <. m (x m)µ(dx) is analyic for m < r and admis a power series expansion C(m) = c k m k, where c 0 = 0 xψ(x)dx, c 1 = 0 ψ(x)dx and c k = 1 k! ψ(k ) (0) for k. k=0 Proof. Due o our assumpions C is seen o be analyic as sum of wo analyic funcions, C(m) = m xψ(x)dx m m ψ(x)dx.

BACHELIER VERSUS BLACK-SCHOLES 9 Indeed, if ψ is analyic around 0, hen he funcions x xψ(x) and m m xψ(x)dx are analyic wih he same radius of convergence r. The same holds rue for he funcion m m ψ(x)dx. The derivaives can be calculaed by he Leibniz rule, m C (m) = mψ(m) = m C (m) = ψ(m), m ψ(x)dx, whence we obain for he k-h derivaive, for k. C (k) (m) = ψ (k ) (m), ψ(x)dx + mψ(m) Remark 1. If we assume ha m C(m) is locally analyic around m = 0 (wihou any assumpion on he densiy ψ), hen he densiy x ψ(x) is analyic around x = 0, oo, by inversion of he above argumen. Remark. In he case when ψ equals he Gaussian disribuion, he calculaion of he Taylor coefficiens yields d dy ( 1 y exp( a 4πa )) = 1 y 1 y 4 π a 3 e 4 πa, d dy ( 1 y exp( a 4πa )) = 1 a y 8 π 3 a 5 e 1 y 4 πa, d 3 dy 3 ( 1 y exp( a 4πa )) = 1 6πa y y 3 16 π 4 a 7 e 1 y 4 πa, d 4 dy 4 ( 1 y exp( a 4πa )) = 1 1 a 4 1y πa + y 4 3 π 5 a 9 e 1 4 y πa, Consequenly ψ(0) = 1 a, ψ (0) = 0, ψ (0) = 1 4π a, ψ (0) = 0 and ψ (0) = 3 3 1 8 π 3 a, hence wih C(0) = a and C (0) = 1 5, (3.4) C(m) = a m + m 4πa m4 96π a 3 + m 6 190π 3 a 5 + O(m8 ) and he series converges for all m, as he Gaussian disribuion is an enire funcion. This is he expansion indicaed by Bachelier in [3]. Since P( m) = C(m), we also obain he expansion for he pu (3.5) P(m) = a + m + m 4πa m4 96π a 3 + m 6 190π 3 a 5 + O(m8 ). Remark 3. Looking once more a Bachelier s series one noes ha i is raher a Taylor expansion in m a han in m. Noe furhermore ha m a is a dimensionless quaniy. The series hen becomes C(m) = a F( m a ) F(x) = 1 x + x 4π x4 96π + x6 190π 3 + O(x8 ). We noe as a curiosiy ha already in he second order erm he number π appears. Whence if we believe in Bachelier s formula for opion pricing we are able o

10 WALTER SCHACHERMAYER AND JOSEF TEICHMANN deermine π a leas approximaely (see (3.6) below) from financial marke daa (compare Georges Louis Leclerc Come de Buffon s mehod o deermine π by using saisical experimens). Le us urn back o Bachelier s original calculaions. He firs runcaed he Taylor series (3.4) afer he quadraic erm, i.e. (3.6) C(m) a m + m 4πa. This (approximae) formula can easily be invered by solving a quadraic equaion, hus yielding an explici formula for m as a funcion of C. Bachelier observes ha he approximaion works well for small values of m a (he cases relevan for his pracical applicaions) and gives some numerical esimaes. We summarize he siuaion. Proposiion 4 (Rule of Thumb 1). For given mauriy T, srike K and σ B > 0, le m = K S 0 and denoe by a = C(0) he Bachelier price of he a he money opion and by C(m) he Bachelier price of he call opion wih srike K = S 0 + m. Define (3.7) (3.8) Ĉ(m) = a m + m 4πa = C(0) m + m 4πC(0), hen we ge an approximaion of he Bachelier price C(m) of order 4, i.e. C(m) Ĉ(m) = O(m 4 ). Remark 4. Noe ha he value of Ĉ(m) only depends on he price a = C(0) of an a he money opion (which is observable a he marke) and he given quaniy m = K S 0. Remark 5. Given any sock price model under a risk neural measure, he above approach of quadraic approximaion can be applied if he densiy ψ of S T S 0 is locally analyic and admis firs momens. The approximaion hen reads (3.9) Ĉ(m) = C(0) B(0)m + ψ(0) m up o a erm of order O(m 3 ). Here C(0) denoes he price of he a he money european call opion (pay-off (S T S 0 ) + ), B(0) he price of he a he money binary opion (pay-off 1 {ST S 0}) and ψ(0) he value of an a he money Dirac opion (pay-off δ S0, wih an appropriae inerpreaion as a limi). Noice ha B(0) can also be inerpreed, in Bachelier s model, as he hedging raion Dela. Example 1. Take for insance he Black-Meron-Scholes model, hen he erms of he quadraic approximaion (3.9) can be calculaed easily, C(0) = S 0 (Φ( 1 σbs T) Φ( 1 σbs T)) B(0) = Φ( 1 σbs T), 1 1 ψ(0) = σ BS exp( 1 T S 0 8 (σbs )T).

