deVeLopment of binomiaL priCing modeL of sHares and bonds for a Life insuranCe CompanY

The process of share price formation is studied in the article as well as a random process, and a model is developed that describes a random process that has characteristics close to the Brownian bridge, and realizes a multiplicative model of the bond price evolution with outpaced repayment time and normal value. The definition of a random process of forming a share price through a geometric Brownian bridge is also given. Since the Ukrainian insurance market is at the initial stage of development, and the country's economy is unstable, insurance companies need practical and reliable tools for calculating and forecasting the expected return on investment activity. And especially urgent for this particular type of activity is the issue of breakeven investment, as insurance companies can’t afford risky investment because of the specifics of their activities. As a research result, the level of investing in a risk-free asset for an insurance company of a cumulative type is calculated.


introduction
With the development of the insurance market, more people are using life insurance to create capital. Thus, such companies get large capital at their disposal, which they then invest in different sources. Today, the issue of the investment activities of insurance companies that have significant differences from other facilities in the market has not yet been fully explored. In this regard, it is relevant to study the features of the investment activities of life insurance companies, to improve their reliability and stability.
This topic is relevant, because insurance companies accumulate significant capital, which can then be increased through various investment tools. Thus, the availability of reliable instruments for calculating the expected profitability of investment instruments can ensure the stability of the insurance company at the market.

the object of research and its technological audit
The object of research is the investment activity of life insurance company. Since insurance companies have a lot of long-term obligations, and dispose а large reserve capital, all of its activities depend on the effectiveness of investment policy in general.
One of the most serious problems in this topic is the issue of forecasting profitability from investment activities and minimizing risks. Since the insurance company cannot afford to invest in risky sources, because it has obligations on the insurance poles.
The largest market for life insurance is the United States, where is the largest number of insurance companies, as well as insurance coverage in the country. In the second place is China, which is actively developing in recent years, and Japan is in the third place.

the aim and objectives of research
The main aim of the paper is assess of the probabilistic distribution of changes in the price of shares and bonds of a life insurance company to increase the effectiveness of the management of the financial flows of the insurance company. This will allow the insurance company assesses more accurately the investment yield, which will lead to increased stability of the company in the future.
To achieve this aim it is necessary: 1. To study the processes of conducting the investment activity of the life insurance company.
2. To define and study the properties of this process. 3. To develop a model that will help predict the price changes the debt market.

research of existing solutions of the problem
Today, the issue of investment activities of insurance companies is not fully disclosed. So in Ukraine this sector has just begun to develop [2]. Most scientists pay attention to the issues of statistical trends in the life insurance market, such as: increasing insurance premiums [3] and insurance payments, covering the insurance market and so on. At the same time, there is no clear methodology for managing the investment activity of an insurance company, which includes all the features of this type of activity. The same questions were raised about the safety of investment activities of insurance companies [1] and the factors that determine it, but such studies are more conducted abroad than in the Ukrainian market.
Abroad, scientists pay attention to such issues as data envelopment analysis in insurance companies, Intellectual capital and performance, productivity and efficiency of the life insurance companies [4][5][6][7][8].

methods of research
The methodological basis of the article is the fundamental positions of the general economic theory, insurance theory, statistics and econometrics and also methods of mathematical modeling

