A trinomial tree with levels is a set of nodes
(representing the underlying price),
where
is the level number and
indexes nodes within a level.
Being at a node
, one can move to one of three nodes (see Figure 6.4a):
(i) to the upper node with value
with probability
;
(ii) to the lower node with value
with probability
;
and (iii) to the middle node with value
with probability
.
For the sake of brevity, we omit the level index
from transition probabilities
unless they refer to a specific level; that is, we write
and
instead of
and
unless the level has to be specified. Similarly, let us denote
the nodes in the new level with capital letters:
(=
),
(=
) and
(=
),
respectively (see Figure 6.4b).
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Starting from a node at time
, there are five unknown
parameters: two transition probabilities
and
and three
prices
,
, and
at new nodes.
To determine them, we need to introduce the notation and main requirements a tree
should satisfy.
First, let
denote the known forward price of the spot price
and
the known Arrow-Debreu price at node
.
The Arrow-Debreu prices for a trinomial tree can be obtained by the
following iterative formulas:
An implied tree provides a discrete representation of the evolution process of underlying prices. To capture and model the underlying price correctly, we desire that an implied tree:
Consequently, we have two constraints (6.12) and (6.13) for
five unknown parameters, and therefore, there is no unique implied trinomial tree.
On the other hand, all trees satisfying these constraints are equivalent
in the sense that as the time spacing tends to zero, all these
trees converge to the same continous process. A common method for constructing an ITT
is to choose first freely the underlying prices and then to solve
equations (6.12) and (6.13) for the transition
probabilities
and
. Afterwards one only has to ensure that these
probabilities do not violate the above mentioned Condition 3. Apparently, using an ITT instead of an IBT gives us additional degrees of freedom.
This allows us to better fit the volatility smile, especially when inconsistent
or arbitrage-violating market option prices make a consistent tree impossible.
Note, however, that even though the constructed tree is consistent, other
difficulties can arise when its local volatility and probability distributions
are jagged and ``implausible.''
There are several methods we can use to construct an initial state space. Let us first discuss a construction of a constant-volatility trinomial tree, which forms a base for an implied trinomial tree. As already mentioned, binomial and trinomial discretization of the constant-volatility Black-Scholes model have the same continous limit, and therefore, are equivalent. Hence, we can start from a constant-volatility CRR binomial tree and then combine two steps of this tree into a single step of a new trinomial tree. This is illustrated in Figure 6.5, where thin lines correspond to the original binomial tree and the thick lines to the constructed trinomial tree.
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Consequently, using formulas (6.2) and (6.3), we can
derive the following expressions for the nodes of the constructed trinomial tree:
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When the implied volatility varies only slowly with strike and expiration, the
regular state space with a uniform mesh size, as described above, is adequate for
constructing ITT models. On the other hand, if the volatility varies significantly
with strike or time to maturity, we should choose a state space reflecting these
properties. Assuming that the volatility is separable in time and stock price,
, an ITT state space with a proper skew and term
structure can be constructed in four steps.
First, we build a regular trinomial lattice with a constant time spacing
and a constant price spacing
as described above. Additionally, we
assume that all interest rates and dividends are equal to zero.
Second, we modify at different time points.
Let us denote the original equally spaced time points
. We can then find the unknown scaled times
by solving the following set
of non-linear equations:
Next, we change at different levels.
Denoting by
the original (known) underlying prices, we solve
for rescaled underlying prices
using
Finally, one can increase all node prices by a sufficiently large growth factor,
which removes forward prices violations, see Section 6.3.4.
Multiplying all zero-rate node prices at time
by
should be always sufficient.
Once the state space of an ITT is fixed, we can compute
the transition probabilities for all nodes at each tree level
.
Let
and
denote today's price of a standard European call
and put option, respectively, struck at
and expiring at
.
These values can be obtained by interpolating the smile surface at various strike
and time points. The values of these options given by the trinomial tree are the discounted
expectations of the pay-off functions:
for the call option and
for the put option at the node
. The expectation is taken
with respect to the probabilities of reaching each node, that is, with respect to
transition probabilities:
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Formulas (6.21)-(6.24) can unfortunately result in
transition probabilities which are negative or greater than one. This is
inconsistent with rational option prices and allows arbitrage. We actually
have to face two forms of this problem, see Figure 6.6 for
examples of such trees. First, we have to check that no forward
price at node
falls outside the range of
its daughter nodes at the level
:
.
This inconsistency is not difficult to overcome since we are free to choose the
state space. Thus, we can overwrite the nodes causing this problem.
Second, extremely small or large values of option prices, which would imply an extreme
value of local volatility, can also result in probabilities that are negative or larger
than one. In such a case, we have to overwrite the option prices which led to the
unacceptable probabilities.
Fortunately, the transition probabilities can be always corrected providing that
the corresponding state space does not violate the forward price condition
. Derman, Kani, and Chriss (1996) proposed to reduce
the troublesome nodes to binomial ones or to set