JP4461245B2 - Stock price fluctuation analysis system - Google Patents

Description

[0001]
BACKGROUND OF THE INVENTION
The present invention relates to a stock price fluctuation numerical analysis device, in particular, sees stock price fluctuation as an advection diffusion type phenomenon, derives a Fokker-Planck equation from a stock price model based on geometric Brownian motion, and uses Galerkin finite element method and BTD method, It belongs to the technical field related to a stock price fluctuation numerical analysis device that numerically analyzes the time evolution of the log stock probability distribution.
[0002]
[Prior art]
Economic and financial phenomena have been considered to be impossible to explore and predict their essential structure because their behavior is so complex. In addition, these phenomena are dominated by limited experience and intuition and have been perceived as being an area where the majority can never be easily involved. Recently, however, researches are actively conducted to approach phenomena that occur in the economic and financial fields using scientific methods. After the Black-Scholes model, which is the price valuation equation for financial derivatives, theories in financial engineering and mathematical finance have rapidly penetrated the market and are still evolving.
[0003]
In the stock price model expressed by the geometric Brownian motion, an uncertain element in stock price fluctuation is expressed by a stochastic process by the Brownian motion, and is described by a stochastic differential equation as a direct sum with a deterministic element in which the stock price moves according to the fluctuation trend. In addition, this model recognizes stock price fluctuations as an advection-diffusion type phenomenon. The parameter for the uncertainties is called volatility. Volatility is measured by annualized conversion of stock price fluctuations over a certain period, and is generally used as a risk index. A parameter for a deterministic element is called a trend and is a parameter indicating a trend component of stock price fluctuation. This trend and volatility are important parameters when dealing with stock price fluctuations.
[0004]
It is known that the stochastic process based on Brownian motion is a diffusion process as a property. The Fokker-Planck equation describes the time evolution of the probability density function for such a stochastic process. This Fokker-Planck equation is characterized by a double structure of probability. That is, it is not only probabilistic in the transition process but also probabilistic in the initial condition. The Fokker-Planck equation is an advection diffusion type equation. Since the Fokker-Planck equation is a time evolution equation of a probability density function, it is not necessary to perform probability statistical processing while dealing with a stochastic process.
[0005]
A stochastic differential equation refers to a stochastic process in which a physical quantity is stochastically fluctuating with time, expressed by an equation including a stochastic derivative, etc., and a solution that can uniquely determine the stochastic process. However, Ito's stochastic differential equation is the only one that can be discussed mathematically and systematically. This is realized by Ito's lemma, which enables differential operations on Brownian motion, which is originally not differentiable. Ito's stochastic differential equation is based on Brownian motion, and for any random variable z = z (t),
dz = g (z, t) dt + h (z, t) dB
It is represented by However, g (z, t) and h (z, t) are arbitrary functions according to the random variable z and time t, and dB is a minute fluctuation of Brownian motion. A stochastic process according to Ito's stochastic differential equation is called an Ito process. For details of Ito's lemma, see Non-Patent Document 2.
[0006]
If the Taylor expansion of an arbitrary two-variable function f = f (z, t) with respect to the random variable z and time t is ignored,
df ＝ {(∂f / ∂t) + (∂f / ∂z) g (z, t) +1/2 (∂²f / ∂z²) h (z, t)²} dt
+ (∂f / ∂z) h (z, t) dB
= (∂f / ∂t) dt + (∂f / ∂z) dz + (1/2) (∂²f / ∂z²) h (z, t)²dt
It becomes. This is Ito's lemma that a random variable z that follows Ito's stochastic process and an arbitrary two-variable function f = f (z, t) with respect to time t are differentiable.
[0007]
When the stock price is S = S (t),
dS = μSdt + σSdB
Is a stock price model expressed by geometric Brownian motion. However, μ represents a trend and σ represents volatility. The stock price model expressed by the geometric Brownian motion is an Ito process that follows Ito's stochastic differential equation. In order to maintain generality for all stocks, the stock price S (t) at time t is changed to the initial stock price S at initial time t = 0.₀= S (0)
x (t) ≡S (t) / S₀
And standardize. Using this standardized stock price x, rewriting the stock price model,
dx = μxdt + σxdB
It becomes.
