Chapter 4: Ten Different Ways to Derive Black–Scholes 265
At the same time the portfolio of option and stock
becomes either
V+−uS or V−−vS.
Having the freedom to choose, we can make the value
of this portfolio the same whether the asset rises or
falls. This is ensured if we make
V+−uS=V−−vS.
This means that we should choose
=
V+−V−
(u−v)S
for hedging. The portfolio value is then
V+−uS=V+−
u(V+−V−)
(u−v)
=V−−vS
=V−−
v(V+−V−)
(u−v)
.
Let’s denote this portfolio value by
+δ.
This just means the original portfolio value plus the
change in value. But we must also haveδ=rδt
to avoid arbitrage opportunities. Bringing all of these
expressions together to eliminate, and after some
rearranging, we get
V=
1
1 +rδt
(
p′V++(1−p′)V−
)
,
where
p′=
1
2
+
r
√
δt
2 σ
.
This is an equation forVgivenV+,andV−, the option
values at the next time step, and the parametersr
andσ.
The right-hand side of the equation forVcan be inter-
preted, rather clearly, as the present value of the expected
