To illustrate the construction of an implied trinomial tree and its use, we consider here ITTs for two artificial implied-volatility functions and an implied-volatility function constructed from real data.
Let us consider a case where the volatility varies only slowly with respect to the
strike price and time to expiration (maturity). Assume that the current index
level is 100 points, the annual riskless interest rate is , and the dividend
yield equals
. The annualized Black-Scholes implied volatility is
assumed to be
, and additionally, it increases (decreases)
linearly by 10 basis points (i.e., 0.1%) with every 10 unit drop (rise) in the
strike price
; that is,
. To keep
the example simple, we consider three one-year steps.
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First, we construct the state space: a constant-volatility trinomial tree as
described in Section 6.3.2.
The first node at time , labeled
in Figure 6.7,
has the value of
, today's spot price.
The next three nodes at time
, are computed from equations
(6.14)-(6.16) and take values
,
, and
, respectively. In order to determine the transition
probabilities, we need to know the price
of a put option
struck at
and expiring one year from now. Since the
implied volatility of this option is
, we calculate its
price using a constant-volatility trinomial tree with the same
state space and find it to be
index points. Further, the forward price
corresponding to node
is
, where
denotes the continuous
interest rate and
the continuous dividend rate.
Hence, the transition probability of a down movement
computed from equation (6.23) is
Let us demonstrate the computation of one further node. Starting from node
in year
of Figure 6.7, the index level at this node is
and its forward price one year later is
. From this
node, the underlying can move to one of three future nodes at time
, with
prices
,
, and
.
The value of a call option struck at
and expiring at time
is
, corresponding to the implied volatility of
interpolated from the smile.
The Arrow-Debreu price computed from equation
(6.8) is
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The complete trees of transition probabilities, Arrow-Debreu prices, and local volatilities for this example are shown in Figures 6.8-6.10.
As already mentioned in Section 6.3.4, the transition
probabilities may fall out of the interval . For example,
let us slightly modify our previous example and assume
that the Black-Scholes volatility increases (decreases) linearly
0.5 percentage points with every 10 unit drop (rise) in the strike price
;
that is,
. In other words,
the volatility smile is now five times steeper than before.
Using the same state space as in the previous example, we find inadmissable
transition probabilities at nodes
and
, see Figures 6.11-6.13.
To overwrite them with plausible values, we used the strategy
described by (6.26) and (6.27) and
obtained reasonable results in the sense of the three conditions
stated on page
.
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Following the artificial examples, let us now demonstrate the ITT modeling for
a real data set, which consists of strike prices for DAX options with
maturities from two weeks to two months on January 4, 1999. Given such data, we
can firstly compute from the Black-Scholes equation (6.1) the
implied volatilities at various combinations of prices and maturities, that is,
we can construct the volatility smile. Next, we build and calibrate an ITT so
that it fits this smile. The procedure is analogous to the examples described
above - the only difference lies in replacing an artificial function
by an estimate of implied volatility
at each point
.
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For the purpose of demonstration, we build a three-level ITT with time step
of two weeks. First, we construct the state space
(Section 6.3.2) starting at time
with the spot price
and riskless interest rate
, see Figure 6.14. Further, we
have to compute the transition probabilities. Because option contracts are not
available for each combination of price and maturity, we use a nonparametric
smoothing procedure to model the whole volatility surface
as
employed by Aït-Sahalia, Wang, and Yared (2001) and Fengler, Härdle, and Villa (2003), for instance.
Given the data, some transition probabilities fall outside interval
;
they are depicted by dashed lines in Figure 6.14. Such
probabilities have to be corrected as described in Section 6.3.4 (there
are no forward price violations). The resulting local volatilities, which
reflect the volatility skew, are on Figure 6.15.
Probably the main result of this ITT model can be summarized by the state price density (the left panel of Figure 6.16). This density describes the price distribution given by the constructed ITT and smoothed by the Nadaraya-Watson estimator. Apparently, the estimated density is rather rough because we used just three steps in our tree. To get a smoother state-price density estimate, we doubled the number of steps; that is, we used six one-week steps instead of three two-week steps (see the right panel of Figure 6.16).
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Finally, it possible to use the constructed ITT to evaluate various DAX
options. For example, a European knock-out option gives the owner the same
rights as a standard European option as long as the index price does not
exceed or fall below some barrier
for the entire life of the knock-out option; see
Härdle et al. (2002) for details. So, let us compute the price of the knock-out-call
DAX option at maturity
weeks, strike price
, and barrier
. The option price at time
(
and
weeks) and stock price
will be denoted
.
At the maturity , the price is known:
. Thus,
and
, for instance. To compute the option price at
, one just has to
discount the conditional expectation of the option price at time