<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
used	O
in	O
corporate	O
finance	O
and	O
mathematical	O
finance	O
to	O
value	O
and	O
analyze	O
(	O
complex	O
)	O
instruments	O
,	O
portfolios	O
and	O
investments	O
by	O
simulating	O
the	O
various	O
sources	O
of	O
uncertainty	O
affecting	O
their	O
value	O
,	O
and	O
then	O
determining	O
the	O
distribution	O
of	O
their	O
value	O
over	O
the	O
range	O
of	O
resultant	O
outcomes	O
.	O
</s>
<s>
This	O
is	O
usually	O
done	O
by	O
help	O
of	O
stochastic	B-Algorithm
asset	I-Algorithm
models	I-Algorithm
.	O
</s>
<s>
The	O
advantage	O
of	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
over	O
other	O
techniques	O
increases	O
as	O
the	O
dimensions	O
(	O
sources	O
of	O
uncertainty	O
)	O
of	O
the	O
problem	O
increase	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
were	O
first	O
introduced	O
to	O
finance	O
in	O
1964	O
by	O
David	O
B	O
.	O
Hertz	O
through	O
his	O
Harvard	O
Business	O
Review	O
article	O
,	O
discussing	O
their	O
application	O
in	O
Corporate	O
Finance	O
.	O
</s>
<s>
This	O
article	O
discusses	O
typical	O
financial	O
problems	O
in	O
which	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
used	O
.	O
</s>
<s>
The	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
encompasses	O
any	O
technique	O
of	O
statistical	O
sampling	O
employed	O
to	O
approximate	O
solutions	O
to	O
quantitative	O
problems	O
.	O
</s>
<s>
Essentially	O
,	O
the	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
solves	O
a	O
problem	O
by	O
directly	O
simulating	O
the	O
underlying	O
(	O
physical	O
)	O
process	O
and	O
then	O
calculating	O
the	O
(	O
average	O
)	O
result	O
of	O
the	O
process	O
.	O
</s>
<s>
This	O
very	O
general	O
approach	O
is	O
valid	O
in	O
areas	O
such	O
as	O
physics	O
,	O
chemistry	O
,	O
computer	B-General_Concept
science	I-General_Concept
etc	O
.	O
</s>
<s>
In	O
finance	O
,	O
the	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
is	O
used	O
to	O
simulate	O
the	O
various	O
sources	O
of	O
uncertainty	O
that	O
affect	O
the	O
value	O
of	O
the	O
instrument	O
,	O
portfolio	O
or	O
investment	O
in	O
question	O
,	O
and	O
to	O
then	O
calculate	O
a	O
representative	O
value	O
given	O
these	O
possible	O
values	O
of	O
the	O
underlying	O
inputs	O
.	O
</s>
<s>
In	O
Corporate	O
Finance	O
,	O
project	O
finance	O
and	O
real	O
options	O
analysis	O
,	O
Monte	B-Algorithm
Carlo	I-Algorithm
Methods	I-Algorithm
are	O
used	O
by	O
financial	O
analysts	O
who	O
wish	O
to	O
construct	O
"	O
stochastic	O
"	O
or	O
probabilistic	O
financial	O
models	O
as	O
opposed	O
to	O
the	O
traditional	O
static	O
and	O
deterministic	O
models	O
.	O
</s>
<s>
Then	O
,	O
these	O
results	O
are	O
combined	O
in	O
a	O
histogram	B-Algorithm
of	O
NPV	O
(	O
i.e.	O
</s>
<s>
Note	O
that	O
whereas	O
equity	O
options	O
are	O
more	O
commonly	O
valued	O
using	O
other	O
pricing	O
models	O
such	O
as	O
lattice	B-Application
based	I-Application
models	I-Application
,	O
for	O
path	O
dependent	O
exotic	O
derivatives	O
–	O
such	O
as	O
Asian	O
options	O
–	O
simulation	O
is	O
