6.2 Trees and Implied Trees

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 $ \sigma(S,t)$. 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:

$\displaystyle \frac{dS_t}{S_t}$ $\displaystyle =$ $\displaystyle \mu dt + \sigma(S,t) dW_t.$ (6.5)

This approach ensures that the valuation of an option remains preference-free, that is, all uncertainty is in the spot price, and thus, we can hedge options using the underlying.

Derman and Kani (1994) show that it is possible to determine $ \sigma(S,t)$ directly from the market prices of liquidly traded options. Further, they use this volatility $ \sigma(S,t)$ 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.

Figure 6.3: Computing the Arrow-Debreu price in a binomial tree. The bold lines with arrows depict all (three) possible path from the root of the tree to point A.

\includegraphics[width=1.0\defpicwidth]{ITT_fig05-ad.ps}

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 $ (n,i)$ denote the $ i$th (highest) node in the $ n$th time level of a tree. The Arrow-Debreu price $ \lambda_{n,i}$ at node $ (n,i)$ 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 $ (n,i)$. 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 $ e^{rT}$.