HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACKMERTONSCHOLES?


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1 HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACKMERTONSCHOLES? WALTER SCHACHERMAYER AND JOSEF TEICHMANN Abstract. We copare the option pricing forulas of Louis Bachelier and BlackMertonScholes and observe theoretically and by typical data that the prices coincide very well. We illustrate Louis Bachelier s efforts to obtain applicable forulas for option pricing in precoputer tie. Furtherore we explain by siple ethods fro chaos expansion why Bachelier s odel yields good shorttie approxiations of prices and volatilities. 1. Introduction It is the pride of Matheatical Finance that L. Bachelier was the first to analyze Brownian otion atheatically, and that he did so in order to develop a theory of option pricing (see [2]). In the present note we shall review soe of the results fro his thesis as well as fro his later textbook on probability theory (see [3]), and we shall explain the rearkable closeness of prices in the Bachelier and Black MertonScholes odel. The fundaental principle underlying Bachelier s approach to option pricing is crystallized in his faous dictu (see [2], p.34): L ésperance atheatique du spéculateur est nul, i.e. the atheatical expectation of a speculator is zero. His arguent in favor of this principle is based on equilibriu considerations (see [2] and [1]), siilar to what in today s terinology is called the efficient arket hypothesis (see [9]), i.e. the use of artingales to describe stochastic tie evolutions of price oveents in ideal arkets (see [2], p. 31): It sees that the arket, the aggregate of speculators, can believe in neither a arket rise nor a arket fall, since, for each quoted price, there are as any buyers as sellers. The reader failiar with today s approach to option pricing ight wonder where the concept of risk free interest rate has disappeared to, which sees crucial in the odern approach of pricing by no arbitrage arguents (recall that the discounted price process should be a artingale under the risk neutral easure). The answer is that L. Bachelier applied his fundaental principle in ters of true prices (this is terinology fro 19 which corresponds to the concept of forward prices in odern terinology). Is is wellknown that the passage to forward prices akes the riskless interest rate disappear: in the context of the BlackMertonScholes forula, this is what aounts to the socalled Black s forula (see [4]). Suing up: Bachelier s fundaental principle yields exactly the sae recipe for option pricing as we use today (for ore details we refer to the first section of 1
2 2 WALTER SCHACHERMAYER AND JOSEF TEICHMANN the St. Flour suer school lecture [1]): using discounted ters ( true prices ) one obtains the prices of options (or of ore general derivatives of European style) by taking expectations. The expectation pertains to a probability easure under which the price process of the underlying security (in discounted ters) satisfies the fundaental principle, i.e. is a artingale in odern terinology. It is iportant to ephasize that, although the recipes for obtaining option prices are the sae for Bachelier s as for the odern approach, the arguents in favour of the are very different: an equilibriu arguent in Bachelier s case as opposed to the no arbitrage arguents in the BlackMertonScholes approach. With all adiration for Bachelier s work, the developent of a theory of hedging and replication by dynaic strategies, which is the crucial ingredient of the Black MertonScholesapproach, was far out of his reach (copare [1]). In order to obtain option prices one has to specify the underlying odel. We fix a tie horizon T >. As is wellknown, Bachelier proposed to use (properly scaled) Brownian otion as a odel for stock prices. In odern terinology this aounts (using true prices ) to (1.1) S B t := S (1 + σw t ), for t T, where (W t ) t T denotes standard Brownian otion and the superscript B stands for Bachelier. The paraeter σ > denotes the volatility in odern σ terinology. In fact, Bachelier used the noralization H = S and called this quantity the coefficient of instability or of nervousness of the security S. The BlackMertonScholes odel (under the riskneutral easure) for the discounted price process is, of course, given by (1.2) S BS t = S exp(σw t σ2 2 t), for t T. This odel was proposed by P. Sauelson in 1965, after he had led by an inquiry of J. Savage for the treatise [3] personally rediscovered the virtually forgotten Bachelier thesis in the library of Harvard University. The difference between the two odels is analogous to the difference between linear and copound interest, as becoes apparent when looking at the associated differential equations ds B t = S B σdw t, ds BS t = S BS t σdw t. This analogy akes us expect that, in the short run, both odels should yield siilar results while, in the long run, the difference should be spectacular. Fortunately, options usually have a relatively short tie to aturity (the options considered by Bachelier had a tie to expiration of less than 2 onths), while in the long run we all are dead (to quote J.M. Keynes). 2. Bachelier versus BlackMertonScholes We now have assebled all the ingredients to recall the derivation of the price of an option in Bachelier s fraework. Fix a strike price K (of course, in true, i.e. discounted ters), a horizon T and consider the European call C, whose payoff at tie T is odeled by the rando variable C B T = (S B T K) +.
