MinMax Fn
MinMax Fn is a direct subtype of Real To Real Function
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TYPE INCLUSION RELATIONSHIPS
AVAILABLE FUNCTIONS
AVAILABLE CREATE FUNCTION KEYS
TYPICAL OBJECTS OF TYPE MinMax Fn
This type represents - in its basic form, but see also variant forms below - a function ƒ that maps u real variable numbers x₁ , x₂ , ... , xᵤ to either their minimum or maximum.
For simplicity, it is assumed in the description here the regular case whereby all input variables x₁ , x₂ , ... , xᵤ are processed in the given order by the function, but it is possible to process only a selected subset of this variables and/or under a different order through the use of the optional key Input Variables
For example, in the minimum case the formula looks like:
ƒ( x₁ , x₂ , ... , xᵤ ) = min{ x₁ , x₂ , ... , xᵤ }
The number of input variables u can be variable, which means this function - in its basic form discussed here - has Domain_Dim = -1
In its basic form, the chart of the output of the function MinMax Fn against the minimum/maximum value is a straight line, as the line A in the diagram below, where the horizontal axis holds the values from 0 to 10 for the minimum/maximum value.
It is possible though to set up the function's parameters in such a way that more interesting charts arise, such as the the line B that resembles the payoff of a call option and the line C that resembles the payoff of a put option.
These function parameters are represented by the following optional keys:
Ref Values
Post Applied Weights
Post Increment
Post Cap
Post Floor
In detail:
1) Calculating the min/max value of the Growth Factors rather than the original numbers:
The minimum or maximum may refer to the growth factors x₁/x⁰₁ , x₂/x⁰₂ , ... , xᵤ/x⁰ᵤ rather than the original input numbers x₁ , x₂ , ... , xᵤ
This is possible by specifying the initial reference values x⁰₁ , x⁰₂ , ... , x⁰ᵤ through the key Ref Values
In the minimum case the formula becomes:
ƒ( x₁ , x₂ , ... , xᵤ ) = min{ x₁/x⁰₁ , x₂/x⁰₂ , ... , xᵤ/x⁰ᵤ }
It is also possible to specify fixed increments c₁ , c₂ , ... , cᵤ through the key Increments
In the minimum case the formula becomes:
x₁/x⁰₁ + c₁ , x₂/x⁰₂ + c₂ , ... , xᵤ/x⁰ᵤ + cᵤ
A practical application of this variant is when x₁ , x₂ , ... , xᵤ represent market prices of u stocks at some particular time and the final payoff depends on the least performer stock.
Then the key Ref Values could be set to hold the reference initial stock prices x⁰₁ , x⁰₂ , ... , x⁰ᵤ
Then the growth factors become x₁/x⁰₁ , x₂/x⁰₂ , ... , xᵤ/x⁰ᵤ which represent the stock performances in that time interval, a fact that means that the function returns the minimum with regard to the performance rather than the spot stock price.
2) Applying weights after having calculated an interim min/max value:
Under this variant, after the minimum or maximum m of the values x₁ , x₂ , ... , xᵤ (or x₁/x⁰₁ , x₂/x⁰₂ , ... , xᵤ/x⁰ᵤ if the key Ref Values applies) has been found, the function returns the product ̅wᵢm rather than m, where:
̅w₁ , ̅w₂ , ... , ̅wᵤ are fixed weights defined through the key Post Applied Weights and
i is the index that corresponds to the found extremum m, i.e. the smallest integer i for which it holds that xᵢ = m (or xᵢ/x⁰ᵢ = m if the key Ref Values applies).
If there are more than one such elements that equate to m so that there exist several candidate indices, i is chosen as the smallest of all candidate indices.
3) Adding an increment to the interim min/max value:
An increment may be optionally added to the function's interim min/max result by adding the optional key Post Increment
4) Imposing an upper or lower bound on the interim min/max value:
An upper (cap) or lower (floor) bound may be optionally imposed on the function's interim min/max result by adding the optional key Post Cap or Post Floor respectively.