[0008]
By using Ito's lemma, for any two-variable function f = f (x, t) with respect to the standardized stock price x and time t,
df ＝ {(∂f / ∂t) + μx (∂f / ∂x) +1/2 (σx)²(∂²f / ∂x²)} dt + σx (∂f / ∂x) dB
The following differential relational expression holds. Arbitrary two-variable function f
f (x, t) = ln (x)
Then,
d (ln (x)) = (μ- (1/2) σ²) dt + σdB
It becomes. further,
y (t) ≡ln (x (t)) (x (t) ≧ 0)
Is called log stock price. From these equations, the relational expression
dy = (μ- (1/2) σ²) dt + σdB
Is obtained. From this equation, it can be seen that the log stock price y follows the Brownian motion.
[0009]
The inventors of the present invention have proposed a “financial derivative evaluation system” in order to improve the accuracy of the evaluation of financial derivatives using the Black-Scholes equation. As disclosed in Patent Document 1, an extended Black-Scholes equation including a trend μ of an underlying asset of a financial derivative product as a parameter is established, and its exact solution is obtained and stored in a storage device. Each statistical data obtained from time series price data in the financial market is input from the input means and stored in the storage means. The price calculation means obtains the price of the financial derivative product from the exact solution based on each statistical data, and outputs it from the output means. By expressing the trend term, it is possible to evaluate the short-term trend of the financial market and to capture the phenomenon of economic physics more accurately than the conventional Black-Scholes equation. It is possible to quantify how the market evaluates asset prices at the current time.
[0010]
[Patent Document 1]
JP 2003-67581 A
[Non-Patent Document 1]
Takahiko Tanahashi “Fine Element Analysis I, II of Flow” (Asakura Shoten 1997)
[Non-Patent Document 2]
“Introduction to numerical probability analysis” by Kunio Yasue (Asakura Shoten 2000)
[0011]
[Problems to be solved by the invention]
However, in the conventional stock price fluctuation numerical analysis method, the stock price fluctuation is recognized as a convection-diffusion type phenomenon, but is analyzed as a diffusion phenomenon that does not consider advection, so there is a problem that the analysis accuracy is insufficient.
[0012]
The present invention solves the above-described conventional problems, automatically estimates stock price fluctuation characteristics with only a small amount of stock price data input, accurately analyzes stock price fluctuations, and provides guidelines for stock investment. Objective.
[0013]
[Means for Solving the Problems]
In order to solve the above problems, in the present invention, the stock price fluctuation is regarded as an advection diffusion type phenomenon, the Fokker-Planck equation is derived from the stock price model based on the geometric Brownian motion, and using the Galerkin finite element method and the BTD method, A stock price fluctuation numerical analysis device that numerically analyzes the time evolution of the log stock price probability distribution, a means of setting initial values of trend and volatility from past price strings, and a trend that is an important parameter of stock price fluctuations within numerical calculations And means for updating the volatility. With this configuration, it is possible to analyze stock price fluctuations over a long period of time with high accuracy.
[0014]
DETAILED DESCRIPTION OF THE INVENTION
Hereinafter, embodiments of the present invention will be described in detail with reference to FIGS.
[0015]
(First embodiment)
The first embodiment of the present invention regards stock price fluctuations as an advection-diffusion type phenomenon, derives a Fokker-Planck equation from a stock price model based on geometric Brownian motion, and calculates trends and volatility from stock prices over the past 10 days. It is a stock price fluctuation numerical analysis device that uses the Galerkin finite element method and the BTD method to perform numerical analysis on the time evolution of the stock price probability distribution while updating the trend and volatility at any time during the numerical calculation as the initial value.
