4 Financial and mathematical modeling of hedging as a way of financial risk management

4.1 Theoretical foundations of financial and mathematical modeling

Responsible decisions in modern purposeful planning and management systems should be, in some sense, extreme or close to them. Deviation from this principle is usually associated with unnecessary costs (often very significant) and reduces the effectiveness of management (often very significant).

When modeling banking activity, one often has to deal with the problem of mathematical programming, which can be formulated as follows: find the values ​​of variables that satisfy inequalities:

(15)

And turn to a minimum (maximum) function :

(16)

The form of the functions and determines the class of mathematical programming problems. If all the functions , , are linear, we get the problem of linear programming. If at least one of the functions is nonlinear, we have the problem of non-linear programming.

The classical methods of searching for an extremum in problems of non-linear programming are closely related to the notion of convex function, saddle point, necessary and sufficient conditions for the extremum (Kuhn-Tucker theorem), Lagrange function and multipliers

A plan, a set of control commands or a project can often be formally represented as a system of numbers or functions that satisfy certain constraints - equalities, inequalities, or logical relationships. The plan, control system or project is optimal if, in addition, they turn a function to the minimum or maximum (depending on the task) on the parameters sought-the quality of the solution.

Records (15), (16), which are quite meaningful with deterministic values ​​of the parameters of the conditions of the problem, lose certainty and require additional explanations for random values ​​of the initial data. Meanwhile, in many applied problems, the coefficients of the objective function, the elements of the condition function, or the components of the constraint vector are random variables.

The initial information for planning, design and management in the economy, as a rule, is not sufficiently reliable. Production planning is usually conducted in conditions of incomplete information about the situation in which the plan will be implemented and the products produced will be realized. In all cases, in mathematical programming models that involve planning, designing, and controlling tasks, some or all of the parameters of the objective function and constraints may turn out to be vague or random. A natural way of analyzing similar problems at first glance is to replace the random parameters with their mean values ​​and calculate Optimal plans of the deterministic models obtained in this way are not always justified. When the parameters of the problem conditions are smoothed out, the adequacy of the model to the phenomenon being studied can be violated. Averaging the original data can lead to the loss of useful information and introduce false information into the model. The solution of a deterministic problem with averaged parameters may not satisfy the constraints of the initial model with admissible realizations of the parameters of the conditions.

Develop plans for an optimal system of financial portfolios of the bank is not easy. A well-coordinated work of a whole group of qualified specialists is needed: a top manager responsible for the strategy and management of the bank's financial resources, a planner or portfolio manager who sets and corrects options for portfolio plans, an analyst of stock market instruments, an analyst-mathematician that provides an algorithmic solution to the optimization task, and a programmer , Which implements financial and mathematical ideas in the form of software. But even if these conditions are met, i.e. Availability of qualified specialists, when introducing tasks in banking, the question arises: Will the plan be useful at all if the majority of model parameters change randomly at all times? The answer to this question is the formulation and solution of the problem of stochastic programming, to the consideration of which we proceed.

Let us consider how to formulate a mathematical model of the optimization problem for the stochastic problem. For a basis we shall take model of linear programming:

(17)

(18)

If the coefficients in the objective function are random variables, then two statements of the optimization problem are possible:

• Maximization (minimization) of the average value of the objective function, which is called the M-statement;

• Maximizing the probability of obtaining the maximum (minimum) value, which is called the P-statement.

If the quantities and appearing in the constraints are random, then restriction is written thus:

, (19)

where - the specified probability with which the constraint must be satisfied.

The problem of stochastic programming in M-statement:

(20)

(21)

To solve the problem, we must pass to its deterministic equivalent. In this case, the objective function is recorded

(22)

The deterministic equivalent of the constraints is as follows:

, (23)

where - the specified probability level with which the constraint must be satisfied;

• - is calculated using the function from

We introduce the notation:

(24)

Then the deterministic equivalent of the problem is as follows:

(25)

(26)

(27)

(28)

.

We will consider the already classic stochastic task of optimizing the bank portfolio as follows:

P = P_S S + P_L L – C_DD DD – C_TD TD ; (29)

S + L = DD + TD + OC; (30)

S + L < 100; (31)

0,7 S + 0,3 L < 0; (32)

L > 35, (33)

где P - profit;

S - securities;

L - loans;

DD - demand deposits;

TD - time deposits;

OC - own capital;

P_S and P_L - profit on securities and loans, respectively;

C_DD and C_TD - the cost of attracting deposits.

The algorithm for solving the stochastic programming problem can be easily obtained using the above relations, as well as the symbolism and methodology for solving the linear programming problem.