research results
Let's consider the time series of share market prices. Schedules of share prices are chaotic in nature. Therefore, the value of the price of the share S t at time t is assumed to be a random variable, which also determines some random process in time t∈[0, +∞).
Let's break the time interval [0, t] on n intervals of the same length Δt: t = n⋅Δt. Let's consider the model of the price evolution of a particular share, when we will be interested only in the price of the opening and the price of closing. Assuming that the price of the opening of the next period is equal to the price of the closing of the previous one, in the framework of our model we will be interested only n+1 value of the share.
Let's assume that the price of a share within each time interval can only either increase with the coefficient u, or fall with the coefficient d. Then the share price S (t+Δt) at the end is either Su or Sd. Let's assume also that from any investor in each interval of time it is possible to get a guaranteed interest r f = r f (Δt). The growth factor when investing in a risk-free asset is denoted by r = 1+r f . To exclude the possibility of arbitration, let's assume that: Let's denote by p the probability that the price will increase. Then 1-p -the probability that the price will decrease: Let's assume that a hypothesis about the risk neutrality of investors with a given current value S and possible future prices Su, taking into account the risk-free coefficient r, is fulfilled. Then the purchase of a share at the price S is equivalent to the operation of investing the sum S into a risk-free asset. This means that the equality is fulfilled: From this let's obtain that r = up+d(1-p) Expressing p from this equality, we finally find that the probabilities p and (1-p) are equal: Let ξ i -an indicator of a random event, consisting of the fact that the price increased during the period [(i-1)Δt, iΔt], i = 1,...,n. Then S iΔt = S (i-1)Δt u ξ d 1-ξ and: Considering the value of the current share price is not a random variable equal to S 0 = S, let's find from the obtained formulas that: Let's denote by v n the number of periods until the time t, in which the price was moving up. Then (n-v n ) will be the number of single periods by the time t, in which the price went down. Where v n is the sum of Bernoulli random variables ξ 1 ,…,ξ n . As a result: Since the random variable v n ∈[0; n], then the possible values of the share price at the end of the i-th period are equal to Su k d n-k , k = 0,…,n.
To obtain the probabilistic distribution of the random variable S t , let's use Bernoulli's scheme. Imagine that every single unit of time is an experiment, the «success» of which is the movement of prices up, and «failure» -the movement of prices down. Then we have n independent experiments with the probability of «success» equal to p. The entered random variable v n will be the number of «successes» in the Bernoulli scheme with the given parameters.
From the formula (7) implies that: The mathematical expectation of a random variable S n equals: Technology audiT and producTion reserves -№ 5/4(37), 2017 Thus, the probability p correlates with the values u, d, and r such that the expected value of the share at time n equals the investment in the risk-free asset.
Let's find the logarithm of equality (7): From the properties of mathematical expectation and dispersion it follows that: Let's consider a certain level of possible values of the price of the share X. Such magnitude is a probabilistic characteristic of the random variable S t , which is called the distribution function. Since the condition S t < X is equivalent to the inequality ln ln , S S (10) and the positivity of the quantity ln u d (due to u > d) it follows that: Let's denote by B(x, n, p) the function of a binomial distribution, which is equal to the probability that the number of «successes» in the Bernoulli scheme will not exceed x. So: where [x] -means the largest integer that does not exceed x. Then the desired probability P{S_t ≤ X} is based on the formula: Consequently, by means of formulas (13) and (15) the share price is determined at time t, using the value of the binomial distribution. Therefore, such model is called binomial. Parameters of this model are: n -number of periods, i and d -coefficients of possible growth and fall in period, respectively, r -coefficient of increase of riskfree asset. All these parameters depend on the value of Δt.
As practice shows, the so-called logarithmic mean a and the volatility σ 2 , which show the average changes in mathe matical expectation and the variance of the random process per unit time, are the most statistically significant values in the study of the evolution of a particular financial series. More strictly, these parameters are determined by the following formulas: Knowing the distribution of the random variable S T value of the price of the financial series at the time T, it is possible to obtain a numerical estimate of the parameters a and σ 2 of the given series. Then, according to formula (15), knowing the parameters of the binomial model u, d, p, n it is possible to construct the share price distribution at time t. There is a simple problem, how the prescribed model parameters should depend on the calculated characteristics a and σ 2 .
Even more important is the dependence of the model parameters on the choice of the length Δt of the time interval. Thus, there is the question of assessing the variability of prices of a particular share. This parameter is precisely the volatility of the action σ 2 . Let's determine the dependence of the coefficients u and d on it: In this case, the binomial model of risk-free growth rate r in one interval is more convenient to express, using continuously accrued by the interest rate μ. By the definition of this value, in time Δt, the risk-free asset increases in exp{μ ⋅ Δt} times. Consequently, r = exp{μ ⋅ Δt}. The probabilities p and 1-p in this case can be calculated as riskneutral using formulas (8). In practice, they usually use more convenient formulas: The constructed model makes it possible to estimate the probable characteristics of the random variable S t . For example, for a given level of risk capital X from the formula (15) let's obtain that the probability of risk is P{S i ≤ X} = B(x, n, p), where x is determined according to (13). By the definition of the quantities u and d we Let's formulate a binomial model more strictly. Let, as before, the time interval [0, t] is divided into n equal intervals of length Δt. For simplicity, let Δt = 1. This means ISSN 2226-3780 that Δt = n. Let's assume that the price of an asset within each single time interval can only either increase with the coefficient u or fall with the coefficient d. Let the choice of direction -up or down -in the first interval is made with a probability of 1/2. Further dynamics depends on the direction of movement on the previous interval. Let's assume that the probability of continuing the price movement in the same direction as in the previous step is equal to 1-c, c∈[0, 1] The probability of changing the direction of motion relative to the previous step is equal to. If c = 1/2, then we have a simple classical binomial model whose probabilistic characteristics are determined only by the parameters u i d. If c = 0 or c = 1, then this random process will be degenerate, since in the next step the previous motion will continue or be necessarily changed. In this case, the behavior of the process will be fully determined by the first step taken. Thus, the dynamics of the probabilistic characteristics of the generalized model is given by the parameters u, d i c.
