We now use the IBT to forecast the future price distribution of
the real stock market data. We use DAX index option prices data at
January 4, 1999, which are included in
M
D
*BASE
, a database located at CASE
(Center for Applied Statistics and Economics) at Humboldt-Universität
zu Berlin, and provide some dataset for
demonstration purposes. In the following program, we estimate the
BS implied volatility surface first, while the quantlet
volsurf
, Fengler, Härdle and Villa (2001), is used to
obtain this estimation from the market option prices, then
construct the IBT using Derman and Kani method and calculate the
interpolated option prices using CRR binomial tree method. Fitting
the function of option prices directly from the market option
prices is hardly ever attempted since the function approaches a
value of zero for very high strike prices and option prices are
bounded by non-arbitrage conditions.
Figure 7.11 shows the price distribution estimation
obtained by the Barle and Cakici IBT, for year.
Obviously, the estimated SPD by the Derman and Kani IBT can be
obtained similarly.
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From Figure 7.12 we conclude that the SPD
estimated by the Derman and Kani IBT and the one
obtained by Barle and Cakici IBT can be used to forecast future
SPD. The SPD estimated by different methods sometimes have
deviations on skewness and kurtosis. In fact the detection of the
difference between the historical time series SPD estimation and
the SPD recovered from daily option prices may be used as trading
rules, see Table 7.1 and Chapter 9. In Table
7.1, SPD estimated from daily option prices data set is
expressed by and the time series SPD is
. A far out of
the money (OTM) call/put is defined as one whose exercise price
is
higher (lower) than the future price. While a near OTM
call/put is defined as one whose exercise price is
higher
(lower) but
lower(higher)than the future price. When
skew(
)
skew(
), agents apparently assign a lower
probability to high outcomes of the underlying than would be
justified by the time series SPD (see Figure 7.13). Since
for call options only the right `tail' of the support determines
the theoretical price the latter is smaller than the price implied
by diffusion process using the time series SPD. That is we buy
calls. The same reason applies to put options.
From the simulations and real data example, we
find that the implied binomial tree is an easy way to assess the
future stock prices, capture the term structure of the underlying
asset, and replicate the volatility smile. But the algorithms
still have some deficiencies. When the time step is chosen too
small, negative transition probabilities are encountered more and
more often. The modification of these values loses the information
about the smile at the corresponding nodes. The Barle and Cakici
algorithm is a better choice when the interest rate is high.Figure
7.15 shows the deviation of the two methods under the
situation that . When the interest rate is a little higher,
Barle and Cakici algorithm still can be used to construct the IBT
while Derman and Kani's cannot work any more. The times of the
negative probabilities appear are fewer than Derman and Kani
construction (see Jackwerth (1999)).
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Besides its basic purpose of pricing derivatives in consistency with the market prices, IBT is useful for other kinds of analysis, such as hedging and calculating of implied probability distributions and volatility surfaces. It estimate the future price distribution according to the historical data. On the practical application aspect, the reliability of the approach depends critically on the quality of the estimation of the dynamics of the underlying price process, such as BS implied volatility surface obtained from the market option prices.
The IBT can be used to produce recombining and arbitrage-free binomial trees to describe stochastic processes with variable volatility. However, some serious limitations such as negative probabilities, even though most of them appeared at the edge of the trees. Overriding them causes loss of the information about the smile at the corresponding nodes. These defects are a consequence of the requirement that a continuous diffusion is approximated by a binomial process. Relaxation of this requirement, using multinomial trees or varinomial trees is possible.