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Market Risk Code 128 Code Set B Decoder In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Making Code 128C In None Using Barcode creation for Software Control to generate, create Code 128A image in Software applications. 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 latticebased 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 latticebased procedure to be applied to option valuation. Schwartz applied it to warrants, and Brennan and Schwartz applied it to Americanstyle 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 latticebased 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 Europeanstyle 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 Recognize Code 128 In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. Code 128 Code Set C Printer In Visual C#.NET Using Barcode printer for Visual Studio .NET Control to generate, create Code 128 image in .NET applications. 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 barrierstyle options, Asianstyle options, Bermudastyle 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 Americanstyle option valuation, in which the decision to exercise early depends on the value of the option that will be forfeit. Compound option approximation. The quasianalytical 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 Americanstyle option.57 The approach is intuitively appealing. An Americanstyle 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 GeskeJohnson 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 pseudoAmerican 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 higherorder 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 higherorder multivariate integral valuations are timeconsuming computationally. Quadratic approximation. BaroneAdesi and Whaley presented a quadratic approximation in 1987.58 Their approach, based on the Creating ANSI/AIM Code 128 In .NET Using Barcode encoder for ASP.NET Control to generate, create ANSI/AIM Code 128 image in ASP.NET applications. Code 128A Maker In .NET Using Barcode maker for VS .NET Control to generate, create Code 128A image in .NET framework applications. Code128 Creator In Visual Basic .NET Using Barcode creator for .NET Control to generate, create Code128 image in .NET framework applications. 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