6.1. model pricing of shares using the geometric brownian motion. Let's be interested in the dynamics of price changes of some financial asset, which is described by the random process S_t, t∈[0, +∞]. Let's call the random process S t a geometric Brownian motion with parameters a and σ 2 if for any non-negative numbers t and s the following conditions are fulfilled: the random variable S t+s /S s is statistically independent of the values of the random process to the time moment s, and ln(S t+s /S s ) has a Gaussian distribution with mean at and a dispersion σ 2 t. In other words, the financial time series is modeled by the geometric Brownian motion, if the price correlation in the future time t to the price in the present time does not depend on the past price dynamics and has a log-normal distribution with the parameters at and σ 2 t.
The process of geometric Brownian motion can be obtained from a binomial model if Δt→0, for n→∞. To prove this, one must use the central boundary theorem. Let's formulate a partial case of the central limit theorem (Muavra-Laplace theorem).
Let's assume that we conduct n independent experiments, each of which with probability p ends with «success» and with the probability of 1-p ends, respectively, «failure».
Then, as before, the number of successful experiments v n has a binomial distribution with the parameters n and p. In the Muavra-Laplace theorem states that for any real number y there is a limiting distribution: lim , where ln(S t /S) is a linear function of v n , which means that it also has an asymptotic Gaussian distribution. Moreover, it is possible to state that: Thus, if Δt→0, then the distribution ln(S t /S) converges to the Gaussian distribution with the parameters at and σ 2 t. Accordingly, when Δt→0, the binomial model leads us to a geometric Brownian motion.
The evolutionary model of the risk asset constructed in this way is often called the lognormal model, since one of the important conclusions of the geometric Brownian motion is that the relation of the future price S t to the present price S is a random variable that has a lognormal distribution with the parameters at and σ 2 t. The latter means that: where η is a standard Gaussian random variable. The resulting formula gives an opportunity to estimate the probabilistic distribution using the Laplace function. In particular, the probability of a price risk not exceeding a given price level X can be estimated by the following formula: P{S t ≤ X} = P{η ≤ z} = Φ(z), where, taking into account the balance equation a = − μ σ 2 2 / : One consequence of the log-normal model is the fact that the expected value of the price increases as a nonrisk asset. Indeed, by virtue of the lognormal distribution properties, the mathematical expectation of the right-hand side in formula (24) is equal to: 6.2. model of bond price evolution with geometric brownian motion. Changes in bonds market prices are more stable than in the share market. Therefore, a model that describes the dynamics of bond prices B t is implemented by a differential equation that does not contain a stochastic component: ISSN 2226-3780 The value r(t) is the interest rate that is continuously calculated and may be random. As a rule, all risks associated with interest rate changes are modeled directly by the random process r(t). In this case, the model of price dynamics of the bond becomes rather complex and difficult to implement. This section proposes a model for pricing bonds, which is a generalization of the Samuelson model, which can be used to market the bonds for an insurance company.
The main difference between share bonds is that the bond has a limited period of validity, and its price at the time of repayment of T equals its nominal value: B T = F. If it is considered that the change in bond prices is something like the geometric Brownian motion, then it is worth considering an arbitrary process with given conditions. Such random processes are known in the theory of probability. For example, the Brownian Bridge belongs to them.
According to the source [2], under the Brownian bridge Y t it is customary to call the Gaussian random process with the function of the mathematical expectation and autocovariance function, which respectively equal: Since, at time T, the mathematical expectation is equal to β, and the dispersion is zero, then with probability 1, the random process Y t takes value β, which in principle corresponds to the dynamics of the price of the risk bonds. However, the use of the Brownian Bridge to simulate the evolution of the price of the insurance company's bonds is quite complicated, since it may take values less than zero, which is impossible for the price. In addition, the Brownian Bridge implements an additive, rather than a multiplicative model. This is evident from the corresponding stochastic differential equation: Since, at time T, the mathematical expectation is equal to β, and the dispersion is zero, then with probability 1, the random process Y t takes value β, which in principle corresponds to the dynamics of the price of the risk bonds. However, the use of the Brownian Bridge to simulate the evolution of the price of the insurance company's bonds is quite complicated, since it may take values less than zero, which is impossible for the price. In addition, the Brownian Bridge implements an additive, rather than a multiplicative model. This is evident from the corresponding stochastic differential equation: Indeed, the increment dY t depends on the value of Y t only in the coefficient with the value of the increment of the argument dt and, consequently, is adaptive. This drawback can be eliminated if instead of the Brownian bridge taking its stochastic exponent.
Let's use the formula of a stochastic differential equation: whereas F t X , ( ) we take the following function: dF t X adt bdW t , , but as an X -Brownian bridge, which is subject to the following differential equation: which has the form: Y W t t 0 0 = . Then, if use the ITO formula for the coefficients: where From here: In addition, Y t 0 can be expressed from equation (32): If substitute the resulting expression in equation (39): Choose f(t) and B such that: With the aid of mathematical transformations, let's find that the function f(t) takes the following form: Substituting (43) and B = σ in (32), let's find that the desired function takes the form: l n .
The stochastic equation for this function is as follows: As a result, we have a stochastic differential equation that defines this random process B t : The solution of the resulting equation will be the following random process, which is determined by the standard Brownian bridge Y t 0 (for it a = β = 0): where B 0 is the current bond price. Thus, a constructed model describing a random process that has characteris tics close to the Brownian bridge, and implements a multiplicative model of the evolution of the bond price with the time of repayment T and the nominal value F. The obtained stochastic differential equation (40) makes it possible to estimate the probability distribution of the growth of dB t the bond price at the time t, as the known stochastic value of the volatility σ 2 .