[0016]
FIG. 1A is a conceptual diagram of a stock price fluctuation numerical value analysis apparatus according to the first embodiment of the present invention. FIG. 1B is a conceptual diagram of an analysis grid used in the stock price fluctuation numerical analysis device. In FIG. 1A, data input means 1 is means for inputting stock price data for the last 10 days immediately before. The data storage means 2 is means for storing stock price data for the last 10 days immediately before. The parameter initial value calculation means 3 is a means for obtaining the parameters μ and σ from the stock price data of the last 10 days immediately before and using them as initial values. The finite element method numerical calculation means 4 is a means for performing numerical analysis on the time evolution of the stock price probability distribution using the Galerkin finite element method and the BTD method. The parameter update means 5 is means for re-evaluating the trend μ and the volatility σ based on the probability distribution regarding the log stock price y obtained by numerical calculation. The data output means 6 is a means for outputting the expected stock price and volatility σ for 20 days. FIG. 2 is a flowchart showing an operation procedure of the stock price fluctuation numerical analysis apparatus.
[0017]
The function and operation of the stock price fluctuation numerical value analysis apparatus in the embodiment of the present invention configured as described above will be described. First, an outline of the operation of the stock price fluctuation numerical analysis device will be described with reference to FIG. Taking the stock price fluctuation as an advection-diffusion type phenomenon, the Fokker-Planck equation is derived from the stock price model based on the geometric Brownian motion. Stock price data for the last 10 days is input from the data input means 1 and stored in the data storage means 2. The parameter initial value calculation means 3 obtains the trend μ and the volatility σ from the stock price data of the last 10 days immediately before and sets them as initial values. Least square method is used for initial trend determination method. The finite element method numerical calculation means 4 solves the Fokker-Planck differential equation using the Galerkin finite element method and the BTD method, and numerically analyzes the time evolution of the log stock probability distribution. Further, correction is performed according to the probability normalization condition. Based on the result, the parameter updating means 5 obtains and updates the trend μ and the volatility σ which are important parameters of the stock price fluctuation. Repeat this for 20 days. The data output means 6 outputs the expected stock price and volatility σ for 20 days.
[0018]
In this way, stock price fluctuations are regarded as advection-diffusion-type phenomena, considering advection, numerical analysis using the advection-diffusion equation, reevaluating and updating parameters in the middle of numerical calculation, Estimation is possible. By simply inputting a small amount of stock price data for the past 10 days, the expected stock price probability distribution for 20 days can be easily obtained, so the stock market that changes from moment to moment can be grasped accurately and over the long term. The risk of stock prices can also be seen from the volatility σ.
[0019]
Next, a method for deriving the Fokker-Planck equation from the logarithmic stock model will be described. Focusing on the fact that the log stock price y directly follows the Brownian motion, the Fokker-Planck equation is derived from the log stock model represented by the log stock price y. The Fokker-Planck equation is derived using the Master equation. Consider a time change of an arbitrary stochastic process z = z (t). Time t₀Value z₀From this state, let P (z, t | z₀, t₀), From the probability normalization condition,
∫_{- ∞} ^∞P (z, t | z₀, t₀) dz = 1
Is established. Consider the case where this stochastic process z is a Markov process. From unit z to state z per unit time₀Transition probability W (z → z₀). Using this transition probability, the equation describing the time evolution of the conditional probability in the Markov process is the Master equation
(∂ / ∂t) P (z, t) ＝-∫_{- ∞} ^∞W (z → z ') P (z, t) dz' + ∫_{- ∞} ^∞W (z '→ z) P (z', t) dz '
It is. However, for the sake of simplicity, the initial time t₀Initial state z in₀Conditional probability P (z, t | z₀, t₀) Is simply written as P (z, t).
[0020]
This Master equation means that the probability distribution of state z at time t decreases with a transition to another state and increases with a transition from another state. Furthermore, this Master equation can be written as a partial differential equation by Kramers-Moyal expansion. K-th moment Ck (z) of transition probability
Ck (z) ≡∫_{- ∞} ^∞r^kW (z → z + r) dr
Define in. The Master equation is
(∂ / ∂t) P (z, t) ＝ Σ_{k = 1} ^∞{(1 / k!) (-∂ / ∂z)^k× Ck (z) P (z, t)}
Can be expressed as This is the Kramers-Moyal expansion. The form of this Kramers-Moyal expansion up to the second order is the Fokker-Planck equation
(∂ / ∂t) P (z, t) ＝-(∂ / ∂z) C₁(z) P (z, t) + (1/2) (∂²/ ∂z²) C₂(z) P (z, t)
It is.
