$7\mathrm{.}2\mathrm{.}1$ Various rainbow optionsPut on a basketChar Count$=0$Dynamic $2$D Problems $197$ Data for a put on a basketValue$181201400\mathrm{.}10\mathrm{.}00\mathrm{.}5$ For options on baskets, at present there is no known analytical solution $($Hunziker and Koch$-$Medina, $1996,$ p$.161).$ Therefore, this option has to be priced with a numerical device or anapproximation $($Huynh$,1994$; Milevsky and Posner, $1998$; Zhang, $1998$b$,$ Chapter $27).$ 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 underly$-$ing. 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 provided$($Hunziker and Koch$-$Medina, $1996,$ p$.163).$ 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$1=0$ and S$2=0$ are the $($time$-$dependent$)$ solution to the basic BlackScholesproblem of pricing a put $($Hunziker and Koch$-$Medina, $1996,$ p$.162)$ with strikes at E and w$2$E $,$ respectively. Together with the data shown in Table $7\mathrm{.}1,$ this becomes the following PDE w$1$problem$\mathrm{.}12$S$22$V$+12$S$22$V$+$SS $2$V $+211$ S $12222$ S $221212$ S $1$ S $2($rD$)$S V $+($rD$)$S V $=$rVV $11$ S$122$ S$2$ tV$($S$1,$S$2,$T$)=$ max$(0,$E $($w$1$S$1+$w$2$S$2))$ V $($ S $1,0,$ t $)=$ g $($ S $1,$ wE $,$ t $)2$V$(0,$S$2,$t$)=$g$($S$2,$wE $,$t$)(7\mathrm{.}9)(7\mathrm{.}10)(7\mathrm{.}11)(7\mathrm{.}12)(7\mathrm{.}13)(7\mathrm{.}14)$V$(100,$S$2,$t$)=0$ V$($S$1,100,$t$)=01$ WY$061-07$drvWY$061-$Topper December $19,200411$:$52$ Char Count$=0198$ Financial Engineering with Finite ElementsTable $7\mathrm{.}2$ Results for a put option on a basket computed on a square domain VolatilityTime to maturity $0\mathrm{.}50\mathrm{.}95430\mathrm{.}95430\mathrm{.}0022\%1\mathrm{.}47561\mathrm{.}47640\mathrm{.}0512\%2\mathrm{.}01862\mathrm{.}01870\mathrm{.}0041\%1\mathrm{.}41201\mathrm{.}41270\mathrm{.}0492\%1\mathrm{.}88351\mathrm{.}88330\mathrm{.}0125\%2\mathrm{.}39412\mathrm{.}39420\mathrm{.}0024\%1\mathrm{.}89411\mathrm{.}89480\mathrm{.}0395\%2\mathrm{.}33012\mathrm{.}32980\mathrm{.}0138\%2\mathrm{.}81122\mathrm{.}81190\mathrm{.}0021\%10\mathrm{.}10\mathrm{.}20\mathrm{.}320\mathrm{.}051\mathrm{.}80250\mathrm{.}11\mathrm{.}80650\mathrm{.}2204\%1\mathrm{.}83330\mathrm{.}21\mathrm{.}83410\mathrm{.}0473\%1\mathrm{.}91180\mathrm{.}31\mathrm{.}91380\mathrm{.}1034\%1\mathrm{.}82710\mathrm{.}11\mathrm{.}82750\mathrm{.}0236\%1\mathrm{.}88590\mathrm{.}21\mathrm{.}88560\mathrm{.}0076\%1\mathrm{.}98160\mathrm{.}31\mathrm{.}98300\mathrm{.}00602\%1\mathrm{.}89060\mathrm{.}11\mathrm{.}89150\mathrm{.}0451\%1\mathrm{.}96830\mathrm{.}21\mathrm{.}96870\mathrm{.}0210\%2\mathrm{.}07390\mathrm{.}32\mathrm{.}07470\mathrm{.}0360\%0\mathrm{.}950\mathrm{.}60430\mathrm{.}60350\mathrm{.}1352\%1\mathrm{.}24081\mathrm{.}24050\mathrm{.}0305\%1\mathrm{.}92651\mathrm{.}92700\mathrm{.}0242\%1\mathrm{.}16071\mathrm{.}16010\mathrm{.}0489\%1\mathrm{.}77581\mathrm{.}77540\mathrm{.}0202\%2\mathrm{.}43892\mathrm{.}43890\mathrm{.}0004\%1\mathrm{.}76491\mathrm{.}76470\mathrm{.}0108\%2\mathrm{.}35572\mathrm{.}35550\mathrm{.}0095\%2\mathrm{.}99852\mathrm{.}99790\mathrm{.}0181\%$PremiumTreeFEM Difference TreeFEM Difference TreeFEM DifferenceTreeFEM Difference TreeFEM Difference TreeFEM DifferenceTreeFEM Difference TreeFEM Difference TreeFEM Difference Ew$1$et al$.(1996).$ The results are shown in Table $7\mathrm{.}2.$As an alternative to pricing this option on a square domain, we also price it on a triangle$($the results are shown in Table $7\mathrm{.}3).$ 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 $(7\mathrm{.}11)$ to $(7\mathrm{.}14)$ are replaced by:Here, the g functions denote a plain vanilla European put with strikes E and w$2$and appropriate volatilities. We compute the cumulative normal distributions in Equations $(7\mathrm{.}11)$ and $(7\mathrm{.}12)$ with an approximation with four$-$digit accuracy from Hull $(2000).$ To compare the results, we also price the put on a basket using a two$-$dimensional binomial tree, as implemented by Haug $(1997).$ This tree can be interpreted as a simple explicit finite difference scheme; see WilmottV$($S$1,0,$t$)=$ g$($S$1,$ wE $,$t$)2$V$(0,$S$2,$t$)=$g$($S$2,$wE $,$t$)1$V$($S$1,$S$2,$t$)=0$ on SmaxSmax $12(7\mathrm{.}15)(7\mathrm{.}16)(7\mathrm{.}17)$ WY$061-07$drvWY$061-$Topper December $19,200411$:$52$ Char Count$=0$Dynamic $2$D Problems Table $7\mathrm{.}3$ Results for a put option on a basket computed on a triangular domain VolatilityTime to maturity $0\mathrm{.}50\mathrm{.}95430\mathrm{.}95450\mathrm{.}0155\%1\mathrm{.}47561\mathrm{.}47700\mathrm{.}0899\%2\mathrm{.}01862\mathrm{.}01870\mathrm{.}0053\%1\mathrm{.}41201\mathrm{.}41190\mathrm{.}0095\%1\mathrm{.}88351\mathrm{.}88340\mathrm{.}0033\%2\mathrm{.}39412\mathrm{.}39400\mathrm{.}0044\%1\mathrm{.}89411\mathrm{.}89370\mathrm{.}0218\%2\mathrm{.}33012\mathrm{.}32990\mathrm{.}0076\%2\mathrm{.}81202\mathrm{.}81200\mathrm{.}0006\%199$PremiumTreeFEM Difference TreeFEM Difference TreeFEM DifferenceTreeFEM Difference TreeFEM Difference TreeFEM DifferenceTreeFEM Difference TreeFEM Difference TreeFEM Difference