sWot analysis of research results
Strengths. The strength of the research is the analysis of the share price evolution process, as well as the creation of a multiplicative model of the evolution of the bond price with a given maturity and a normal value that describes a random process with characteristics close to the Brownian Bridge.
Weaknesses. The weak point is that the model can't include all the risks that may arise in the process of investment and business activities of the insurance company.
Opportunities. Opportunities for further research are the further improvement of the model, the inclusion of higher risk factors for it. And also the use of this model for a real insurance company Threats. Threats to the results are difficulty in using the model in real terms.

Conclusions
As a result of the studies, the following results are obtained: 1. The analysis of the evolution of the price of shares as a random process. The scheme for calculating the expected share price as an investment in a risk-free asset.
2. It is reviewed and summarized lognormal and binomial model to calculate the expected stock price. The constructed binomial model makes it possible to estimate the probabilistic characteristics of the expected share price. The definition of the random process of formation of the price of shares through the geometric Brownian bridge is given. 3. A model is developed that describes a random process that has characteristics close to the Brownian Bridge and implements a multiplicative model of the evolution of the bond price with the time of repayment of T and the nominal value of F. The resulting stochastic differential equation gives an opportunity to estimate the probabilistic distribution of the bond price rise at a given time point, as the known stochastic value of volatility.
This model can be used to effectively manage the investment activity of an insurance company to prevent loss-making from activities, and maximize the company's returns. references