The transition from discrete to continuous time
SØK460 Finance Theory, D. Lund, Spring semester 2001
On the top of p. 207 in Cox and Rubinstein, it is claimed that
1 1
p→ +
2 2
Ã
log r − 12 σ 2
σ
!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:
¶
√ µ
log r − 12 σ 2 √
1
=
t.
lim 2 n p −
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
√
√
(t/n) log r+σ t/n
2σ t/n
2e
−e
−1
µ √
¶
lim
.
n→∞
n−1/2 e2σ t/n − 1
Both numerator and denominator approach zero, and we use L’Hôpital’s rule, giving
√ ³
√
ë
√
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
√ ³
lim
´
√
n→∞
e2σ t/n n−2 σ t + 12 n−3/2 − 12 n−3/2
√ ³
ë
√
2e(t/n) log r−σ t/n n−1/2 t log r + 12 σ t − σ t
√
= lim
q
n→∞
1
1 −2nσ t/n
σ t/n + 2 − 2 e
√ h³
i
√ ´³
ë
(t/n) log r−σ t/n
2e
−n−2 t log r + 12 n−3/2 σ t n−1/2 t log r + 12 σ t − 12 n−3/2 t log r
L0 H
√
= 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
2n 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
√ ¶
√ µ
n→∞
−2σ t/n
−σ t 1 + e
3/2
=
4
³
1
tσ 2
4
− 12 t log r
√
−2σ t
´
log r − 12 σ 2 √
=
t, q.e.d.
σ
The proof is complete, and it follows directly (as stated further down on p. 207) that
µ
¶
1
µ̂p n → log r − σ 2 t.
2