5.2 The Solution
51
c(x, τ ) =
K e
−r f τ
√
2πσ 2 τ
⎧
⎨
⎩
e
x+σ
2 τ/2
∞
0
exp
−
(x
− x − σ
2
τ )
2
2σ 2 τ
dx
−
∞
0
exp
−
(x
− x)
2
2σ 2 τ
dx
⎫
⎬
⎭
,
(5.14)
where we completed the square in the exponent of the first integral. After shifting
the integration variable to simplify the exponent and substituting y
= x
/σ
√ τ , we
express the integrals in terms of cumulative distribution functions N (x), already
defined in (4.39)
c(x, τ ) = K e
x+σ
2 τ/2−r f τ
1 − N
−
x + σ
2
τ
σ
√ τ
− K e
−r f τ
1 − N
−
x
σ
√ τ
= K e
x+σ
2 τ/2−r f τ N
x + σ
2
τ
σ
√
τ
− K e
−r f τ N
x
σ
√ τ
.
(5.15)
Now we can substitute back the original variables x = z + ˆ
r τ and z = ln S/K . Using
ˆ
r = r f − σ
2
/2 to simplify the previous equation, we arrive at
c(S, τ ) = S N
ln(S/K ) + (r f + σ
2
/2)τ
σ
√ τ
(5.16)
−K e
−r f τ N
ln(S/K ) + (r f − σ
2
/2)τ
σ
√ τ
.
After replacing τ = T − t we find the price of the call option c(S, t)
c(S, t) = S N
ln(S/K ) + (r f + σ
2
/2)(T − t)
σ
√
T − t
(5.17)
−K e
−r f (T −t) N
ln(S/K ) + (r f − σ
2
/2)(T − t)
σ
√
T − t
,
which almost coincides with the result found in Sect. 4.6 in (4.40). Only the growth
rate ρ is replaced by the risk-free rate r f in (4.40).
It is remarkable that the different initial assumptions about the option price used
in the derivation in Sect. 4.6 and in this section lead to similar results. In the former
case we calculated the expectation value of the future profit back-propagated to
today’s value using the expected growth rate of the asset. In this section we derived
a differential equation that describes a hedged portfolio (5.3) that is risk-free and
therefore grows with the risk-free rate r f . It links the temporal evolution of the stocks
S to that of the option c. The solution to the differential equation with the boundary
conditions, pertinent to the type of option, resulted in the same option price, including
its temporal evolution towards maturity (T − t), albeit with ρ replaced by r f .
51
c(x, τ ) =
K e
−r f τ
√
2πσ 2 τ
⎧
⎨
⎩
e
x+σ
2 τ/2
∞
0
exp
−
(x
− x − σ
2
τ )
2
2σ 2 τ
dx
−
∞
0
exp
−
(x
− x)
2
2σ 2 τ
dx
⎫
⎬
⎭
,
(5.14)
where we completed the square in the exponent of the first integral. After shifting
the integration variable to simplify the exponent and substituting y
= x
/σ
√ τ , we
express the integrals in terms of cumulative distribution functions N (x), already
defined in (4.39)
c(x, τ ) = K e
x+σ
2 τ/2−r f τ
1 − N
−
x + σ
2
τ
σ
√ τ
− K e
−r f τ
1 − N
−
x
σ
√ τ
= K e
x+σ
2 τ/2−r f τ N
x + σ
2
τ
σ
√
τ
− K e
−r f τ N
x
σ
√ τ
.
(5.15)
Now we can substitute back the original variables x = z + ˆ
r τ and z = ln S/K . Using
ˆ
r = r f − σ
2
/2 to simplify the previous equation, we arrive at
c(S, τ ) = S N
ln(S/K ) + (r f + σ
2
/2)τ
σ
√ τ
(5.16)
−K e
−r f τ N
ln(S/K ) + (r f − σ
2
/2)τ
σ
√ τ
.
After replacing τ = T − t we find the price of the call option c(S, t)
c(S, t) = S N
ln(S/K ) + (r f + σ
2
/2)(T − t)
σ
√
T − t
(5.17)
−K e
−r f (T −t) N
ln(S/K ) + (r f − σ
2
/2)(T − t)
σ
√
T − t
,
which almost coincides with the result found in Sect. 4.6 in (4.40). Only the growth
rate ρ is replaced by the risk-free rate r f in (4.40).
It is remarkable that the different initial assumptions about the option price used
in the derivation in Sect. 4.6 and in this section lead to similar results. In the former
case we calculated the expectation value of the future profit back-propagated to
today’s value using the expected growth rate of the asset. In this section we derived
a differential equation that describes a hedged portfolio (5.3) that is risk-free and
therefore grows with the risk-free rate r f . It links the temporal evolution of the stocks
S to that of the option c. The solution to the differential equation with the boundary
conditions, pertinent to the type of option, resulted in the same option price, including
its temporal evolution towards maturity (T − t), albeit with ρ replaced by r f .