Composite Payoff
Composite Payoff is a direct subtype of Contingent Payoff
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TYPE INCLUSION RELATIONSHIPS
AVAILABLE FUNCTIONS
AVAILABLE CREATE FUNCTION KEYS
TYPICAL OBJECTS OF TYPE Composite Payoff
This type represents several different payoffs, i.e. objects of type Contingent Payoff acting concurrently in the context of Structured Product
The payoffs are supplied as input through the key Payoffs
The order by which the payoffs are defined is important because in the case where the regimes of any two payoffs overlap, the prevailing payoff is determined by a priority rule according to which a payoff with a smaller index takes precedence.
Note, the Payoff Regime of a Composite Payoff is the whole real axis (-∞ , +∞), but the payoff is not zero only on the union of the regimes of the constituent payoffs.
A Contingent Payoff object is constructed by supplying through the key Payoffs a 1D-array of u payoffs P₁ , P₂ , ... Pᵤ, i.e. objects of type Contingent Payoff
Then a 1D-array of v payoffs P'₁ , P'₂ , ... , P'ᵥ is automatically generated and held by the read-only key _Cascaded Payoffs
Each payoff Pᵢ applies on a certain regime Rᵢ, which is a real interval represented as an object of type Real Interval
The collection of all these regimes are thus denoted as R₁ , R₂ , ... Rᵤ and may overlap with each other.
The question then arises, which payoff prevails on the overlapping regions.
The answer is given by assuming that the given payoffs P₁ , P₂ , ... Pᵤ obey a priority rule from left to right, whereby the payoff Pᵢ takes precedence over the payoff Pᵢ₊₁ , for every integer i ≥ 1.
Concretely, this means that if the regimes of any two payoffs Pᵢ and Pⱼ - with i < j - overlap, then only the payoff Pᵢ applies on the overlapping region.
Next denote with R the union of R₁ , R₂ , ... Rᵤ.
Taking into account the mentioned priority rule, it is possible to partition R into v distinct subintervals S₁ , S₂ , ... Sᵥ such that on each of them exactly one payoff applies, as determined by the priority rule.
Parenthetically, the creation of S₁ , S₂ , ... Sᵥ out of R₁ , R₂ , ... Rᵤ is implemented by the function Stacked Covered Subintervals
Furthermore, the distinct subintervals S₁ , S₂ , ... Sᵥ are assumed to be ordered in an increasing sense with respect to their left boundary.
So, the following correspondence between subintervals and payoffs holds:
S₁ → P'₁
S₂ → P'₂
...
Sᵥ → P'ᵥ
Note that each of the P'₁ , P'₂ , ... , P'ᵥ equals one of the original given payoffs P₁ , P₂ , ... Pᵤ.
Stated differently, the regimes of P'₁ , P'₂ , ... , P'ᵥ are the distinct, increasing subintervals S₁ , S₂ , ... Sᵥ.
Note, the subintervals S₁ , S₂ , ... Sᵥ are displayed through the read-only key _Cascaded Intervals