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within the framework, the binomial method can be used to value many types of exotic options. Knockout options, for example, can be valued using this technique. One simply imposes a different check on the calculated option values at the nodes of the intermediate time steps between 0 and n, i.e., if the underlying asset price falls below the option s barrier, the option value at that node is set equal to 0. The method can also be extended to handle multiple sources of asset price uncertainty. Boyle, Evnine, and Gibbs adapt the binomial procedure to handle exotics with multiple sources of uncertainty, including options on the minimum and maximum, spread options, and so on.53 Trinomial method. The trinomial method is another popular lattice-based method. As outlined by Boyle, this method allows the asset to move up, move down, or stay the same at each time increment.54 Again, the parameters of the discrete distribution are chosen in a manner consistent with the lognormal distribution, and the procedure begins at the end of the option s life and works backward. By having three branches instead of two, the trinomial method provides greater accuracy than the binomial method for a given number of time steps. The cost, of course, is that the greater the number of branches, the slower the computational speed. Finite difference method. The explicit finite difference method was the first lattice-based procedure to be applied to option valuation. Schwartz applied it to warrants, and Brennan and Schwartz applied it to American-style put options on common stocks.55 The finite difference method is similar to the trinomial method in the sense that the asset price moves up, moves down, or stays the same at each time step during the option s life. The difference in the techniques arise only from how the price increments and the probabilities are set. In addition, finite difference methods calculate an entire rectangle of node values rather than simply a tree. Monte Carlo simulation. Boyle introduced Monte Carlo simulation to option valuation.56 Like the lattice-based procedures, the technique involves simulating possible paths that the asset price may take over the life of the option. Again, the simulation is performed in a manner consistent with the lognormal asset price process. To value a European-style option, each sample run is used to produce a terminal asset price, which, in turn, is used to determine the terminal option value. With repeated sample runs, a distribution of terminal options values is obtained, and the expected terminal option value may be calculated. This expected value is then discounted to the present to obtain the option valuation. An advantage of the Monte Carlo method is that the
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degree of valuation error can be assessed directly, using the standard error of the estimate. The standard error equals the standard deviation of the terminal option values divided by the square root of the number of trials. Another advantage of the Monte Carlo technique is its flexibility. Because the path of the asset price beginning at time 0 and continuing throughout the life of the option is observed, the technique is well suited for handling barrier-style options, Asian-style options, Bermuda-style options, and the like. Moreover, it can easily be adapted to handle multiple sources of price uncertainty. The technique s chief disadvantage is that it can be applied only when the option payout does not depend on its value at future points in time. This eliminates the possibility of applying the technique to American-style option valuation, in which the decision to exercise early depends on the value of the option that will be forfeit. Compound option approximation. The quasi-analytical methods for option valuation are quite different from the procedures that attempt to describe asset price paths. Geske and Johnson, for example, use a Geske compound option model to develop an approximate value for an American-style option.57 The approach is intuitively appealing. An American-style option, after all, is a compound option with an infinite number of early exercise opportunities. While valuing an option in this way makes intuitive sense, the problem is intractable from a computational standpoint. The Geske-Johnson insight is that although we cannot value an option with an infinite number of early exercise opportunities, we can extrapolate its value by valuing a sequence of pseudo-American options with zero, one, two, and perhaps more early exercise opportunities at discrete, equally spaced intervals during the option s life. The advantage that this offers is that each of these options can be valued analytically. With each new option added to the sequence, however, the valuation of a higher-order multivariate normal integral is required. With no early exercise opportunities, only a univariate function is required. However, with one early exercise opportunity, a bivariate function is required; with two opportunities, a trivariate function is required, and so on. The more of these options used in the series, the greater the precision in approximating the limiting value of the sequence. The cost of increased precision is that higher-order multivariate integral valuations are time-consuming computationally. Quadratic approximation. Barone-Adesi and Whaley presented a quadratic approximation in 1987.58 Their approach, based on the
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