[0021]
Apply the Fokker-Planck equation to the stock price model.
dy = (μ- (1/2) σ²) dt + σdB
When calculating the first moment of the transition probability for
C₁(y) = lim_{Δ t → 0}(1 / Δt) E [Δy] ＝ μ- (1/2) σ²
Is obtained. The second moment of transition probability is
C₂(y) = lim_{Δ t → 0}(1 / Δt) E [(Δy)²] = Σ²
It becomes. Since the third and higher moments are zero, the Fokker-Planck equation for the log stock model is
(∂ / ∂t) P (y, t) =-(∂ / ∂y) (μ- (1/2) σ²) P (y, t) + (1/2) (∂²/ ∂y²) σ²P (y, t)
It becomes.
[0022]
Third, the stock price fluctuation parameter will be described. Logarithmic stock model
dy = (μ- (1/2) σ²) dt + σdB
The volatility σ is measured by the standard deviation of price movements in the financial market, and is generally a concept of risk and is a very important parameter. For simplicity, an assumption is made that the trend μ and the volatility σ are constant values over the entire space region y. This assumption is reasonable enough to assume actual market movements.
[0023]
A method for calculating the initial values of the trend μ and the volatility σ will be described. A method of calculating volatility σ using historical volatility is adopted. The trend μ is calculated by using the least square method as a method of measuring the stock price fluctuation trend. The newer the information obtained from past price sequences, the higher the importance. Therefore, past price strings should not be treated uniformly. In order to place importance on the latest information, weights are used for price sequences with equal time intervals, and initial values of trend μ and volatility σ are determined.
[0024]
Trend μ and volatility σ are parameters that change with time. Trend μ and volatility σ are re-evaluated based on the probability distribution regarding logarithmic stock price y obtained by numerical calculation. Further, the trend μ and the volatility σ are updated by progressing time according to the parameter update time step m and the parameter update time step Δtm. In response to the numerical calculation results of the Fokker-Planck equation, the trend μ is updated as needed. formula
μ^{m + 1}= (1/2) (σ^{m + 1})²+ (1 / Δtm) (E [y]^{m + 1}-E [y]^m)
To update the trend μ. Volatility σ at parameter update time step m + 1^{m + 1}Is a value calculated using the volatility σ update formula. The update formula for volatility σ is
σ^{m + 1}= √ (Π^m/ (2Δtm))
It is. However, the coefficient Π is
Π^m≡ (E [y]^m)²+ (E [y]^{m + 1})²-2E [y]^mE [y]^{m + 1}−2Var [y]^m+ 2Var [y]^{m + 1}
It is. The shouldered m in these equations is a subscript indicating a time step and does not mean a power. The same applies to the following similar expressions.
[0025]
Fourth, a method for solving the Fokker-Planck equation by the finite element method will be described. For details of the finite element method, refer to Non-Patent Document 1. Logarithmic stock price model expression expressed by logarithmic stock price y
dy = (μ− (1/2) σ²) dt + σdB
Fokker-Planck equation
(∂ / ∂t) P (y, t) = − (∂ / ∂y) (μ− (1/2) σ²) P (y, t) + (1/2) (∂²/ ∂y²) σ²P (y, t)
The time evolution of the logarithmic stock price probability distribution is followed. However, the probability density function that becomes the logarithmic stock price y at time t is P (y, t), and will be displayed as P for simplicity. In this equation, trend μ and volatility σ use constant assumptions in the entire spatial domain, so
(∂ / ∂t) P = −u (∂ / ∂y) P + ν (∂²/ ∂y²) P
Can be rewritten as Also, the advection velocity u and the diffusion coefficient ν
u≡μ− (1/2) σ²
ν≡ (1/2) σ²
Define as follows. The Fokker-Planck equation is numerically calculated as a governing equation in numerical analysis.
