60 4 Bachelier and Black-Scholes
of the value of the option. By Itˆo’s formula (see, e.g., [RY 91])
dC(St,T−t)=∂C
∂S
dSt+(
∂C
∂t+
1
2
∂^2 C
∂S^2
σ^2)
dt, (4.10)wherewehaveuseddSt=σdWt. One readily deduces from formula (4.7) that
Cverifies the heat equation with parameterσ
2
2 displayed in (4.11) below (time
is running into the negative direction in the present setting). In particular,
for the processCdefined in (4.7), the drift term in (4.10) vanishes as it must
be the case according to the general theory (the option price process is a
martingale by (4.1)). Hence (4.10) reduces to the formula
C(St,T−t)=C(S 0 ,T)+(H·S)t,whereHis given by (4.9). Rephrasing this result once more we have shown
that the trading strategyH, whose existence was guaranteed by the martin-
gale representation (Theorem 4.2.1), is of the form (4.9).
One more word on the fact thatC(S, T−t) satisfies the heat equation
(4.11) below, which was known to L. Bachelier in 1900 and may be verified by
simply calculating the partial derivatives in (4.7). Admitting this calculation,
we concluded above that the drift term in (4.10) vanishes. One may also turn
the argument around to conclude from (4.1) that the drift term in (4.10) must
vanish, which thenimplies thatC(S, T−t) must satisfy the heat equation
(time running inversely)
∂C
∂t(S, T−t)=−σ^2
2∂^2 C
∂S^2
(S, T−t). (4.11)Imposing the boundary conditionC(S, T−T)=C(S,0) = (S−K)+one may
deriveby standard methods the solution (4.7) of this p.d.e.. This is, in fact,
how F. Black and M. Scholes originally proceeded (in the framework of their
model) to derive their option pricing formula, which we shall now analyze.
4.4 The Black-ScholesModel
This model of a stock market was proposed by the famous economist P. Samuel-
son in 1965 ([S 65]), who was aware of Bachelier’s work. In fact, triggered by
a question of J. Savage, it was P. Samuelson who had rediscovered Bachelier’s
work for the economic literature some years before 1965.
The model is usually called the Black-Scholes model today and became
the standard reference model in the context of option pricing:
B̂t=ert,Ŝt=S 0 eσWt+(
μ−σ 22)
t, 0 ≤t≤T. (4.12)AgainWis a standard Brownian motion with natural base (Ω,(Ft)t∈[0,T],P).