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Trinomial Tree Algorithm for Barrier Option A trinomial tree can be used for pricing particular types of barrier options. We consider particular types of single barrier and double barrier options. The single barrier options include certain types A D_IN call option specification, for example, includes the exercise type (i.e., either American or European), an exercise time, , a strike level, , a rebate value, (where , and a barrier level, , which depends continuously on time over the interval . Here the underlying security is any security whose price, S , can be modeled as a piecewise geometric Brownian motion over the life of the option. In addition we require that the initial spot level for the underlying security, , lie above the initial barrier level, . In contrast a long, American D_IN call can be exercised immediately after it has knocked in (i.e., after the underlying security has crossed the lower barrier level); if the option does not knock in, then the fixed rebate, , is received at maturity. As mentioned above, we also consider certain types of double barrier options. These options include particular types We also consider two types of knockout annuities, Down and Out and Up and Out. If we are long such a knockout annuity, we receive a fixed coupon annuity until the price of the underlying security crosses a preset barrier level; we then receive the accrued annuity since the last pay date. Note that only European exercise is permitted for the knockout annuities above, and no rebates are allowed. Analytic formulas for pricing barrier options do not exist for the case where the barrier is an arbitrary, continuous function of time or where the exercise type is American. Tree methods (e.g., trinomial or binomial) can, however, be used to approximate the price of barrier options. Unfortunately standard tree methods, when applied to price barrier options, suffer from several drawbacks, that is, these methods may converge very slowly and/or display a persistent bias in the price. The disadvantages above are due to the inability of standard tree methods to ensure, for example, for a single barrier option, that a layer of tree nodes always coincides with the barrier. In such a case, then, the tree method effectively prices a different option (i.e., with a new barrier). An interesting, new trinomial tree method is presented for overcoming the above specification error in the barrier. The idea of the method is to construct a tree lattice, for example, for a single barrier option, by ensuring that certain nodes near the barrier always branch onto the barrier.
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