the	O
valuation	O
method	O
most	O
commonly	O
employed	O
;	O
see	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
for	I-Algorithm
option	I-Algorithm
pricing	I-Algorithm
for	O
discussion	O
as	O
to	O
further	O
–	O
and	O
more	O
complex	O
–	O
option	O
modelling	O
.	O
</s>
<s>
For	O
example	O
,	O
for	O
bonds	O
,	O
and	O
bond	O
options	O
,	O
under	O
each	O
possible	O
evolution	O
of	O
interest	O
rates	O
we	O
observe	O
a	O
different	O
yield	B-Algorithm
curve	I-Algorithm
and	O
a	O
different	O
resultant	O
bond	O
price	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
Methods	I-Algorithm
are	O
used	O
for	O
portfolio	O
evaluation	O
.	O
</s>
<s>
As	O
for	O
corporate	O
finance	O
,	O
above	O
,	O
the	O
various	O
portfolio	O
values	O
are	O
then	O
combined	O
in	O
a	O
histogram	B-Algorithm
,	O
and	O
the	O
statistical	B-General_Concept
characteristics	I-General_Concept
of	O
the	O
portfolio	O
are	O
observed	O
,	O
and	O
the	O
portfolio	O
assessed	O
as	O
required	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
Methods	I-Algorithm
are	O
used	O
for	O
personal	O
financial	O
planning	O
.	O
</s>
<s>
Discrete	B-General_Concept
event	I-General_Concept
simulation	I-General_Concept
can	O
be	O
used	O
in	O
evaluating	O
a	O
proposed	O
capital	O
investment	O
's	O
impact	O
on	O
existing	O
operations	O
.	O
</s>
<s>
return	O
)	O
relative	O
to	O
the	O
cost	O
(	O
via	O
histogram	B-Algorithm
as	O
above	O
)	O
;	O
it	O
may	O
also	O
be	O
used	O
in	O
stress	O
testing	O
the	O
design	O
.	O
</s>
<s>
Although	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
provide	O
flexibility	O
,	O
and	O
can	O
handle	O
multiple	O
sources	O
of	O
uncertainty	O
,	O
the	O
use	O
of	O
these	O
techniques	O
is	O
nevertheless	O
not	O
always	O
appropriate	O
.	O
</s>
<s>
In	O
many	O
cases	O
these	O
integrals	O
can	O
be	O
valued	O
analytically	O
,	O
and	O
in	O
still	O
more	O
cases	O
they	O
can	O
be	O
valued	O
using	O
numerical	B-Algorithm
integration	I-Algorithm
,	O
or	O
computed	O
using	O
a	O
partial	O
differential	O
equation	O
(	O
PDE	O
)	O
.	O
</s>
<s>
However	O
,	O
when	O
the	O
number	O
of	O
dimensions	O
(	O
or	O
degrees	O
of	O
freedom	O
)	O
in	O
the	O
problem	O
is	O
large	O
,	O
PDEs	O
and	O
numerical	O
integrals	O
become	O
intractable	O
,	O
and	O
in	O
these	O
cases	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
often	O
give	O
better	O
results	O
.	O
</s>
<s>
analytic	O
solutions	O
)	O
do	O
not	O
exist	O
,	O
while	O
other	O
numerical	B-Algorithm
methods	I-Algorithm
such	O
as	O
the	O
Binomial	B-Application
options	I-Application
pricing	I-Application
model	I-Application
and	O
finite	B-Algorithm
difference	I-Algorithm
methods	I-Algorithm
face	O
several	O
difficulties	O
and	O
are	O
not	O
practical	O
.	O
</s>
<s>
In	O
these	O
cases	O
,	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
converge	O
to	O
the	O
solution	O
more	O
quickly	O
than	O
numerical	B-Algorithm
methods	I-Algorithm
,	O
require	O
less	O
memory	O
and	O
are	O
easier	O
to	O
program	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
can	O