BACHELIER VERSUS BLACK-SCHOLES 11 Noice ha he densiy ψ of S T S 0 in he Black-Meron-Scholes model is no an enire funcion, hence he Taylor expansion only converges wih a finie radius of convergence. Alhough Bachelier had achieved wih formula (3.7) a pracically saisfacory soluion, which allowed o calculae (approximaely) m as a funcion of C by only using pre-compuer echnology, he was no enirely saisfied. Following he reflexes of a rue mahemaician he ried o obain beer approximaions (yielding sill easily compuable quaniies) han simply runcaing he Taylor series afer he quadraic erm. He observed ha, using he series expansion for he pu opion (see formula 3.5) P(m) = a + m + m 4πa m4 96π a 3 +... and compuing he produc funcion C(m)P(m) or, somewha more sophisicaedly, he riple produc funcion C(m)P(m) C(m)+P(m), one obains ineresing cancellaions in he corresponding Taylor series, C(m)P(m) = a m 4 + m + O(m4 ), C(m) + P(m) C(m)P(m) = a 3 m a + 3m a 4 4π + O(m4 ). Observe ha (C(m)P(m)) 1 is he geomeric mean of he corresponding call and pu price, while (C(m)P(m) C(m)+P(m) ) 1 3 is he geomeric mean of he call, he pu and he arihmeic mean of he call and pu price. The laer equaion yields he approximae ideniy (C(m) + P(m))C(m)P(m) a 3 which Bachelier rephrases as a cooking book recipe (see [3], p.01): On addiionne l imporance de la prime e son écar. On muliplie l imporance de la prime par son écar. On fai le produi des deux résulas. Ce produi doi êre le même pour oues les primes qui on même échéance, i.e. One adds up he call and pu price, one muliplies he call and he pu price, one muliplies he wo resuls. The produc has o be he same for all premia, which correspond o opions of he same mauriy a. This recipe allows o approximaely calculae for m m in a quadruple (C(m), P(m), C(m ), P(m )) any one of hese four quaniies as an easy (from he poin of view of pre-compuer echnology) funcion of he oher hree. Noe ha, in he case m = 0, we have C(0) = P(0) = a, which makes he resuling calculaion even easier. We now inerpre hese equaions in a more conemporary language (bu, of course, only rephrasing Bachelier s insigh in his way). Proposiion 5 (Rule of Thumb ). For given T > 0, σ B > 0 and m = K S 0 denoe by C(m) and P(m) he prices of he corresponding call and pu opions in

