A well known model for financial option pricing is a GBM with constant volatility, it has a log-normal price distribution with density,
However, the market
implied volatilities of stock index options often show "the
volatility smile", which decreases with the strike level, and
increases with the time to maturity . There are various
proposed extensions of this GBM model to account for "the
volatility smile". One approach is to incorporate a stochastic
volatility factor, Hull and White (1987); another allows for
discontinuous jumps in the stock price, Merton (1976).
However, these extensions cause several practical difficulties.
For example, they violate the risk-neutral condition. The IBT
technique proposed by Rubinstein (1994), Derman and Kani (1994),
Dupire (1994), and Barle and Cakici (1998) account for this
phenomenon. These papers assume the stock prices in the future are
generated by a modified random walk where the underlying asset has
a variable volatility that depends on both stock price and time.
Since the implied binomial trees allow for non-constant volatility
, they are in fact modifications of
the original Cox, Ross and Rubinstein (1979) binomial trees. The IBT
construction uses the observable market option prices in order to
estimate the implied distribution. It is therefore nonparametric
in nature. Alternative approaches may be based on the kernel
method, Aït-Sahalia, and Lo (1998), nonparametric constrained least
squares, Härdle and Yatchew (2001), and curve-fitting methods,
Jackwerth and Rubinstein (1996).
The CRR binomial tree is the discrete implementation of the GBM process
In the implied binomial tree framework, stock prices, transition probabilities, and Arrow-Debreu prices (discounted risk-neutral probabilities, see Chapter 8) at each node are calculated iteratively level by level.
Suppose we want to build an IBT on the time interval with equally spaced
levels,
apart. At
, is the current
price of the underlying, and there are
nodes at the
th
level of the tree. Let
be the stock price of the
th
node at the
th level,
and
the forward price at level
of
at
level
, and
the transition probability of making a
transition from node
to node
.
Figure 7.1 illustrates the construction of an IBT.
We assume the forward price
satisfies the risk-neutral condition:
Thus the transition probability can be obtained from the following equation:
The Arrow-Debreu price
,
is the price of an option that pays 1 unit payoff in one and only
one state
at
th level, and otherwise pays 0. In general,
Arrow-Debreu prices can be obtained by the iterative formula,
where
as a definition.
We give an example to illustrate the calculation of Arrow-Debreu
prices in a CRR Binomial tree. Suppose that the current value of
the underlying , time to maturity
years,
year, constant volatility
, and riskless
interest rate
, and
. The Arrow-Debreu price tree
can be calculated from the stock price tree:
stock price 122.15 110.52 100.00 100.00 90.48 81.88 Arrow-Debreu price 0.37 0.61 1.00 0.44 0.36 0.13For example, using the CRR method,
Option prices in the Black-Scholes framework are given by:
There are parameters which define the transition from
the
th to the
th level of the tree, i.e.,
stock
prices of the nodes at the
th level, and
transition
probabilities. Suppose
parameters corresponding to the
th level are known, the
and
corresponding
to the
th level can be calculated depending on the
following principles:
We always start from the center nodes in one level, if is
even, define
, for
, and if
is
odd, start from the two central nodes
and
for
, and suppose
, which
adjusts the logarithmic spacing between
and
to be the same as that between
and
. This principle yields the calculation formula of
, see Derman and Kani (1994),
Once we have the initial nodes' stock prices, according to the
relationships among the different parameters, we can continue to
calculate those at higher nodes
and
transition probabilities one by one using the formula:
Similarly, we are able to continue to calculate the parameters at
lower nodes
according to the following
recursion:
In order to avoid arbitrage, the transition probability
at any node should lie between 0 and 1, it makes therefore sense
to limit the estimated stock prices
In fact, the product of the Arrow-Debreu prices
at the
th
level with the influence of interest rate
can be considered as a discrete estimation of the implied
distribution, the SPD,
at
. In the case of the GBM model with
constant volatility, this density is corresponding to
(7.1).
After the construction of an IBT, we know
all stock prices, transition probabilities, and Arrow-Debreu
prices at any node in the tree. We are thus able to calculate the
implied local volatility
(which describes the structure of the second moment of the
underlying process) at any level
as a discrete estimation of
the following conditional variance at
. Under the risk-neutral assumption
In the IBT construction, the discrete estimation can be calculated as:
Analogously, we can calculate the implied local volatility at different
times. In general, if we have calculated the
transition probabilities
from the node
to the nodes
, then with
Notice that the instantaneous volatility function used in (7.3) is different from the BS implied volatility function defined in (7.16), but in the GBM they are identical.
If we choose
small enough, we obtain the estimated SPD
at fixed time to maturity, and the distribution of implied local volatility
. Notice that the BS implied volatility
(which assumes Black-Scholes model is established (at least
locally)) and implied local volatility
is
different, they have different parameters, and describe different
characteristics of the second moment.
Barle and Cakici (1998) proposed an improvement of the Derman and
Kani construction. The major modification is the choice of the
stock price of the central nodes in the tree: their algorithm
takes the riskless interest rate into account. If is odd,
then
for
, if
is even, then start from the two central
nodes
and
for
, and suppose
. Thus
can be
calculated as:
![]() |
(7.18) |
where is defined as in the Derman and Kani algorithm,
and the
is
After stock prices of the initial nodes are obtained, then continue to calculate
those at higher nodes
and transition
probabilities one by one using the following recursion:
![]() |
(7.20) |
where is as in (7.19),
is defined as
in (7.5).
Similarly, continue to calculate the parameters iteratively at
lower nodes
.
![]() |
(7.21) |
The balancing inequality (7.15) and a redefinition are still used in the Barle and Cakici
algorithm for avoiding arbitrage: the algorithm uses the average
of and
as the re-estimation of
.