64 4 Bachelier and Black-Scholes
Expandingφ(x) into a Taylor series around zero and usingφ′′(0) =−√^12 π
we get the asymptotic expression
CB−CBS=S 0
[
1
24
√
2 π(
σBS√
T
) 3 ]
+o((
σBS√
T
) 3 )
,
which indicates a very small difference between the option valuesCBandCBS
in the Bachelier and Black-Scholes model respectively, providedσBS
√
T is
small. Evaluating this expression for the empirical data reported by Bachelier
(see [B 00] or [S 03, Chap. 1]), i.e.,σBS≈ 2 .4 % on a yearly basis, andT≈
2months =^16 year (this is a generous upper bound for the periods considered
by Bachelier which were ranging between 10 and 45 days) we find
CB−CBS≈S 0
1
24
√
2 π(
0. 024
√
1
6
) 3
≈ 1. 56 ∗ 10 −^8 S 0.
Hence for this data the difference between the option value obtained using
Bachelier’s and the Black-Scholes model is of order 10−^8 times the valueS 0
of the underlying; observing that for Bachelier’s data, the price of an option
was of the order of 100 S^0 , we find that the difference is of the order 10−^6 of the
price of the option.
In view of all the uncertainties involved in option pricing, in particular
regarding the estimation ofσ, one might be tempted to call this quantity
“completely negligible, a priori” (this expression was used by Bachelier when
discussing the drawbacks of the normal distribution giving positive probability
to negative stock prices).
For more information we refer to the introductory chapter of [S 03, Chap. 1]
as well as to [ST 05].
3.)Let us now comment on the role of the riskless rate of interestr, appearing
in the Black-Scholes formula and on the reason why this variable does not show
up in Bachelier’s formula: noting the obvious fact that
ln(
S 0
K
)
+rT=ln(
S 0
Ke−rT)
,
one readily observes that this quantity only appears in the Black-Scholes for-
mula (4.16) via the discounting of the strike price, i.e., transformingKunits
ofet=T intoKe−rT units ofet=0. When comparing the setting of Black-
Scholes to that of Bachelier one should recall that the option premium in
Bachelier’s days pertained to a payment at timeTrather than at time 0.
Under the assumption of a constant riskless interest rate — as is the case
in the Black-Scholes model — this amounts to considering the present day
quantities upcounted byerT. This was perfectly taken into account by Bache-
lier, who stressed that the quantities appearing in his formulae have to be
understood in terms of forward prices in modern terminology, which amounts
to upcounting byerTin the present setting.