1 WALTER SCHACHERMAYER AND JOSEF TEICHMANN he Bachelier model. Denoe by a(m) := C(m)P(m) b(m) = C(m)P(m)( C(m) + P(m) ) he producs considered by Bachelier, hen we have a(0) = a and b(0) = a 3, and a(m) a(0) = 1 (π )( m a ) + O(m 4 ), 4π b(m) b(0) = 1 (π 3)( m a ) + O(m 4 ). 4π Remark 6. We rediscover an approximaion of he reciprociy relaion of Proposiion 3.1 in he firs of he wo rules of he humb. Noice also ha his rule of humb only holds up o order ( m a ). Finally noe ha π 3 0.1416 while π 1.1416, so ha he coefficien of he quadraic erm in he above expressions is smaller for b(m) a(m) b(0) by a facor of 8 as compared o a(0). This is why Bachelier recommended his slighly more sophisicaed produc. 4. Bachelier versus oher models Bachelier s model yields very good approximaion resuls wih respec o he Black-Scholes-model for a he money opions. This corresponds o noions of weak (absolue or relaive) errors of approximaion in shor ime asympoics, i.e. esimaes of he quaniiy E(f(S B T )) E(f(S BS T )) for T 0. Compare [6] for he noion of weak and srond errors of approximaion in he realm of numerical mehods for SDE. This can be very delicae as seen in Proposiion and he remarks hereafer. However, in his secion we concenrae on he easier noion of L -srong errors, which means esimaes on he L -disance beween models a given poins in ime T. We ask wo quesions: firs, how o generalize Bachelier s model in order o obain beer approximaions of he Black-Scholes-model and second, how o exend his approach beyond he Black-Scholes model. Boh quesions can be answered by he mehods from chaos expansion. There are possible exensions of he Bachelier model in several direcions, bu exensions, which improve in an opimal way he L -disance and he shor-ime asympoics (wih respec o a given model), are favorable from he poin of view of applicaions. This observaion in mind we aim for bes (in he sense of L -disance) approximaions of a given general process (S ) 0 T by ieraed Wiener-Io inegrals up o a cerain order. We demand ha he approximaing processes are maringales o mainain no arbirage properies. The mehodology resuls ino he one of chaos expansion or Sroock-Taylor Theorems (see [8]). Mehods from chaos expansion for he (explici) consrucion of price processes have already proved o be very useful, see for [7]. In he sequel we shall work on a probabiliy space (Ω, F, P) carrying a one-dimensional Brownian moion (W ) 0 T wih is naural filraion (F ) 0 T. For all necessary deails on Gaussian spaces, chaos expansion, n-h Wiener chaos, ec, see [7]. Cerainly, he following consideraions easily generalize o muli-dimensional Brownian moions. We shall call any Gaussian maringale in his seing a (general) Bachelier model.

BACHELIER VERSUS BLACK-SCHOLES 13 Definiion 1. Fix N 1. Denoe by (M (n) ) 0 T maringales wih coninuous rajecories for 0 n N, such ha M (n) H n for 0 T, where H n denoes he n-h Wiener chaos in L (Ω). Then we call he process S (N) := N n=0 M (n) an exension of degree N of he Bachelier model. Noe ha M (0) consan, and ha S (1) = M (0) + M (1) is a (general) Bachelier model. = M (0) is Given an L -maringale (S ) 0 T wih S 0 > 0 and N 1. There exiss a unique exension of degree N of he Bachelier model (in he sense ha he maringales (M (n) ) 0 T are uniquely defined for 0 n N) minimizing he L -norm E[(S S (N) ) ] for all 0 T. Furhermore S (N) S in he L -norm as N, uniformly for 0 T. Indeed, since he orhogonal projecions p n : L (Ω, F T, P) L (Ω, F T, P) ono he n-h Wiener chaos commue wih condiional expecaions E(. F ) (see for insance [8]), we obain ha M (n) := p n (S ), for 0 T, is a maringale wih coninuous rajecories, because for 0 T. Consequenly E(p n (S T ) F ) = p n (E(S T F ) = p n (S ) S (N) := N n=0 M (n) is a process minimizing he disance o S for any [0, T]. Clearly we have ha S := in he L -opology. n=0 M (n) Example. For he Black-Meron-Scholes model wih σ BS = σ we obain ha M (n) = S 0 σ n n Hn ( W ), where H n denoes he n-h Hermie polynomial, i.e. (n + 1)H n+1 (x) = xh n (x) H n 1 (x) and H 0 (x) = 1, H 1 (x) = x for n 1 (for deails see [9]). Hence M (0) = S 0, M (1) = S 0 σw, M () = S 0 σ (W ). We recover Bachelier s model as exension of degree 1 minimizing he disance o he Black-Meron-Scholes model. Noe ha we have S S (N) C N S 0 σ N+1 N+1,

