# 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

**and**

*P***, both containing**

*C**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**,

**, and**

*C**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*in this case denotes a return value of the function.

_{ij}### Definition 3.2: *S*-Matrix

All ...

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