3 BACHELIER VERSUS BLACKSCHOLES 3 Applying Bachelier s fundaental principle and using that ST B is norally distributed with ean S and variance Sσ 2 2 T, we obtain for the price of the option at tie t = (2.1a) (2.1b) C B = E[(S B T K) + ] = 1 (S + x K) K S S σ T exp( x2 2σ 2 S 2T )dx = (S K)Φ( S K S σ T ) + S σ T φ( S K S σ T ), where φ(x) = 1 exp( x2 2 ) denotes the density of the standard noral distribution. We applied the relation φ (x) = xφ(x) to pass fro (2.1a) to (2.1b). For further use we shall need the BlackMertonScholes price, too, (2.2a) (2.2b) (2.2c) C BS = E[(S BS K) + ] = = (S exp( σ2 T 2 + σ 1 T x) K) + exp( x2 2 )dx log K S + σ2 T 2 σ T (S exp( σ2 T 2 + σ 1 T x) K) exp( x2 2 )dx S log K = S Φ( σ2 S T σ log K ) KΦ( 1 2 σ2 T T σ ). T Interestingly, Bachelier explicitly wrote down forula (2.1a), but did not bother to spell out forula (2.1b). The ain reason sees to be that at his tie option prices at least in Paris were quoted the other way around: while today the strike prices K is fixed and the option price fluctuates according to supply and deand, at Bachelier s ties the option prices were fixed (at 1, 2 and 5 Centies for a rente, i.e., a perpetual bond with par value of 1 Francs) and therefore the strike price K fluctuates. Hence what Bachelier really needed was the inverse version of the above relation between C = C B and K. Apparently there is no siple forula to express this inverse relationship. This is soewhat analogous to the situation in the BlackMertonScholes odel, where there is also no forula for the inverse proble of calculating the iplied volatility as a function of the given option price. We shall see below that L. Bachelier designed a clever series expansion for C B as a function of the strike price K in order to derive (very) easy forulae which approxiate this inverse relation and which were well suited to precoputer technology At the oney options. Bachelier first specializes to this case (called siple options in the terinology of 19), when S = K. In this case (2.1b) reduces to the siple and beautiful relation T C B = S σ. As explicitly noticed by Bachelier, this forula can also be used, for a given price C of the at the oney option with aturity T, to deterine the coefficient of nervousness of the security H = Sσ, i.e., to deterine the iplied volatility in odern technology. Indeed, it suffices to noralize the price C B by T to obtain
4 4 WALTER SCHACHERMAYER AND JOSEF TEICHMANN H = CB T. This leads us to our first result. For convenience we phrase it rather in ters of σ than of H. Proposition 1. The volatility σ in the Bachelier odel is deterined by the price C B of an at the oney option with aturity T by (2.3) σ = CB S T. In the subsequent Proposition, we copare, for fixed volatility σ > and tie to aturity T, the price of an at the oney call option as obtained fro the Black MertonScholes and Bachelier s forula respectively. Furtherore we also copare the iplied volatilities, for given price C of an at the oney call, in the Bachelier and BlackMertonScholes odel. Proposition 2. Fix σ >, T > and S = K, and denote by C B and C BS the corresponding prices for a European call option in the Bachelier (1.1) and Black MertonScholes odel (1.2) respectively. Then (2.4) C B C BS S 12 σ3 T 3 2 = O((σ T ) 3 ). Conversely, fix the price < C < S of an at the oney option and denote by σ B := σ the iplied Bachelier volatility and by σ BS the iplied BlackMerton Scholes volatility, then (2.5) σ BS σ B T 12 (σbs ) 3. Proof. (copare [1]). For S = K, we obtain in the Bachelier and BlackMerton Scholes odel the following prices, respectively, Hence C B = S σ T C BS = S (Φ( 1 2 σ T ) Φ( 1 2 σ T )). C B C BS = ( S x S (Φ( x 2 ) Φ( x 2 ))) x=σ T S x 2 x 2 y 2 2 dy x=σ T = = S x 3 12 x=σ T = S 12 σ3 T 3 2 = O((σ T ) 3 ), since e y 1 + y for all y, so that y2 2 1 e y 2 2 for all y. For the second assertion note that solving equation C = σb T = Φ( 1 2 σbs T ) Φ( 1 2 σbs T )
