4.3 Bachelier’s Model 59It is straightforward to derive from (4.4) an “option pricing formula” by
calculating the integral in (4.4) (compare, e.g., [Sh 99]): denoting the standard
normal density function byφ(x), i.e.,φ(x) equals (4.5) forσ^2 T=1,denoting
the corresponding distribution function byΦ(x) and using the relationφ′(x)=
−xφ(x), an elementary calculation reveals that
C(S 0 ,T)=E[f]=(S 0 −K)Φ(
S 0 −K
σ√
T
)
+σ√
Tφ(
S 0 −K
σ√
T
)
. (4.6)
By the same token we obtain, for every 0≤t≤T, and conditionally on
the stock price having the valueStat timet,
C(St,T−t) (4.7)=E[f|St]=(St−K)Φ(
St−K
σ√
T−t)
+σ√
T−tφ(
St−K
σ√
T−t)
.
This solves the pricing problem in Bachelier’s model, based on no-arbitrage
arguments, as we have the “replication formula” (4.3).
ButwhatisthetradingstrategyH, in other words, the recipe to replicate
the option by trading dynamically? Economic intuition suggests that we have
H(S, t)=∂
∂S
C(S, T−t).Indeed, consider the following heuristic reasoning using infinitesimals: sup-
pose that at timetthe stock price equalsSt, so that the value of the option
equalsC(St,T−t). During the infinitesimal interval (t, t+dt) the Brownian
motionWtwill move bydWt=Wt+dt−Wt=εt
√
dt,whereP[εt=1]=P[εt=
−1] =^12 ,sothatStwill move bydSt=St+dt−St=εtσ
√
dt. Hence the value of
the optionC(St,T−t) will move bydCt=C(St+dt,T−(t+dt))−C(St,T−t)≈
εt∂C∂S(St,T−t)σ
√
dt, where we neglect terms of smaller order than√
dt.In
other words, the ratio between the up or down movement of the underlying
stockSand the option is
dCt
dSt=
∂C
∂S
(St,T−t)
εtσ√
dt
εtσ√
dt(4.8)
=
∂C
∂S
(St,T−t).If we want to replicate the option by investing the proper quantityHof
the underlying stock, formula (4.8) suggests that this quantity should equal
∂C
∂S(St,T−t).
After these motivating remarks, let us deduce the equation
H(St,t)=∂C
∂S
(St,T−t) (4.9)more formally. Consider the stochastic process
C(St,T−t)=C(S 0 +σWt,T−t), 0 ≤t≤T,