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7 . 2 . 1 Various rainbow optionsPut on a basketChar Count = 0 Dynamic 2 D Problems 1 9 7 Data for a put
Various rainbow optionsPut on a basketChar CountDynamic D Problems Data for a put on a basketValue For options on baskets, at present there is no known analytical solution Hunziker and KochMedina, p Therefore, this option has to be priced with a numerical device or anapproximation Huynh; Milevsky and Posner, ; Zhang, b Chapter Thebasic idea of these approximations is to combine the volatilities of the underlying and theircorrelations to a single volatility of the basket. This basket is then treated as a single underlying. Using this approach, the problem of pricing an option on a basket is reduced to pricingan option on a single equity. Accordingly, the models to price options with exotic featurescan also be applied to options on baskets. Precise error estimates are generally not providedHunziker and KochMedina, p Here, however, we price options on baskets usinga multidimensional PDE. For a plain vanilla put we first derive the boundary conditions. Asone or both underlyings become worth much more than the strike, the price of the options goesto zero. As the price of the first underlying is zero, while the second is positive, the value of theoption behaves like the value of a plain vanilla put on a single equity. Therefore, the boundaryconditions at S and S are the timedependent solution to the basic BlackScholesproblem of pricing a put Hunziker and KochMedina, p with strikes at E and wE respectively. Together with the data shown in Table this becomes the following PDE wproblemSVSVSS V S S S S rDS V rDS V rVV S S tVSST maxE wSwS V S t g S wE t VStgSwE tVSt VSt WYdrvWYTopper December : Char Count Financial Engineering with Finite ElementsTable Results for a put option on a basket computed on a square domain VolatilityTime to maturity PremiumTreeFEM Difference TreeFEM Difference TreeFEM DifferenceTreeFEM Difference TreeFEM Difference TreeFEM DifferenceTreeFEM Difference TreeFEM Difference TreeFEM Difference Ewet al The results are shown in Table As an alternative to pricing this option on a square domain, we also price it on a trianglethe results are shown in Table Basically, we cut off the section of the domain where the option is totally out of the money and therefore worthless. This, of course, reduces computing time. The boundary conditions to are replaced by:Here, the g functions denote a plain vanilla European put with strikes E and wand appropriate volatilities. We compute the cumulative normal distributions in Equations and with an approximation with fourdigit accuracy from Hull To compare the results, we also price the put on a basket using a twodimensional binomial tree, as implemented by Haug This tree can be interpreted as a simple explicit finite difference scheme; see WilmottVSt gS wE tVStgSwE tVSSt on SmaxSmax WYdrvWYTopper December : Char CountDynamic D Problems Table Results for a put option on a basket computed on a triangular domain VolatilityTime to maturity PremiumTreeFEM Difference TreeFEM Difference TreeFEM DifferenceTreeFEM Difference TreeFEM Difference TreeFEM DifferenceTreeFEM Difference TreeFEM Difference TreeFEM Difference
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