<s>
In	O
finance	O
,	O
the	O
binomial	B-Application
options	I-Application
pricing	I-Application
model	I-Application
(	O
BOPM	B-Application
)	O
provides	O
a	O
generalizable	O
numerical	B-General_Concept
method	I-General_Concept
for	O
the	O
valuation	O
of	O
options	O
.	O
</s>
<s>
Essentially	O
,	O
the	O
model	O
uses	O
a	O
"	O
discrete-time	O
"	O
(	O
lattice	B-Application
based	I-Application
)	O
model	O
of	O
the	O
varying	O
price	O
over	O
time	O
of	O
the	O
underlying	O
financial	O
instrument	O
,	O
addressing	O
cases	O
where	O
the	O
closed-form	O
Black	O
–	O
Scholes	O
formula	O
is	O
wanting	O
.	O
</s>
<s>
The	O
Binomial	B-Application
options	I-Application
pricing	I-Application
model	I-Application
approach	O
has	O
been	O
widely	O
used	O
since	O
it	O
is	O
able	O
to	O
handle	O
a	O
variety	O
of	O
conditions	O
for	O
which	O
other	O
models	O
cannot	O
easily	O
be	O
applied	O
.	O
</s>
<s>
This	O
is	O
largely	O
because	O
the	O
BOPM	B-Application
is	O
based	O
on	O
the	O
description	O
of	O
an	O
underlying	O
instrument	O
over	O
a	O
period	O
of	O
time	O
rather	O
than	O
a	O
single	O
point	O
.	O
</s>
<s>
Being	O
relatively	O
simple	O
,	O
the	O
model	O
is	O
readily	O
implementable	O
in	O
computer	O
software	O
(	O
including	O
a	O
spreadsheet	B-Application
)	O
.	O
</s>
<s>
For	O
options	O
with	O
several	O
sources	O
of	O
uncertainty	O
(	O
e.g.	O
,	O
real	O
options	O
)	O
and	O
for	O
options	O
with	O
complicated	O
features	O
(	O
e.g.	O
,	O
Asian	O
options	O
)	O
,	O
binomial	O
methods	O
are	O
less	O
practical	O
due	O
to	O
several	O
difficulties	O
,	O
and	O
Monte	B-Algorithm
Carlo	I-Algorithm
option	I-Algorithm
models	I-Algorithm
are	O
commonly	O
used	O
instead	O
.	O
</s>
<s>
When	O
simulating	O
a	O
small	O
number	O
of	O
time	O
steps	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
will	O
be	O
more	O
computationally	O
time-consuming	O
than	O
BOPM	B-Application
(	O
cf	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
in	I-Algorithm
finance	I-Algorithm
)	O
.	O
</s>
<s>
However	O
,	O
the	O
worst-case	O
runtime	O
of	O
BOPM	B-Application
will	O
be	O
O(2n )	O
,	O
where	O
n	O
is	O
the	O
number	O
of	O
time	O
steps	O
in	O
the	O
simulation	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
simulations	I-Algorithm
will	O
generally	O
have	O
a	O
polynomial	O
time	O
complexity	O
,	O
and	O
will	O
be	O
faster	O
for	O
large	O
numbers	O
of	O
simulation	O
steps	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
simulations	I-Algorithm
are	O
also	O
less	O
susceptible	O
to	O
sampling	O
errors	O
,	O
since	O
binomial	O
techniques	O
use	O
discrete	O
time	O
units	O
.	O
</s>