5 BACHELIER VERSUS BLACKSCHOLES 5 given σ B > yields the BlackMertonScholes iplied volatility σ BS. We obtain siilarly as above σ BS σ B = σ BS (Φ( 1 T 2 σbs T ) + Φ( 1 2 σbs T )) 1 = ( x (Φ( x T 2 ) Φ( x 2 ))) x=σ BS T 1 T 12 (σbs ) 3 T 3 (σ BS ) 3 T 2 =. 12 Proposition 1 and 2 yield in particular the wellknown asyptotic behaviour of an at the oney call price in the BlackMertonScholes odel as described in [1]. Proposition 2 tells us that for the case when (σ T ) 1 (which typically holds true in applications), forula (2.3) gives a satisfactory approxiation of the iplied BlackMertonScholes volatility, and is very easy to calculate. We note that for the data reported by Bachelier (see [2] and [1]), the yearly volatility was of the order of 2.4% p.a. and T in the order of one onth, i.e T = 1 12 years so that T.3. Consequently we get (σ T ) 3 (.8) The estiate in Proposition 2 S yields a right hand side of = S, i.e. the difference of the Bachelier and BlackMertonScholes price (when using the sae volatility σ = 2.4% p.a.) is of the order 1 8 of the price S of the underlying security. Reark 1. Inequality (2.4) is an estiate of third order, whereas inequality (2.5) only yields an estiate of the relative error of second order σ BS σ B σ BS 1 12 (σbs T ) 2. On the other hand, for the tiestandardized volatilities at aturity T we obtain an estiate of third order σ BS T σ B T 1 12 (σbs T ) Further results of L. Bachelier We now proceed to a ore detailed analysis of the option pricing forula (2.1b) for general strike prices K. We shall introduce soe notation used by L. Bachelier for the following two reasons: firstly, it should ake the task easier for the interested reader to look up the original texts by Bachelier; secondly, and ore iportantly, we shall see that his notation has its own erits and allows for intuitive and econoically eaningful interpretations (as we have already seen for the noralization H = Sσ of the volatility, which equals the tiestandardized price of an at the oney option). L. Bachelier found it convenient to use a parallel shift of the coordinate syste oving S to, so that the Gaussian distribution will be centered at. We write (3.1) a = S σ T, := K S, P := + C. The paraeter a equals, up to the noralizing factor S, the tiestandardized volatility σ T at aturity T. Readers failiar, e.g. with the HullWhite odel of
6 6 WALTER SCHACHERMAYER AND JOSEF TEICHMANN stochastic volatility, will realize that this is a very natural paraetrization for an option with aturity T. In any case, the quantity a was a natural paraetrization for L. Bachelier, as it is the price of the at the oney option with the aturity T (see forula 2.3), so that it can be observed fro arket data. The quantity is the strike price K shifted by S and needs no further explanation. P has a natural interpretation (in Bachelier s ties it was called écart, i.e. the spread of an option): it is the price P of a european put at aturity T with strike price K, as was explicitly noted by Bachelier (using, of course, different terinology): as nicely explained in [2], a speculator á la hausse, i.e. hoping for a rise of S T ay buy a forward contract with aturity T. Using the fundaental principle, which in this case boils down to eleentary no arbitrage arguents, one concludes that the forward price ust equal S, so that the total gain or loss of this operation is given by the rando variable S T S at tie T. On the other hand, a ore prudent speculator ight want to liit the axial loss by a quantity K >. She thus would buy a call option with price C, which would correspond to a strike price K (here we see very nicely the above entioned fact that in Bachelier s ties the strike price was considered as a function of the option price C la prie in french and not vice versa as today). Her total gain or loss would then be given by the rando variable (S T K) + C. If at tie T it indeed turns out that S T K, then the buyer of the forward contract is, of course, better off than the option buyer. The difference equals (S T S ) [(S T K) C] = K S + C = P, which therefore ay be interpreted as a cost of insurance. If S T K, we obtain (S T S ) [ C] = (S T K) + K S + C = (S T K) + P. By the Bachelier s fundaental principle we obtain P = E[(S T K) ]. Hence Bachelier was led by noarbitrage considerations to the putcall parity. For further considerations we denote the put price in the Bachelier odel at tie t = by P B or P () respectively. Clearly, the higher the potential loss C is, which the option buyer is ready to accept, the lower the costs of insurance P should be and vice versa, so that we expect a onotone dependence of these two quantities. In fact, Bachelier observed that the following pretty result holds true in his odel (see [3], p.295): Proposition 3 (Theore of reciprocity). For fixed σ > and T > the quantities C and P are reciprocal in Bachelier s odel, i.e. there is onotone strictly decreasing and selfinverse, i.e. I = I 1 function I : R > R > such that P = I(C). Proof. Denote by ψ the density of S T S, then C() = P () = (x )ψ(x)dx, ( x)ψ(x)dx.