[0026]
By dealing with stochastic processes,
∫_{- ∞} ^∞P (y, t) dy = 1
It is also necessary to satisfy the probability normalization conditions shown in (1). Furthermore, using expected value E [y] and variance Var [y] calculated from logarithmic stock probability distribution obtained by numerical calculation, trend μ and volatility according to parameter update time step m and parameter update time step size Δtm σ is an update formula for each
μ^{m + 1}= (1/2) (σ^{m + 1})²+ (1 / Δtm) (E [y]^{m + 1}-E [y]^m)
σ^{m + 1}= √ (Π^m/ (2Δtm))
To advance and update. Perform the time progression of the governing equation using the forward Euler method. By using the forward Euler method to advance the Fokker-Planck equation derived from the log stock model,
(P^{n + 1}-Pⁿ) / Δt = −uⁿ(∂ / ∂y) Pⁿ+ Νⁿ(∂²/ ∂y²) Pⁿ
Get. The shouldered n in these equations is a subscript indicating a time step and does not mean a power. The same applies to the following similar expressions.
[0027]
The Galerkin finite element method is adopted for the discretization of space. As shown in FIG. 1B as an example, the analysis grid is an unequal division grid using an exponential function. The governing equation is multiplied by the weight δω and integrated over the entire space region Ω with respect to the equation that progresses in time by the forward Euler method. Furthermore, when the entire region Ω is divided into a number of elements and attention is paid to one element Ωe,
∫_{Ω e}δω ((P^{n + 1}-Pⁿ) / Δt) dΩ
= -Uⁿ∫_{Ω e}δω (∂ / ∂y) PⁿdΩ + νⁿ∫_{Ω e}δω (∂²/ ∂y²) PⁿdΩ
It becomes. In this equation, partial integration is performed on the last term on the right side, and Green's theorem is applied.
∫_{Ω e}δω ((P^{n + 1}-Pⁿ) / Δt) dΩ
= -Uⁿ∫_{Ω e}δω (∂ / ∂y) (PⁿdΩ
+ Νⁿ∫_{Γ e}δω (∂ / ∂y) (PⁿdΓ
−νⁿ∫_{Ω e}(∂ / ∂y) (δω) (∂ / ∂y) (PⁿdΩ
It becomes. When the probability density function P is interpolated within the element by a bilinear function,
P = N_αP_α
Get a relationship. Where α is the local node number within the element, N_αIs a shape function, and the sum rule applies to α.
[0028]
Using this shape function, the weight function δω
δω = N_αδω_α
Interpolate as follows. δω_αEach component of is an arbitrary constant, and δω under the basic boundary condition Γ1_αThe component of is zero. δω_αUsing the arbitraryness of each component of
∫_{Ω e}N_αN_βdΩ (P_β ^{n + 1}-P_β ⁿ) / (Δt)
= -Uⁿ∫_{Ω e}N_α(∂ / ∂y) (N_βdΩP_β ⁿ
+ Νⁿ∫_{Γ e}N_α(∂ / ∂y) (PⁿdΓ
−νⁿ∫_{Ω e}(∂ / ∂y) (N_α) (∂ / ∂y) (N_βdΩP_β ⁿ
Can guide you. In addition, when mass concentration is performed using a coefficient matrix, the discretization formula
M^- _αβ(P_β ^{n + 1}-P_β ⁿ) / (Δt)
= -UⁿA_αβP_β ⁿ−νⁿD_αβP_β ⁿ+ Λ_α
Get. However, each coefficient matrix is
M^- _αβ= ∫_{Ω e}N_αN_βdΩ (mass matrix)
A_αβ= ∫_{Ω e}N_α(∂ / ∂y) (N_β) dΩ (Advection matrix)
D_αβ= ∫_{Ω e}(∂ / ∂y) (N_α) (∂ / ∂y) (N_β) dΩ (diffusion matrix)
Defined by M^- _αβIs the concentrated diagonal matrix, Λ_αRepresents the boundary integral term.
