<s>
In	O
finance	O
,	O
a	O
lattice	B-Application
model	I-Application
is	O
a	O
technique	O
applied	O
to	O
the	O
valuation	O
of	O
derivatives	O
,	O
where	O
a	O
discrete	O
time	O
model	O
is	O
required	O
.	O
</s>
<s>
The	O
method	O
is	O
also	O
used	O
for	O
valuing	O
certain	O
exotic	O
options	O
,	O
where	O
because	O
of	O
path	O
dependence	O
in	O
the	O
payoff	O
,	O
Monte	B-Algorithm
Carlo	I-Algorithm
methods	I-Algorithm
for	I-Algorithm
option	I-Algorithm
pricing	I-Algorithm
fail	O
to	O
account	O
for	O
optimal	O
decisions	O
to	O
terminate	O
the	O
derivative	O
by	O
early	O
exercise	O
,	O
though	O
methods	O
now	O
exist	O
for	O
solving	O
this	O
problem	O
.	O
</s>
<s>
Some	O
exotic	O
options	O
,	O
such	O
as	O
barrier	O
options	O
,	O
are	O
also	O
easily	O
modeled	O
here	O
;	O
for	O
other	O
Path-Dependent	O
Options	O
,	O
simulation	B-Algorithm
would	O
be	O
preferred	O
.	O
</s>
<s>
The	O
simplest	O
lattice	B-Application
model	I-Application
is	O
the	O
binomial	B-Application
options	I-Application
pricing	I-Application
model	I-Application
;	O
the	O
standard	O
(	O
"	O
canonical	O
"	O
)	O
method	O
is	O
that	O
proposed	O
by	O
Cox	O
,	O
Ross	O
and	O
Rubinstein	O
(	O
CRR	O
)	O
in	O
1979	O
;	O
see	O
diagram	O
for	O
formulae	O
.	O
</s>
<s>
In	B-Algorithm
the	I-Algorithm
limit	I-Algorithm
,	O
as	O
the	O
number	O
of	O
time-steps	O
increases	O
,	O
these	O
converge	B-Algorithm
to	O
the	O
Log-normal	O
distribution	O
,	O
and	O
hence	O
produce	O
the	O
"	O
same	O
"	O
option	O
price	O
as	O
Black-Scholes	O
:	O
to	O
achieve	O
this	O
,	O
these	O
will	O
variously	O
seek	O
to	O
agree	O
with	O
the	O
underlying	O
's	O
central	B-General_Concept
moments	I-General_Concept
,	O
raw	O
moments	O
and	O
/	O
or	O
log-moments	O
at	O
each	O
time-step	O
,	O
as	O
measured	O
discretely	O
.	O
</s>
<s>
More	O
recent	O
models	O
,	O
in	O
fact	O
,	O
are	O
designed	O
around	O
direct	O
convergence	B-Algorithm
to	O
Black-Scholes	O
.	O
</s>
<s>
A	O
variant	O
on	O
the	O
Binomial	O
,	O
is	O
the	O
Trinomial	B-Application
tree	I-Application
,	O
developed	O
by	O
Phelim	O
Boyle	O
in	O
1986	O
.	O
</s>
<s>
For	O
vanilla	O
options	O
,	O
as	O
the	O
number	O
of	O
steps	O
increases	O
,	O
the	O
results	O
rapidly	O
converge	B-Algorithm
,	O
and	O
the	O
binomial	O
model	O
is	O
then	O
preferred	O
due	O
to	O
its	O
simpler	O
implementation	O
.	O
</s>
<s>
There	O
exist	O
both	O
implied	O
binomial	O
trees	O
,	O
often	O
Rubinstein	O
IBTs	O
(	O
R-IBT	O
)	O
,	O
and	O
implied	O
trinomial	B-Application
trees	I-Application
,	O
often	O
Derman-Kani-Chriss	O
(	O
DKC	O
;	O
superseding	O
the	O
DK-IBT	O
)	O
.	O
</s>
<s>
allow	O
for	O
an	O
analyst-specified	O
skew	B-General_Concept
and	O
kurtosis	B-Error_Name
in	O
spot	O
price	O
returns	O
;	O
see	O
Edgeworth	O
series	O
.	O
</s>
<s>
This	O
approach	O
is	O
limited	O
as	O
to	O
the	O
set	O
of	O
skewness	B-General_Concept
and	O
kurtosis	B-Error_Name
pairs	O
for	O
which	O
valid	O
distributions	O
are	O
available	O
.	O
</s>
<s>
As	O
for	O
equity	O
,	O
trinomial	B-Application
trees	I-Application
may	O
also	O
be	O
employed	O
for	O
these	O
models	O
;	O
this	O
is	O
usually	O
the	O
case	O
for	O
Hull-White	O
trees	O
.	O
</s>
<s>
This	O
distinction	O
:	O
for	O
equilibrium-based	O
models	O
the	O
yield	B-Algorithm
curve	I-Algorithm
is	O
an	O
output	O
from	O
the	O
model	O
,	O
while	O
for	O
arbitrage-free	O
models	O
the	O
yield	B-Algorithm
curve	I-Algorithm
is	O
an	O
input	O
to	O
the	O
model	O
.	O
</s>
<s>
In	O
the	O
former	O
case	O
,	O
the	O
approach	O
is	O
to	O
"	O
calibrate	O
"	O
the	O
model	O
parameters	O
,	O
such	O
that	O
bond	O
prices	O
produced	O
by	O
the	O
model	O
,	O
in	O
its	O
continuous	O
form	O
,	O
best	B-Algorithm
fit	I-Algorithm
observed	O
market	O
prices	O
.	O
</s>
<s>
the	O
yield	B-Algorithm
curve	I-Algorithm
)	O
,	O
and	O
the	O
corresponding	O
volatility	O
structure	O
.	O
</s>
<s>
Here	O
,	O
calibration	O
means	O
that	O
the	O
interest-rate-tree	O
reproduces	O
the	O
prices	O
of	O
the	O
zero-coupon	O
bonds	O
—	O
and	O
any	O
other	O
interest-rate	O
sensitive	O
securities	O
—	O
used	O
in	O
constructing	O
the	O
yield	B-Algorithm
curve	I-Algorithm
;	O
note	O
the	O
parallel	O
to	O
the	O
implied	O
trees	O
for	O
equity	O
above	O
,	O
and	O
compare	O
Bootstrapping	O
(	O
finance	O
)	O
.	O
</s>
<s>
Lattice	B-Application
models	I-Application
have	O
been	O
developed	O
for	O
equity	O
analysis	O
here	O
,	O
particularly	O
as	O
relates	O
to	O
distressed	O
firms	O
.	O
</s>
<s>
Here	O
,	O
similar	O
to	O
rho	O
and	O
vega	O
above	O
,	O
the	O
interest	O
rate	O
tree	O
is	O
rebuilt	O
for	O
an	O
upward	O
and	O
then	O
downward	O
parallel	O
shift	O
in	O
the	O
yield	B-Algorithm
curve	I-Algorithm
and	O
these	O
measures	O
are	O
calculated	O
numerically	O
given	O
the	O
corresponding	O
changes	O
in	O
bond	O
value	O
.	O
</s>
