# CHAPTER 3

# R- and L-Algorithms for Maximum Profit Strategy

**T**he 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 *S*_{ij} in this case denotes a return value of the function.

### Definition 3.2: *S*-Matrix

All ...