7 BACHELIER VERSUS BLACKSCHOLES 7 Hence we obtain that C( ) = P (). We note in passing that this is only due to the syetry of the density ψ with respect to reflection at. Since C () < (see the proof of Proposition 1) we obtain P = P ((C)) := I(C), where C (C) inverts the function C(). C aps R in a strictly decreasing way to R > and P aps R in a strictly increasing way to R >. The resulting ap I is therefore strictly decreasing, and due to syetry we obtain I(P ) = P ((P )) = P ( (C)) = C, so I is selfinverse. Using the above notations (3.1), equation (2.1b) for the option price C B (which we now write as C() to stress the dependence on the strike price) obtained fro the fundaental principles becoes (3.2) C() = (x )µ(dx), where µ denotes the distribution of S T S, which has the Gaussian density µ(dx) = ψ(x)dx with 1 ψ(x) = S σ T exp( x2 2σ 2 S 2T ) = 1 a exp( x2 4πa 2 ). As entioned above, Bachelier does not siply calculate the integral (3.2). He rather does soething ore interesting (see [3], Nr.445, p.294): Si l on développe l integrale en série, on obtient (3.3) C() = a πa 4 96π 2 a a In the subsequent theore we justify this step. It is worth noting that the ethod for developing this series expansion is not restricted to Bachelier s odel, but holds true in general (provided that µ, the probability distribution of S T S, adits a density function ψ, which is analytic in a neighborhood of ). Theore 1. Suppose that the law µ of the rando variable S T adits a density µ(dx) = ψ(x)dx, such that ψ is analytic in a ball of radius r > around, and that Then the function C() = xψ(x)dx <. (x )µ(dx) is analytic for < r and adits a power series expansion C() = c k k, where c = xψ(x)dx, c 1 = ψ(x)dx and c k = 1 k! ψ(k 2) () for k 2. k=
8 8 WALTER SCHACHERMAYER AND JOSEF TEICHMANN Proof. Due to our assuptions C is seen to be analytic as su of two analytic functions, C() = xψ(x)dx ψ(x)dx. Indeed, if ψ is analytic around, then the functions x xψ(x) and xψ(x)dx are analytic with the sae radius of convergence r. The sae holds true for the function ψ(x)dx. The derivatives can be calculated by the Leibniz rule, C () = ψ() = C () = ψ(), ψ(x)dx, whence we obtain for the kth derivative, C (k) () = ψ (k 2) (), ψ(x)dx + ψ() for k 2. Reark 2. If we assue that C() is locally analytic around = (without any assuption on the density ψ), then the density x ψ(x) is analytic around x =, too, by inversion of the above arguent. Reark 3. Arguing forally, the forulae for the coefficients c k becoe rather obvious on an intuitive level: denote by H (x) = (x ) + the hockey stick function with kink at. Note the syetry up to the sign in the variables and x. In particular H k (x) = ( 1) k H (k) (x), where the derivatives have to be interpreted in the distributional sense. We write H (k) for H x k (x) and observe that H (x) = 1 {x }, which is the Heaviside function centered at, and H (x) = δ (x), the Dirac δfunction centered at. Applying this foral arguent again, we obtain fro 2 ψ, H 2 = ψ, H (2) = ψ(), and consequently, k k ψ, H = ψ (k 2) () for any Schwartz test function ψ. Hence (under the assuption that the Taylor series akes sense) with ψ, H () C() = = = c and k= 1 k= ψ, H (1) k k = ψ, H k k! c k k k! + = c 1. k=2 ψ (k 2) () k k!,
