4.4 The Black-Scholes Model 67Keeping in mind that this was achieved during an interval of total length
2 dt(which corresponds to the passage fromσ^2 in (4.20) to σ
2
2 in (4.19))
we have found a heuristic explanation for the Black-Scholes equation (4.19).
We also note that the same argument applied to Bachelier’s model, yields a
heuristic explanation of the heat equation (4.11). The general phenomenon
behind this fact is that, in the case of convexity, the “wobbling” of Brownian
motion, which is of order
√
dtin an interval of lengthdt, causes the hedger
to have systematic losses, which are proportional to ∂
(^2) C
∂S^2 as well as to the
incrementd〈S〉tof the quadratic variation process〈S〉t=
∫t
0 σ
(^2) S 2
uduof the
stock price processS.
5.)When deriving the Black-Scholes formula (4.16) we did not go through the
(elementary but tedious) trouble of explicitly calculating (4.15). We shall now
provide an explicit derivation of the formula which has the merit of yielding
an interpretation of the two probabilities appearing in (4.16). It also allows
for a better understanding of the formula (for example, for the remarkable
fact, that the parameterμhas disappeared) and dispenses us of cumbersome
calculation.
As observed in (4.15), the contingent claimf(ω)=(ST(ω)−Ke−rT)+
(expressed in terms of the num ́eraireBt) splits into
(
ST−Ke−rT
)
+=STχ{ST≥Ke−rT}−Ke−rTχ
{ST≥Ke−rT}
=f 1 −f 2.
We have to calculateEQ[f 1 ]andEQ[f 2 ] under the risk-neutral measureQ
defined in (4.14). This is easy forf 2 and we do not have to use the explicit form
of the density (4.14) provided by Girsanov’s theorem. It suffices to observe
thatST=S 0 exp(σ ̃WT−σ
2
2 T)where
̃Wis a Brownian motion underQ.SoX:=
ln(
ST
S 0)
+σ2
2 T
σ√
T
∼N(0,1) underQ,whence
EQ[f 2 ]=e−rTKQ[
ST≥e−rTK]
(4.21)
=e−rTKQ⎧
⎨
⎩
ln(
ST
S 0)
+σ2
2 T
σ√
T
≥
ln(
e−rTK
S 0)
+σ2
2 T
σ√
T
⎫
⎬
⎭
=e−rTKQ⎧
⎨
⎩
X≥
ln(
e−rTK
S 0)
+σ2
2 T
σ√
T
⎫
⎬
⎭
=e−rTKΦ⎛
⎝
ln(S 0
K)
+
(
r−σ2
2)
T
σ√
T
⎞
⎠,
which yields the second term of the Black-Scholes formula.