[0029]
In order to obtain second-order time accuracy, the second-order time differential term is evaluated by an advection equation. This corresponds to the BTD method in CFD. First, in the governing equation, the advection equation considering only the advection term is
(∂ / ∂t) (P) =-u (∂ / ∂y) (P)
It is represented by By performing Taylor expansion around the time step of interest for the probability density function P,
(∂ / ∂t) (P) |ⁿ= (P^{n + 1}-Pⁿ) / Δt
− (Δt / 2) (∂²/ ∂t²) (P) |ⁿ+ O (Δt²)
Can be obtained. Using this equation, in addition to the time progression by the forward Euler method, the time accuracy can be made second order by considering the second-order time differential term. Here, by evaluating the second-order time differential term with the advection equation,
(∂²/ ∂t²) (P) =-(du / dt) (∂ / ∂y) (P) + u²(∂²/ ∂y²) (P)
Get. However, since the trend μ and the volatility σ are time variables, the advection velocity u can also be regarded as a time variable from the definition formula.
[0030]
Therefore, the time derivative of the advection velocity u is
(du / dt) ≒ Δu / Δt = (uⁿ−u^n-1) / Δt
To approximate. Therefore, the BTD term at time step n is
BTDⁿ= (Δt / 2) {-(Δu / Δt) (∂ / ∂y) (Pⁿ) + (Uⁿ)²(∂²/ ∂y²) (Pⁿ)}
Can be obtained as follows. Applying the Galerkin finite element method to the BTD term and expressing it using a coefficient matrix,
BTDⁿ= (Δt / 2) {-(Δu / Δt) A_αβP_β ⁿ− (Uⁿ)²D_αβP_β ⁿ}
+ (Δt / 2) νⁿ∫_{Γ e}N_α(∂ / ∂y) (PⁿdΓ
It becomes. Adding the BTD term to the discretization formula gives the discretization formula
M^- _αβ(P_β ^{n + 1}-P_β ⁿ) / (Δt)
=-(Uⁿ+ (1/2) Δu) A_αβP_β ⁿ
− {Νⁿ+ (1/2) Δt (uⁿ)²} D_αβP_β ⁿ+ Λ_α
It becomes. Numerical calculation is performed according to the discretization formula including this BTD term. However, the boundary integral terms are collectively_αIt is displayed as.
[0031]
In dealing with stochastic processes, it is very important to meet the probability normalization condition. Therefore, when performing numerical calculations, it is necessary to store the total probability amount in the calculation region and satisfy the probability normalization condition. The boundary condition is fixed in the initial state at both ends of the calculation area. That is, since the basic boundary condition Γ1 is adopted, it is not necessary to calculate the boundary integral term from the nature of the Galerkin finite element method. In addition, by taking a calculation area to some extent, the influence of the method of fixing the boundary on the numerical calculation can be minimized. When numerically analyzing a random variable, it is necessary to proceed with a numerical calculation while satisfying a probability normalization condition. The total probability is stored by the following method.
[0032]
Consider the state after an arbitrary time step. The probability amount at each element is P_eThen, the total probability amount P in the entire calculation domain_ΩIs
P_Ω= ∫_ΩP_edΩ
It is obtained by. Therefore, from the probability normalization condition, the error amount Pε of the probability is
P_ε= 1-P_Ω
It is obtained. Error amount P of this probability_εIs corrected in the entire calculation region according to the probability amount in each element. Therefore, the probability amount P in each element after applying the probability normalization condition_e ^*Is
P_e ^*= P_e+ P_ε(P_e/ P_Ω)
Can be calculated as The series of calculations described above are performed after each step calculation to satisfy the probability normalization condition.
[0033]
In order to carry out numerical calculations stably for the unsteady advection-diffusion equation discretized by the explicit method, there are restrictions on the Courant number and the diffusion number. When the Courant number is written as c and the diffusion number as d with respect to the advection velocity u and the diffusion coefficient ν, each condition is
c = | uΔt / Δy | ≦ 1
d = νΔt / (Δy)²≦ 1/2
Given in. Therefore, this condition must be satisfied when numerically calculating the discretization formula.
