The Mathematics of Arbitrage

4.4 The Black-Scholes Model 63

always remains positive, while Brownian motion may also attain negative
values. This fact has strong effects for very largeσor — what amounts roughly
to the same — for very largeT. Nevertheless we shall presently see that —
for reasonable values ofσandT— the Black-Scholes formula and Bachelier’s
formula (4.18) are very close. This seems to be the essential fact, keeping in
mind Keynes’ famous dictum telling us,notto look at the limitT→∞:in
the long run we are all dead.

2.)Let us compare the Black-Scholes formula (4.16) and Bachelier’s formula
(4.18) more systematically. To do so we specialise in the Black-Scholes formula
tor=0andS 0 =K, and we have to let the volatility in the Black-Scholes
formula, which we now denote byσBS, correspond to the “parameter of ner-
vousness”σ(this wording was used by Bachelier) appearing in Bachelier’s for-
mula, which we denote byσB. As the former pertains to the relative standard
deviation of stock prices and the latter to the absolute standard deviation, we
roughly find the correspondence — at least for small values ofT—

σB≈σBSS 0

In the subsequent calculations we therefore suppose thatσB≈σBSS 0 .Inthis
case, the Black-Scholes and Bachelier option prices to be compared are

CBS=S 0

[

Φ

(

σBS

√

T

2

)

−Φ

(

−

σBS

√

T

2

)]

,

while

CB=

σB

√

2 π

√

T≈S 0

σBS

√

2 π

√

T.

The difference of the two quantities is best understood by looking at the

shaded area in the subsequent graph involving the densityφ(x)=√^12 πe−

x 22

of

the standard normal distribution, and noting thatφ(0) =√^12 π. This shaded

area, let’s call itA,equalsσ

BS√T

√

2 π −

[

Φ

(

σBS

√

T

2

)

−Φ

(

−σ

BS√T

2

)]

which — up

to the factorS 0 — is just the difference betweenCBandCBS





Fig. 4.1.Comparison of the Bachelier with the Black-Scholes formula.