deal	O
with	O
derivatives	O
which	O
have	O
path	O
dependent	O
payoffs	O
in	O
a	O
fairly	O
straightforward	O
manner	O
.	O
</s>
<s>
Monte-Carlo	B-Algorithm
methods	I-Algorithm
are	O
harder	O
to	O
use	O
with	O
American	O
options	O
.	O
</s>
<s>
This	O
is	O
because	O
,	O
in	O
contrast	O
to	O
a	O
partial	O
differential	O
equation	O
,	O
the	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
really	O
only	O
estimates	O
the	O
option	O
value	O
assuming	O
a	O
given	O
starting	O
point	O
and	O
time	O
.	O
</s>
<s>
In	O
Monte-Carlo	O
this	O
information	O
is	O
harder	O
to	O
obtain	O
,	O
but	O
it	O
can	O
be	O
done	O
for	O
example	O
using	O
the	O
least	B-Algorithm
squares	I-Algorithm
algorithm	O
of	O
Carriere	O
(	O
see	O
link	O
to	O
original	O
paper	O
)	O
which	O
was	O
made	O
popular	O
a	O
few	O
years	O
later	O
by	O
Longstaff	O
and	O
Schwartz	O
(	O
see	O
link	O
to	O
original	O
paper	O
)	O
.	O
</s>
<s>
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
ideally	O
suited	O
to	O
evaluating	O
difficult	O
integrals	O
(	O
see	O
also	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
)	O
.	O
</s>
<s>
In	O
practice	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
are	O
used	O
for	O
European-style	O
derivatives	O
involving	O
at	O
least	O
three	O
variables	O
(	O
more	O
direct	O
methods	O
involving	O
numerical	B-Algorithm
integration	I-Algorithm
can	O
usually	O
be	O
used	O
for	O
those	O
problems	O
with	O
only	O
one	O
or	O
two	O
underlyings	O
.	O
</s>
<s>
See	O
Monte	B-Algorithm
Carlo	I-Algorithm
option	I-Algorithm
model	I-Algorithm
.	O
</s>
<s>
This	O
can	O
be	O
a	O
time-consuming	O
process	O
(	O
an	O
entire	O
Monte	B-Algorithm
Carlo	I-Algorithm
run	I-Algorithm
must	O
be	O
performed	O
for	O
each	O
"	O
bump	O
"	O
or	O
small	O
change	O
in	O
input	O
parameters	O
)	O
.	O
</s>
<s>
Practitioners	O
regard	O
these	O
points	O
as	O
a	O
key	O
problem	O
with	O
using	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
This	O
state	O
of	O
affairs	O
can	O
be	O
mitigated	O
by	O
variance	B-Algorithm
reduction	I-Algorithm
techniques	O
.	O
</s>
<s>
It	O
is	O
also	O
natural	O
to	O
use	O
a	O
control	B-Algorithm
variate	I-Algorithm
.	O
</s>
<s>
Therefore	O
,	O
a	O
standard	O
way	O
of	O
choosing	O
the	O
derivative	O
I	O
consists	O
in	O
choosing	O
a	O
replicating	O
portfolios	O
of	O
options	O
for	O
H	O
.	O
In	O
practice	O
,	O
one	O
will	O
price	O
H	O
without	O
variance	B-Algorithm
reduction	I-Algorithm
,	O
calculate	O
deltas	O
and	O
vegas	O
,	O
and	O
then	O
use	O
a	O
combination	O
of	O
calls	O
and	O
puts	O
that	O
have	O
the	O
same	O
deltas	O
and	O
vegas	O
as	O
control	B-Algorithm
variate	I-Algorithm
.	O
</s>
<s>
When	O
calculating	O
the	O
delta	O
using	O
a	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
,	O
the	O
most	O
straightforward	O
way	O
is	O
the	O
black-box	O
technique	O
consisting	O
in	O
doing	O
a	O
Monte	O
Carlo	O
on	O
the	O
original	O
market	O
data	O
and	O
another	O
one	O
on	O
the	O
changed	O
market	O
data	O
,	O
and	O
calculate	O
the	O
risk	O
by	O
doing	O
the	O
difference	O
.	O
</s>
