SØK460/ECON460 Finance Theory, Fall semester, 2002
The transition from discrete to continuous time
On the top of p. 207 in Cox and Rubinstein, it is claimed that
log r − 12 σ 2
σ
1 1
p→ +
2 2
!s
t
n
as n → ∞. We shall prove this.
The limit contains n. This means that p, being a function of n, approaches another function
of n, which equals 1/2 plus a constant
multiplied by n−1/2 . It is easier to subtract 1/2 from
√
both sides, and multiply by 2 n, giving a limit which does not vary with n, like this:
√ log r − 12 σ 2 √
1
lim 2 n p −
=
t.
n→∞
2
σ
The proof is based on using L’Hôpital’s rule twice. Enter the expressions for p, u, and d,
and rewrite rt/n as e(t/n) log r . The limit becomes
√
√
2e(t/n) log r+σ t/n − e2σ t/n − 1
√
lim
.
n→∞
2σ t/n
−1/2
n
e
−1
Both numerator and denominator approach zero, and we use L’Hôpital’s rule, giving
√ √
√
√
2e(t/n) log r+σ t/n n−2 t log r + 12 n−3/2 σ t − e2σ t/n n−3/2 σ t
√ lim
√
n→∞
e2σ t/n n−2 σ t + 12 n−3/2 − 12 n−3/2
√ √
√
2e(t/n) log r−σ t/n n−1/2 t log r + 12 σ t − σ t
√
= lim
q
n→∞
σ t/n + 12 − 12 e−2nσ t/n
√ h
i
√ −1/2
√ 1 −3/2
(t/n) log r−σ t/n
1 −3/2
1
−2
2e
−n
t
log
r
+
n
σ
t
n
t
log
r
+
σ
t
−
n
t
log
r
0
LH
2
2
2
√
= lim
.
√
√
n→∞
− 12 n−3/2 σ t − 12 e−2σ t/n n−3/2 σ t
L’Hôpital was used once more in order to arrive at the last expression. Multiply by 2n3/2
in numerator and denominator. This gives
√ h
i
4e(t/n) log r−σ t/n n−1 (t log r)2 − n1/2 t3/2 σ log r + 14 tσ 2 − 12 t log r
lim
√ √ n→∞
−2σ t/n
−σ t 1 + e
=
4
1
tσ 2
4
− 12 t log r
√
−2σ t
log r − 21 σ 2 √
=
t, q.e.d.
σ
The proof is complete, and it follows directly (as stated further down on p. 207) that
1
µ̂p n → log r − σ 2 t.
2
1