62 4 Bachelier and Black-Scholes
After an elementary calculation (see, e.g., [LL 96]) this yields the famous
Black-Scholes formula
C(S 0 ,T)=S 0 Φ
⎛
⎝
ln(S 0
K)
+
(
r+σ2
2)
T
σ√
T
⎞
⎠ (4.16)
−Ke−rTΦ⎛
⎝
ln(S 0
K)
+
(
r−σ
2
2)
T
σ√
T
⎞
⎠
and, by the same token, for 0≤t≤TandSt>0,
C(St,T−t)=S 0 Φ⎛
⎝
ln(St
K)
+
(
r+σ2
2)
(T−t)
σ√
T−t⎞
⎠ (4.17)
−Ke−rTΦ⎛
⎝
ln(St
K)
+
(
r−σ2
2)
(T−t)
σ√
T−t⎞
⎠.
Let us take some time to contemplate on this truly remarkable formula
(for which R. Merton and M. Scholes received the Nobel prize in economics
in 1997; F. Black unfortunately had passed away in 1995).
1.)As a warm-up consider the limits asσ→∞(which yieldsC(S 0 ,T)=S 0 )
andσ→0 (which yieldsC(S 0 ,T)=(S 0 −Ke−rT)+). The reader should
convince herself that this does make sense economically. For an extremely
risky underlyingS, an option on one unit ofSis almost as valuable as one
unit ofSitself (think, for example, of a call option on a lottery ticket with
strike priceK= 100 and exercise timeT, such thatTis later than the drawing
at which it is decided whether the ticket wins a million or nothing). Intuitively,
it is quite obvious that the option on the lottery ticket is almost as valuable
as the lottery ticket itself). On the other hand, if the underlyingSis (almost)
riskless a similar consideration reveals that the value of an option is almost
equal to its “inner value” (S 0 −Ke−rT)+.
This behaviour of the Black-Scholes formula should be contrasted to
Bachelier’s formula (specialising to the caseS 0 =Kandr=0)
CBachelier(S 0 ,T)=σ
√
2 π√
T (4.18)
obtained in (4.6) above, which tends to infinity asσ→∞; this limiting
behaviour is economically absurd and contradicts an obvious no-arbitrage
argument which — using the fact thatSTis non-negative — shows that the
value of a call option always must be less than the value of the underlying
stock.
The reason for this difference in the behaviour of the Black-Scholes formula
and Bachelier’s one, for large values ofσ, is that geometric Brownian motion