Copyright
c
2007 by Karl Sigman
Binomial lattice model for stock prices
Here we model the price of a stock in discrete time by a Markov chain of the recursive form
S
n
+1
=
S
n
Y
n
+1
, n
≥
0
,
where the
{
Y
i
}
are iid with distribution
P
(
Y
=
u
) =
p, P
(
Y
=
d
) =
1

p
. Here 0
< d <
1 +
r < u
are constants with
r
the riskfree interest rate ((1 +
r
)
x
is the
payoff you would receive one unit of time later if you bought $
x
worth of the riskfree asset
(a bond for example, or placed money in a savings account at that fixed rate) at time
n
= 0).
Given the value of
S
n
,
S
n
+1
=
uS
n
,
w.p.
p
;
dS
n
,
w.p. 1

p
,
n
≥
0
,
independent of the past. Thus the stock either goes up (“u”) or down (“d”) in each time period,
and the randomness is due to iid Bernoulli (
p
) rvs (flips of a coin so to speak) where we can
view “up=success”, and “down=failure”.
Expanding the recursion yields
S
n
=
S
0
×
Y
1
× · · · ×
Y
n
, n
≥
1
,
(1)
where
S
0
is the initial price per share and
S
n
is the price per share at time
n
.
1
It follows from (1) that for a given
n
,
S
n
=
u
i
d
n

i
S
0
for some
i
∈ {
0
, . . . n
}
, meaning that
the stock went up
i
times and down
n

i
times during the first
n
time periods (
i
“successes”
and
n

i
“failures” out of
n
independent Bernoulli (
p
) trials). The corresponding probabilities
are thus determined by the binomial(
n, p
) distribution;
P
(
S
n
=
u
i
d
n

i
S
0
) =
n
i
p
i
(1

p
)
n

i
,
0
≤
i
≤
n,
which is why we refer to this model as the
binomial lattice model (BLM)
. The lattice is the set
of points
{
u
i
d
n

i
S
0
: 0
≤
i
≤
n <
∞}
, which is the state space for this Markov chain. Note
that this lattice depends on the initial price
S
0
and the values of
u, d
.
Portfolios of stock and a riskfree asset
In addition to our stock there is a
riskfree
asset (money) with fixed interest rate 0
< r <
1 that
costs $1
.
00 per share;
x
shares bought now (at time
t
= 0) would be worth the deterministic
amount
x
(1 +
r
)
n
at time
t
=
n, n
≥
1 (interest is compounded each time unit). Buying this
asset is lending money. Selling this asset is borrowing money (shorting this asset).
We must have 1 +
r < u
for otherwise there would be no reason to invest in the stock: you
could instead obtain a riskless payoff of
S
0
(1+
r
)
≥
S
0
u
at time
t
= 1 by buying
S
0
shares of the
1
This model is meant to approximate the continuoustime geometric Brownian motion (GBM)
S
(
t
) =
S
0
e
X
(
t
)
model for stock, where
X
(
t
) =
σB
(
t
)+
μt
is Brownian motion (BM) with drift
μ
and variance term
σ
2
. The idea is
to break up the time interval (0
, t
] into
n
small subintervals of length
h
=
t/n
, (0
, h
]
,
(
h,
2
h
]
, . . .
((
n

1)
h, nh
=
t
],
and rewrite
S
(
t
) =
S
(0)
×
H
1
× · · · ×
H
n
,
where
H
i
=
S
(
ih
)
/S
((
i

1)
h
)
, i
≥
1 are the succesive price ratios, and are in fact iid (due to the stationary and
independent increments of the BM
X
(
t
)). Then we find an appropriate
p
,
u
,
d
so that the distribution of
H
is
well approximated by the twopoint distribution of
Y
(typically done by fitting the first two moments of
H
with
those of
Y
). As
h
↓
0 the approximation becomes exact.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Whitt
 Operations Research, Strike price

Click to edit the document details