The example used here to show the procedure of generating the IBT,
is taken from Derman and Kani (1994). Assume that the current
value of the stock is , the annually compounded riskless
interest rate is
per year for all time expirations, the
stock has zero dividend. The annual BS implied volatility of an
at-the-money call is assumed to be
, and the BS
implied volatility increases (decreases) linearly by 0.5
percentage points with every 10 point drop (rise) in the strike.
From the assumptions, we see that
.
In order to investigate the precision of the SPD estimation obtained from
the IBT, we give a simulation example assuming that the stock
price process is generated by the stochastic differential equation
model (7.3), with an instantaneous local volatility function
,
. We may
then easily compare the SPD estimations obtained from the two
different methods.
With the XploRe quantlet XFGIBT01.xpl, using the assumption on the BS implied volatility surface, we obtain the following one year stock price implied binomial tree, transition probability tree, and Arrow-Debreu price tree.
Derman and Kani one year (four step) implied binomial tree stock price 119.91 115.06 110.04 110.06 105.13 105.13 100.00 100.00 100.00 95.12 95.12 89.93 89.92 85.22 80.01
transition probability 0.60 0.58 0.59 0.59 0.56 0.56 0.59 0.59 0.54 0.59
Arrow-Debreu price 0.111 0.187 0.327 0.312 0.559 0.405 1.000 0.480 0.343 0.434 0.305 0.178 0.172 0.080 0.033
This IBT is corresponding to year, and
year,
which shows the stock prices, and the elements at the
th column
are corresponding to the stock prices of the nodes at the
th level in the tree. The second one, its
element
is corresponding to the transition probability from the node
to the nodes
. The third tree contains the
Arrow-Debreu prices of the nodes. Using the stock prices together
with Arrow-Debreu prices of the nodes at the final level, a
discrete approximation of the implied distribution can be
obtained. Notice that by the definition of the Arrow-Debreu price,
the risk neutral probability corresponding to each node should be
calculated as the product of the Arrow-Debreu price and the factor
.
If we choose small enough time steps, we obtain
the estimation of the implied price distribution and the implied
local volatility surface
. We still use the
same assumption on the BS implied volatility surface as above
here, which means
, and
assume
year.
Two figures are generated by running the quantlet
XFGIBT02.xpl, Figure 7.2 shows
the plot of the SPD estimation resulting
from fitting an implied five-year tree with 20 levels. The implied
local volatilities
in the implied tree at
different time to maturity and stock price
levels is shown in Figure 7.3, which obviously
decreases with the stock price and increases with time to maturity as expected.
The Barle and Cakici algorithm can be applied in analogy to
Derman and Kani's. The
XploRe
quantlets used here are similar to
those presented in Section 7.2.1, one has to replace the
quantlet
IBTdk
by
IBTdc
. The
following figure displays the one-year (four step) stock price
tree, transition probability tree, and Arrow-Debreu tree. Figure
7.4 presents the plot of the estimated SPD by fitting
a five year implied binomial tree with 20 levels to the
volatility smile using Barle and Cakici algorithm, and Figure
7.5, shows the characteristics of the implied
local volatility surface of the generated IBT, decreases with the
stock price, and increases with time.
Barle and Cakici one year implied binomial tree stock price 123.85 117.02 112.23 112.93 104.84 107.03 100.00 101.51 103.05 96.83 97.73 90.53 93.08 87.60 82.00 transition probability 0.46 0.61 0.38 0.48 0.49 0.49 0.64 0.54 0.36 0.57 Arrow-Debreu price 0.050 0.111 0.185 0.240 0.486 0.373 1.000 0.619 0.394 0.506 0.378 0.181 0.237 0.116 0.050
We now compare the SPD estimation at the fifth year obtained by
the two IBT methods with the estimated density function of the
Monte-Carlo simulation of
generated from the model
(7.3), where
,
. We use the Milstein scheme,
Kloeden, Platen and Schurz (1994) to perform the discrete time
approximation in (7.3). It has strong convergence rate
. We have set the time step with
here.
In order to construct the IBT, we calculate the option prices corresponding to each node at the implied tree according to their definition by Monte-Carlo simulation.
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Here we use the quantlets
XFGIBTcdk.xpl
and
XFGIBTcbc.xpl
. These two are used to construct the IBT directly
from the option price function, not starting from the BS implied
volatility surface as in quantlets
IBTdk
and
IBTbc
. In the data file
"IBTmcsimulation20.dat", there are 1000 Monte-Carlo
simulation samples for each
in the diffusion model
(7.3), for
year,
, from which we
calculate the simulated values of the option prices according to
its theoretical definition and estimate the density of
year as the SPD estimation at the fifth year.
From the estimated distribution shown in the Figures 7.2.3, we observe their deviation from the log-normal characteristics according to their skewness and kurtosis. The SPD estimation obtained from the two IBT methods coincides with the estimation obtained from the Monte-Carlo simulation well, the difference between the estimations obtained from the two IBTs is not very large.
On the other hand, we can also estimate the implied local volatility surface from the implied binomial tree, and compare it with the one obtained by the simulation. Compare Figure 7.7 and Figure 7.8 with Figure 7.9, and notice that in the first two figures, some edge values cannot be obtained directly from the five-year IBT. However, the three implied local volatility surface plots all actually coincide with the volatility smile characteristic, the implied local volatility of the out-the-money options decreases with the increasing stock price, and increase with time.
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We use the data file "IBTmcsimulation50.dat" to obtain an
estimated BS implied volatility surface. There are 1000
Monte-Carlo simulation samples for each in the diffusion
model (7.3), for
year in it,
, because
we can calculate the BS implied volatility corresponding to
different strike prices and time to maturities after we have the
estimated option prices corresponding to these strike price and
time to maturity levels. Figure 7.10 shows that
the BS implied volatility surface of our example reflects the
characteristics that the BS implied volatility decrease with the
strike price. But this BS implied volatility surface does not
change with time a lot, which is probably due to our assumption
about the local instantaneous volatility function, which only
changes with the stock price.