[0034]
Fifth, a numerical analysis procedure will be described with reference to FIG. The analysis target is daily fluctuation. Price sequence used to determine initial parameters {S_i} Etc. are daily data. Therefore, the parameter update in the numerical calculation is performed every day. First, in step 1, a past price string before the calculation start time is input. In step 2, initial parameters used for numerical analysis are calculated from a past price string before the calculation start time. Thereafter, initial conditions are set, and repeated calculation of time step n is started. In the process of time progression by iterative calculation, first, in step 3, a discretized governing equation is calculated by the Galerkin finite element method and the BTD method. In step 4, the obtained probability density function is corrected according to the probability normalization condition. In step 5, it is checked whether the calculation for each set parameter update time step Δtm is completed. If completed, in step 6, the volatility σ is recalculated based on the update formula. In step 7, the trend μ is recalculated based on the update formula. The time progresses by repeating this series of calculations, and the calculation is performed until a specified time.
[0035]
As described above, in the first embodiment of the present invention, the stock price fluctuation numerical analysis apparatus regards stock price fluctuation as an advection diffusion type phenomenon, derives the Fokker-Planck equation from the stock price model based on the geometric Brownian motion, Calculate the trend and volatility from the 10-day stock price as initial values, and update the trend and volatility as needed during the numerical calculation, and perform numerical analysis on the time evolution of the stock price probability distribution using the Galerkin finite element method and the BTD method Since it is configured to perform this, it is possible to automatically estimate the fluctuation characteristics of the stock price only by inputting a small amount of stock price data, accurately analyze the stock price fluctuation, and provide a guideline for stock investment.
[0036]
This stock price fluctuation numerical analysis device can estimate stock price fluctuation characteristics and provide guidelines for stock investment. By obtaining the temporal development of the probability distribution and the transition of the trend μ and the volatility σ, various information can be shown for specific investment activities. By estimating the stock price fluctuation, it is possible to obtain a probability distribution of the future stock price, and to show how much the stock price falls within a certain range. Useful for risk estimation and management in stock price fluctuations.
[0037]
(Second Embodiment)
In the second embodiment of the present invention, the stock price fluctuation is regarded as an advection diffusion type phenomenon, the Fokker-Planck equation is derived from the stock price model based on the geometric Brownian motion, and the trend and volatility are calculated from the stock price of the past 10 days. Using the Galerkin finite element method and the BTD method, the Levy distribution is reproduced in the numerical calculation using the Galerkin finite element method and the BTD method, while the trend and volatility are updated during the numerical calculation. It is a stock price fluctuation numerical analysis device that performs numerical analysis on development.
[0038]
FIG. 3A is a conceptual diagram of a stock price fluctuation numerical value analyzing apparatus according to the second embodiment of the present invention. FIG. 3B is a conceptual diagram of a shape function used in the stock price fluctuation numerical analysis device. In FIG. 3 (a), the Levy distribution finite element method numerical calculation means 7 reproduces the Levy distribution in the numerical calculation using the Galerkin finite element method and the BTD method, using the shape function. It is a means for numerical analysis of development. Others are the same as those in the first embodiment. FIG. 4 is a flowchart showing an operation procedure of the stock price fluctuation numerical analysis apparatus.
[0039]
The function and operation of the stock price fluctuation numerical value analysis apparatus according to the second embodiment of the present invention configured as described above will be described. First, an outline of the operation of the stock price fluctuation numerical analysis apparatus will be described with reference to FIG. Taking the stock price fluctuation as an advection-diffusion type phenomenon, the Fokker-Planck equation is derived from the stock price model based on the geometric Brownian motion. Stock price data for the last 10 days is input from the data input means 1 and stored in the data storage means 2. The parameter initial value calculation means 3 obtains the trend μ and the volatility σ from the stock price data of the last 10 days immediately before and sets them as initial values. Least square method is used for initial trend determination method. Up to this point, the process is the same as in the first embodiment. The Levy distribution finite element method numerical calculation means 7 reproduces the Fokker-Planck differential equation in the numerical calculation using the shape function, solves it using the Galerkin finite element method and the BTD method, and the log stock probability distribution Numerical analysis of the time evolution of Further, correction is performed according to the probability normalization condition. Based on the result, the parameter updating means 5 obtains and updates the trend μ and the volatility σ which are important parameters of the stock price fluctuation. Repeat this for 20 days. The data output means 6 outputs the expected stock price and volatility σ for 20 days.