9 BACHELIER VERSUS BLACKSCHOLES 9 Reark 4. In the case when ψ equals the Gaussian distribution, the calculation of the Taylor coefficients yields d dy ( 1 y2 exp( a 4πa 2 )) = 1 y 1 y 2 4 π 2 a 3 e 4 πa 2, d 2 dy 2 ( 1 y2 exp( a 4πa 2 )) = 1 a 2 y 2 8 π 3 a 5 e 1 y 2 4 πa 2, d 3 dy 3 ( 1 y2 exp( a 4πa 2 )) = 1 6πa 2 y y 3 16 π 4 a 7 e 1 y 2 4 πa 2, d 4 dy 4 ( 1 y2 exp( a 4πa 2 )) = a 4 12y 2 πa 2 + y 4 32 π 5 a 9 e 1 4 y 2 πa 2, Consequently ψ() = 1 a, ψ () =, ψ () = 1 4π 2 a, ψ () = and ψ () = π 3 a, hence with C() = a and C () = 1 5 2, (3.4) C() = a πa 4 96π 2 a a 5 + O(8 ) and the series converges for all, as the Gaussian distribution is an entire function. This is the expansion indicated by Bachelier in [3]. Since P ( ) = C(), we also obtain the expansion for the put (3.5) P () = a πa 4 96π 2 a a 5 + O(8 ). Reark 5. Looking once ore at Bachelier s series one notes that it is rather a Taylor expansion in a than in. Note furtherore that a is a diensionless quantity. The series then becoes C() = a F ( a ) F (x) = 1 x 2 + x2 4π x4 96π 2 + x O(x8 ). We note as a curiosity that already in the second order ter the nuber π appears. Whence if we believe in Bachelier s forula for option pricing we are able to deterine π at least approxiately (see (3.6) below) fro financial arket data (copare Georges Louis Leclerc Cote de Buffon s ethod to deterine π by using statistical experients). Reark 6. Denoting by C(, S ) the price of the call in Bachelier s odel as a function of = K S and S, observe that there is another reciprocity relation naely C( + b, S ) = C(, S b) for all b R. This is due to the linear dependence of S T on S. Hence we obtain k k C(, S ) = ( 1) k k S k C(, S ). Note that the derivatives on the right hand side are Greeks of the option in today s terinology. Hence in the Bachelier odel as in any other Markovian odel with linear dependence on the initial value S the derivatives with respect to the strike price correspond to the calculation of certain Greeks. Letting S = K, we find that in odern terinology in Bachelier s odel for at the oney European call options the delta equals and the gaa equals. At this point, however,
10 1 WALTER SCHACHERMAYER AND JOSEF TEICHMANN we have to be careful not to overdo our interpretation of Bachelier s findings: the idea of Greeks and its relation to dynaic replication can to the best of our knowledge not be found in Bachelier s work. Let us turn back to Bachelier s original calculations. Taylor series (3.4) after the quadratic ter, i.e. He first truncated the (3.6) C() a πa. This (approxiate) forula can easily be inverted by solving a quadratic equation, thus yielding an explicit forula for as a function of C. Bachelier observes that the approxiation works well for sall values of a (the cases relevant for his practical applications) and gives soe nuerical estiates. We suarize the situation. Proposition 4 (Rule of Thub 1). For a given aturity T denote by a = C() the price of the at the oney option and by C() the price of the call option with strike K = S +. Letting (3.7) (3.8) Ĉ() = a πa = C() πC(), we get an approxiation of the Bachelier price C() up to order 3. Reark 7. Note that the value of Ĉ() only depends on the price a of an at the oney option (which is observable at the arket) and the quantity = K S. Reark 8. Given any stock price odel under a risk neutral easure, the above approach of quadratic approxiation can be applied if the density ψ of S T S is locally analytic and adits first oents. The approxiation then reads Ĉ() = C() B() + ψ() 2 2 up to a ter of order O( 3 ). Here C() denotes the price of the at the oney european call option (payoff (S T S ) + ), B() the price of the at the oney binary option (payoff 1 {ST S }) and ψ() the value of an at the oney Dirac option (payoff δ S, with an appropriate interpretation as a liit). Exaple 1. Take for instance the BlackMertonScholes odel, then the quadratic approxiation can be calculated easily, C() = S (Φ( 1 2 σ T ) Φ( 1 2 σ T )) B() = Φ( 1 2 σ T ), 1 ψ() = σ 1 exp( 1 T S 8 σ2 T ). Notice that the density ψ of S T S in the BlackMertonScholes odel is not an entire function, hence the expansion only holds with a finite radius of convergence.
