While the Black-Scholes model assumes that an underlying asset follows
a geometric Brownian motion (6.1) with a constant volatility,
more complex models assume that the underlying
follows a process with a price- and time-varying volatility
.
See Dupire (1994) and Fengler, Härdle, and Villa (2003) for details and related evidence.
Such a process can be expressed by the following stochastic differential equation:
Derman and Kani (1994) show that it is possible to determine
directly from the market prices of liquidly traded options. Further, they
use this volatility
to construct an implied binomial tree (IBT), which
is a natural discrete representation of a non-lognormal evolution process
of the underlying prices.
In general, we can use - instead of an IBT - any (higher-order)
multinomial tree for the discretization of process (6.5).
Nevertheless, as the time step tends towards zero, all of them converge to the
same continuous process (Hull and White; 1990). Thus, IBTs are among all implied
multinomial trees minimal in the sense that they have only one degree of
freedom - the arbitrary choice of the central node at each level of the tree.
Although one may feel now that binomial trees are sufficient, some higher-order
trees could be more useful because they allow for a more flexible
discretization in the sense that transition probabilities and probability
distributions can vary as smoothly as possible across a tree. This is
especially important when the market option prices are inaccurate because of
inefficiency, market frictions, and so on.
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At the end of this section, let us to recall the concept of Arrow-Debreu
prices, which is closely related to multinomial trees and becomes very useful
in subsequent derivations (Section 6.3). Let denote the
th (highest) node in the
th time level of a tree. The Arrow-Debreu price
at node
of a tree is computed as the sum of the product
of the risklessly discounted transition probabilities over all paths starting
in the root of the tree and leading to node
. Hence, the Arrow-Debreu
price of the root is equal to one and the Arrow-Debreu prices at the final
level of a (multinomial) tree form a discrete approximation of the state price
density. Notice that these prices are discounted, and thus, the
risk-neutral probability corresponding to each node (at the final level) should
be calculated as the product of the Arrow-Debreu price and the capitalizing factor
.