14 WALTER SCHACHERMAYER AND JOSEF TEICHMANN for 0 T. Furhermore we can calculae a sharp consan C N, namely CN = sup (σ ) (m N 1) E[(H m ( W )) ] 0 T m N+1 = (σ ) (m N 1) 1 m!. m N+1 since E[(H m ( W )) ] = 1 m! for 0 < T and m 0. Due o he paricular srucure of he chaos decomposiion we can prove he desired shor-ime asympoics: Theorem. Given an L -maringale (S ) 0 T, assume ha S T = i=0 W i(f i ) wih symmeric L -funcions f i : [0, T] i R and ieraed Wiener-Io inegrals W i (f i) := f i ( 1,..., i )dw 1 dw i. 0 1 i If here is an index i 0 such ha for i i 0 he funcions f i are bounded and K := (T) i i i i 0 i! f i <, hen we obain S S (N) K n+1 for each N i 0 and 0 T. Proof. We apply ha E[(E(W i T (f i) F )) ] = 1 i! 1 i [0,] f i L ([0,T] i ) i i! f i for i i 0. Hence he resul by applying S T S (N) T i i 0 E[(E(W i T (f i ) F T )) ] KT N+1. Remark 7. Noice ha he Sroock-Taylor Theorem (see [8], p.161) ells ha for S T D, he series WT(( i 1,..., i ) E(D 1,..., i S T )) = S T i=0 converges in D,. Hence he above condiion is a saemen abou boundedness of higher Malliavin derivaives. The well-known case of he Black-Meron-Scholes model yields = 1 i [0,T] D 1,..., i ST BS for i 0, so he condiion of he previous heorem is saisfied. References [1] Ananhanarayanan, A.L. and Boyle, Ph., The Impac of Variance Esimaion on Opion Valuaion Models, Journal of Financial Economics 6, 4, 375-387, 1977. [] Bachelier, L., Theorie de la Speculaion, Paris, 1900, see also: hp://www.numdam.org/en/ [3] Bachelier, L., Calcul de Probabiliés, Paris, 191, Les Grands Classiques Gauhier-Villars, Ediions Jacques Gabay 199. [4] Black, F., The Pricing of Commodiy Conracs, Journal of Financial Economics 3, 167 179 (1976). [5] Delbaen, F. and Schachermayer, W., The mahemaics of Arbirage, Springer Finance (005). [6] P. E. Kloeden and E. Plaen, Numerical soluion of sochasic differenial equaions, Applicaions of Mahemaics, 3, Springer-Verlag, Berlin (199). [7] Malliavin, P. and Thalmaier, A., Sochasic Calculus of Variaions in Mahemaical Finance, Springer Finance, Springer Verlag (005).

BACHELIER VERSUS BLACK-SCHOLES 15 [8] Malliavin, P., Sochasic Analysis, Grundlehren der mahemaischen Wissenschafen 313, Springer Verlag (1995). [9] Nualar, D., The Malliavin Calculus and Relaed Topics, Springer-Verlag (1995). [10] Samuelson, P. A., Proof ha Properly Anicipaed Prices Flucuae Randomly, Indusrial Managemen Review 6, 41 49 (1965). [11] Schachermayer, W., Inroducion o Mahemaics of financial markes, Lecure Noes in Mahemaics 1816 - Lecures on Probabiliy Theory and Saisics, Sain-Flour summer school 000 (Pierre Bernard, edior), Springer Verlag, Heidelberg, pp. 111-177. 5. Appendix We provide ables wih Bachelier and Black-Scholes implied volailiies for differen imes o mauriy, where we have used Nasdaq-index-opion daa in discouned prices in order o compare he resuls. Circles represen he Black-Scholes implied volailiy of daa poins above a cerain level of rade volume. The doed line represens he implied Bachelier (relaive) volailiy σ B = S 0 σ rel. Black Scholes and Bachelier (doed) Volailiy for an index opion mauring in 10 days IV 0.6 0.8 0.30 0.3 0.34 0.36 33 34 35 36 37 38 39 40 Srike

16 WALTER SCHACHERMAYER AND JOSEF TEICHMANN Black Scholes and Bachelier (doed) Volailiy for an index opion mauring in 73 days IV 0.18 0.0 0. 0.4 0.6 33 34 35 36 37 38 39 40 Srike Black Scholes and Bachelier (doed) Volailiy for an index opion mauring in 19 days IV 0.0 0.5 0.30 0.35 33 34 35 36 37 38 39 40 Srike

BACHELIER VERSUS BLACK-SCHOLES 17 Insiue of Mahemaical Mehods in Economics, Vienna Universiy of Technology, Wiedner Haupsrasse 8-10, A-1040 Vienna, Ausria E-mail address: wschach@fam.uwien.ac.a, jeichma@fam.uwien.ac.a