11 BACHELIER VERSUS BLACKSCHOLES 11 Although Bachelier had achieved with forula (3.7) a practically satisfactory solution, which allowed to calculate (approxiately) as a function of C by only using precoputer technology, he was not entirely satisfied. Following the reflexes of a true atheatician he tried to obtain better approxiations (yielding still easily coputable quantities) than siply truncating the Taylor series after the quadratic ter. He observed that, using the series expansion for the put option (see forula 3.5) P () = a πa 4 96π 2 a and coputing the product function C()P () or, soewhat ore sophisticatedly, C()+P () the triple product function C()P () 2, one obtains interesting cancellations in the corresponding Taylor series, C()P () = a O(4 ), C() + P () C()P () = a 3 2 a + 32 a 2 4 4π + O(4 ). Observe that (C()P ()) 1 2 is the geoetric ean of the corresponding call and C()+P () put price, while (C()P () 2 ) 1 3 is the geoetric ean of the call, the put and the arithetic ean of the call and put price. The latter equation yields the approxiate identity (C() + P ())C()P () 2a 3 which Bachelier rephrases as a cooking book recipe (see [3], p.21): On additionne l iportance de la prie et son écart. On ultiplie l iportance de la prie par son écart. On fait le produit des deux résultats. Ce produit doit être le êe pour toutes les pries qui ont êe échéance. This recipe allows to approxiately calculate for in a quadruple (C(), P (), C( ), P ( )) any one of these four quantities as an easy (fro the point of view of precoputer technology) function of the other three. Note that, in the case =, we have C() = P () = a, which akes the resulting calculation even easier. We now interpret these equations in a ore conteporary language (but, of course, only rephrasing Bachelier s insight in this way). Proposition 5 (Rule of Thub 2). For given T >, σ B > and = K S denote by C() and P () the prices of the corresponding call and put options in the Bachelier odel. Denote by a() := C()P () b() = C()P ()( C() + P () ) 2
12 12 WALTER SCHACHERMAYER AND JOSEF TEICHMANN the products considered by Bachelier, then we have a() = a 2 and b() = a 3, and a() a() = 1 (π 2)( a )2 + O( 4 ), 4π b() b() = 1 (π 3)( a )2 + O( 4 ). 4π Reark 9. We rediscover an approxiation of the reciprocity relation of Proposition 3.1 in the first of the two rules of the thub. Notice also that this rule of thub only holds up to order ( a )2. Finally note that π while π , so that the coefficient of the quadratic ter in the above expressions is saller for b() a() b() by a factor of 8 as copared to a(). This is why Bachelier recoended this slightly ore sophisticated product. 4. Bachelier versus other odels We now ai to explain the rearkable asyptotic closeness encountered in Proposition 2 of the BlackMertonScholes and the Bachelier price for at the oney options in a ore systeatic way. We therefore consider a general odel (S t ) t T in true (i.e. discounted) prices (under a artingale easure) and provide conditions for asyptotic closeness to the Bachelier odel (St B ) t T, and systeatic generalizations of these considerations. We shall show that under ild technical conditions  we can always find a Bachelier odel (St B ) t T, i.e. a paraeter σ >, such that the shorttie asyptotics of the difference of the European style option prices in the general odel and in the Bachelier odel is of order O(S (σ t) 2 ) as (σ t). Since (σ t) is typically sall, this yields rearkably good approxiations for prices. We assue that (S t ) t T is odeled on a filtered probability space (Ω, (F t ) t T, P ) such that (F t ) t T is generated by a standard Brownian otion (W t ) t T. We assue that (S t ) t T is of the for S t := S + t α(s)dw s, where (α t ) t T is a predictable L 2 integrator, so that (S t ) t T is an L 2 artingale with continuous trajectories. First we analyse a condition, such that Bachelier s odel yields a shorttie asyptotics of order O(S (σ t) 2 ). Proposition 6. Assue that for soe σ > and fixed tie horizon T > we have E[ t (α(s) σs ) 2 ds] 1 2 CS σ 2 t for t [, T ] and soe constant C > (which ay depend on the horizon T ). Then for any globally Lipschitz function g : R R we have E[g(St B )] E[g(S t )] C g Lip S σ 2 t, for t T, where (S B t ) t T denotes the Bachelier odel with constants σ > and S B = S.
