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Chapter 5: Valuation of Complex Barrier ... > 3 Simple, Intuitive and Accurate Val... - Pg. 132

132 DERIVATIVES where L is the lower barrier and U is the upper barrier level. These symmetry relationships also hold for partial-time single and double barrier options, described by Heynen and Kat (1994) and Hui (1997). These new symmetry relationships give new insight and should be useful when calculating bar- rier option values. If one has a formula for a barrier call, the relationship will give the value for the barrier put and vice versa. The relationship also gives new opportunities for static hedging and val- uation of "second-generation" exotic options. An example of this would be a first-down-then-up- and-in call. In a first-down-then-up-and-in call C t dui (S t , X, L, U ) the option holder gets a standard up-and-in call with barrier U (U > S t ) and strike X if the asset first hit a lower barrier L(L < S t < U ). Using the up-and-in call down-and-in put barrier symmetry described above we can simply construct a static hedge and thereby a valuation formula for this new type of barrier option; C t dui (S t , X, L, U, r, 0, ) = X di L 2 L 2 S t , P t , , r, b, L X U (5.12) In other words to hedge a first-down-then-up-and-in barrier call option all we need to do is buy X L 2 L 2 L number of standard down-and-in puts with strike X and barrier U . If the asset price never touches L both the first-down-then-up-and-in call and the standard down-and-in put will expire worthless. On the other hand, if the asset price hits the lower barrier L the value of the X down- L and-in puts will be exactly equal to the value of the up-and-in call. So in that case all we need