R- and L-Algorithms for Maximum Profit Strategy
The profit-and-loss function that has the arguments prices, strategy, and transaction costs and returns the profit or loss value helps in understanding the concepts of s-function, s-matrix, and s-interval. Having developed these three concepts, they will be useful for construction of the algorithm that evaluates potential or maximum profit and for building the corresponding strategy.
S-FUNCTION AND S-MATRIX
Let me define the following scalar function, S = S(P, C, i, j, k), where P and C, both containing n elements, are the vectors of prices and transaction costs, respectively; i and j are indices taking arbitrary integer values from the closed interval [1, n]; and k is a coefficient converting contract prices into absolute dollar amounts. The vector of transaction costs contains elements expressed as absolute dollar amounts paid per contract per transaction. The coef'ficient k can be computed as the tick value/tick, which is equivalent to the value of a full-point move.
Definition 3.1: S-Function
The following equation defines the s-function:
If P, C, and k are constant, meaning that a historical interval of prices and costs is selected for a contract with given specifications, then S is a function of the indices i and j only. The symbol Sij in this case denotes a return value of the function.
Definition 3.2: S-Matrix