13 BACHELIER VERSUS BLACKSCHOLES 13 Proof. Fro the basic L 2 isoetry of stochastic integration we obtain E[g(S B t )] E[g(S t )] g Lip E[ S B t S t ] for t T. g Lip E[(S B t S t ) 2 ] 1 2 = g Lip E[ t C g Lip S σ 2 t, (α(s) σs ) 2 ds] 1 2 Exaple 2. If we copare the Bachelier and the BlackMertonScholes odel, we observe that there is a constant C = C(T ) > such that E( t (σs exp(σw s σ2 s 2 ) σs ) 2 ds) C 2 S 2 σ 4 t 2, hence the estiate holds for t T. As in Exaple 5 below we obtain an explicit sharp estiate, naely C 2 = 2 T 2! = exp(t ) 1 T T 2. Exaple 3. Given the HobsonRogers odel (see [6]), i.e. S HR t = S + β(s) = η(y s ), t S η(y u ) exp( 1 2 u dy s = ( η(y s) 2 + λy s )ds + η(y s )dw s, 2 β(s) 2 ds + u β(s)dw s )dw u, with η : R R a C bounded function and η(y ) = σ. Hence we obtain a siilar estiate as in the BlackMertonScholes case (with a different constant C of course) in general, but we have to choose the Bachelier odel with appropriate volatility σ = η(y ). Now we focus on possibly higher orders in the shorttie asyptotics. We first replace Bachelier s original odel by a general Gaussian artingale odel and ask for the best Gaussian artingale odel to approxiate a given L 2 artingale. We then consider extensions of this Gaussian artingale odel into higher chaos coponents to obtain higher order tie asyptotics. We finally prove that, under a ild technical assuption on the general process (S t ) t T, we are able to provide shorttie asyptotics of any order. If we analyse the crucial condition E( t (α(s) σs ) 2 ds) 1 2 CS σ 2 t of Proposition 6 ore carefully, we realize that we can iprove this estiate if we allow for deterinistic, tiedependent volatility functions σ : [, T ] R instead of a constant σ >, i.e. if we iniize E( t (α(s) σ(s)s ) 2 ds) along all possible choices of deterinistic functions s σ(s). Fro the basic inertia identity E[(X E[X]) 2 ] = in a R E[(X a) 2 ], which holds true for any rando
14 14 WALTER SCHACHERMAYER AND JOSEF TEICHMANN variable X with finite second oent, we infer that the best possible choice is given by (4.1) σ(s) := E(α(s)) S. If we are already given an estiate E( t (α(s) σs ) 2 ds) 1 2 CS σ 2 t for soe constant σ >, then E( t (α(s) σ(s)s ) 2 ds) 1 2 C S σ 2 t, with C C, for σ(s) defined by (4.1). Instead of solving the onediensional proble of finding the best Bachelier odel with inial L 2 distance to (S t ) t T, we hence ask for the best Gaussian artingale odel with respect to the L 2 distance, which will iprove the constants in the shorttie asyptotics. Notice, however, that for the BlackMertonScholes odel with constant volatility σ, the choice σ(s) = σ for s T is the optial. Exaple 4. In the HobsonRogers odel (4.1) leads to σ(s) := E[η(Y s ) exp( 1 2 s β(v) 2 dv + s β(v)dw v )] for s T and hence the best (in the sense of L 2 distance) Gaussian approxiation of (St HR ) t T is given through (S (1 + t σ(s)dw s )) t T. There are possible extensions of the Bachelier odel in several directions, but extensions, which iprove in an optial way the L 2 distance and the shorttie asyptotics, are favorable fro the point of applications. This observation in ind we ai for best (in the sense of L 2 distance) approxiations of a given general process (S t ) t T by iterated WienerIto integrals up to a certain order. We deand that the approxiating processes are artingales. The ethodology results into the one of chaos expansion or StroockTaylor theores (see [8]). Methods fro chaos expansion for the (explicit) construction of price processes have proved to be very useful, see for instance [5]. In the sequel we shall call any Gaussian artingale a (general) Bachelier odel. Definition 1. Fix N 1. Denote by (M (n) t ) t T artingales with continuous trajectories for n N, such that M (n) t H n for t T, where H n denotes the nth Wiener chaos in L 2 (Ω). Then we call the process S (N) t := N n= M (n) t an extension of degree N of the Bachelier odel. Note that M () t and that S (1) t = M + M (1) t is a (general) Bachelier odel. = M is constant, Given an L 2 artingale (S t ) t T and N 1. There exists a unique extension of degree N of the Bachelier odel (in the sense that the artingales (M (n) t ) t T are uniquely defined for n N) iniizing the L 2 nor E[(S t S (N) t ) 2 ] for all t T. Furtherore S (N) t S t in the L 2 nor as N, uniforly for t T.
15 BACHELIER VERSUS BLACKSCHOLES 15 Indeed, since the orthogonal projections p n : L 2 (Ω, F T, P ) L 2 (Ω, F T, P ) onto the nth Wiener chaos coute with conditional expectations E(. F t ) (see for instance [8]), we obtain that M (n) t := p n (S t ), for t T, is a artingale with continuous trajectories, because for t T. Consequently E(p n (S T ) F t ) = p n (E(S T F t ) = p n (S t ) S (N) t := N n= M (n) t is a process iniizing the distance to S t for any t [, T ]. Clearly we have that S t := in the L 2 topology. n= M (n) t Exaple 5. For the BlackMertonScholes odel we obtain that M (n) t = S σ n t n 2 Hn ( W t t ), where H n denotes the nth Herite polynoial, i.e. (n + 1)H n+1 (x) = xh n (x) H n 1 (x) and H (x) = 1, H 1 (x) = x for n 1 (for details see [7]). Hence M () t = S, M (1) t = S σw t, M (2) t = S σ 2 2 (W 2 t t). We recover Bachelier s odel as extension of degree 1 iniizing the distance to the BlackMertonScholes odel. Note that we have S t S (N) t 2 CS σ N+1 2 t N+1 2, for t T. Furtherore we can calculate a sharp constant C, naely C 2 = t N 1 E[(H ( W t )) 2 ] t sup t T = N+1 N+1 T N 1 1!. since E[(H ( Wt t )) 2 ] = 1! for < t T and. Due to the particular structure of the chaos decoposition we can prove the desired shorttie asyptotics: Theore 2. Given an L 2 artingale (S t ) t T, assue that S T = i= W i(f i ) with syetric L 2 functions f i : [, T ] i R and iterated WienerIto integrals Wt i (f i ) := f i (t 1,..., t i )dw t1 dw ti t 1 t i t
16 16 WALTER SCHACHERMAYER AND JOSEF TEICHMANN. If there is an index i such that for i i the functions f i are bounded and K 2 := T i i i i i! f i 2 <, then we obtain S t S (N) t 2 Kt n+1 2 for each N i and t T. Proof. We apply that E[(E(WT i (f i) F t )) 2 ] = 1 i! 1 i [,t] f i 2 L 2 ([,T ] i ) i i. Hence the result by applying S t S (N) t 2 i i E[(E(W i T (f i ) F t )) 2 ] Kt N+1 2. ti i! f i 2 for Reark 1. Notice that the StroockTaylor Theore (see [8], p.161) tells that for S T D 2, the series WT i ((t 1,..., t i ) E(D t1,...,t i S T )) = S T i= converges in D 2,. Hence the above condition is a stateent about boundedness of higher Malliavin derivatives. The wellknown case of the BlackMertonScholes odel yields D t1,...,t i ST BS for i, so the condition of the previous theore is satisfied. = 1 i [,T ] 5. Appendix We provide tables with iplied BlackMertonScholes and Bachelier volatilities on the British Pound/ US $ foreign exchange arket. The data refer to February 11, 22. The spot price on this day was approxiately 1.41 US$. In the graphical illustrations we indicate by K the strike price in true prices. We easure tie to aturity in Business days (BD). The graphs show the close atch between iplied volatilities for at the oney options; this atch becoes less precise when the options tends out of or into the oney, and tie to aturity T increases, as we have seen in Section 2.
17 BACHELIER VERSUS BLACKSCHOLES 17 References [1] Ananthanarayanan, A.L. and Boyle, Ph., The Ipact of Variance Estiation on Option Valuation Models, Journal of Financial Econoics 6, 4, , [2] Bachelier, L., Theorie de la Speculation, Paris, 19. [3] Bachelier, L., Calcul de Probabilités, Paris, [4] Black, F., The Pricing of Coodity Contracts, Journal of Financial Econoics 3, (1976). [5] Flesaker, B. and Hughston, L. P., Positive Interest, Risk 9, No. 1 (1996). [6] Hobson D. and Rogers, C., Coplete Models with stochastic volatility, Matheatical Finance 8, (1998). [7] Nualart, D., The Malliavin Calculus and Related Topics, SpringerVerlag (1995). [8] Malliavin, P., Stochastic Analysis, Grundlehren der atheatischen Wissenschaften 313, Springer Verlag (1995).
18 18 WALTER SCHACHERMAYER AND JOSEF TEICHMANN [9] Sauelson, P. A., Proof that Properly Anticipated Prices Fluctuate Randoly, Industrial Manageent Review 6, (1965). [1] Schacherayer, W., Introduction to Matheatics of financial arkets, Lecture Notes in Matheatics Lectures on Probability Theory and Statistics, SaintFlour suer school 2 (Pierre Bernard, editor), Springer Verlag, Heidelberg, pp Institute of Matheatical Methods in Econoics, Vienna University of Technology, Wiedner Hauptstrasse 81, A14 Vienna, Austria Eail address:
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