[0040]
By using the shape function shown in FIG. 3B, the Levy distribution is reproduced in the numerical calculation. In the model based on the geometric Brownian motion, it is assumed that the distribution of stock price returns is given by a normal distribution. However, in the actual market, a normal distribution is rarely confirmed, and it is known that the distribution is distributed in a “high peak fat tail” shape as compared with the normal distribution. The distribution follows the Levy distribution. The Levy distribution is characterized by a higher kurtosis at the center of the distribution than at the normal distribution and a thicker tail at the distribution. In other words, for stocks with a small fluctuation range, fluctuations are suppressed compared to the normal distribution, and conversely for stocks with a large fluctuation range, it can be regarded as a phenomenon in which fluctuations are promoted compared to the normal distribution. .
[0041]
The point corresponding to the boundary where the opposing forces of suppression and promotion work is the value of ± 1σ centered on the expected value. This is because volatility σ is given by the standard deviation of stock price fluctuations, and the value of ± 1 standard deviation centered on the expected value in the normal distribution is the inflection point of the distribution function. The force is acting in the direction that makes the distribution convex upward (downward). That is, in the Levy distribution, these forces become even stronger at each point, thereby forming a “high peak / fat tail” shaped distribution. By introducing the factors that form these Levy distributions into the stock price model, it is possible to capture stock price fluctuations with higher accuracy.
[0042]
The Galerkin finite element method is adopted as a numerical analysis method. One of the features of this Galerkin finite element method is the shape function used for intra-element interpolation and weighting functions. Using this shape function, the Levy distribution is reproduced in the numerical calculation. In the Levy distribution, forces in different directions act inside and outside ± 1σ centered on the expected value, forming the distribution. A shape function that increases the weight in the center direction is selected inside ± 1σ around the expected value, and a shape function that increases the weight in the outer direction is selected outside ± 1σ. By constructing, the Levy distribution is reproduced.
[0043]
N_α= (1/2) (1 + ξ_αξ) + γξ_α
Is used. The shape of the shape function for reproducing the Levy distribution is shown in FIG. This shape function has the same shape as one of the weight functions used in upstream in the finite element method. If γ is positive (negative), the weight on the left side (right side) becomes large in FIG. 3B. Using this shape function, the value of γ is adjusted as appropriate according to the positional relationship with ± 1σ centered on the expected value, and the weight function in the finite element method is configured to reproduce the Levy distribution in the numerical calculation. it can.
[0044]
When this shape function is used as a weight function and the governing equation of numerical analysis is discretized by the finite element method and the BTD method, the discretization formula is equivalent to the form already shown, but the coefficient matrix is
M_αβ= ∫_{Ω e}N_αN_βdΩ = J_e((1/6) ξ_αξ_β+ (1/2) + γξ_α) (Mass matrix)
A_αβ= ∫_{Ω e}N_α(∂ / ∂y) N_βdΩ = ((1/2) + γξ_α) ξ_β(Advection matrix)
D_αβ= ∫_{Ω e}(∂ / ∂y) (N_α) (∂ / ∂y) (N_βdΩ
= (1 / J_e) ((1/2) ξ_αξ_β) (Diffusion matrix)
It becomes. Je is Jacobian which is calculated by element average.
[0045]
The procedure of numerical analysis will be described with reference to FIG. The analysis target is daily fluctuation. Price sequence used to determine initial parameters {S_i} Etc. are daily data. Therefore, the parameter update in the numerical calculation is performed every day. First, in step 11, a past price string before the calculation start time is input. In step 12, initial parameters used for numerical analysis are calculated from a past price string before the calculation start time. Thereafter, initial conditions are set, and repeated calculation of time step n is started. In the process of time progression by iterative calculation, first, in Step 13, the Levy distribution is reproduced in the numerical calculation using the shape function, and the discretized governing equation is calculated by the Galerkin finite element method and the BTD method. In step 14, the obtained probability density function is corrected according to the probability normalization condition. In step 15, it is checked whether the calculation for each set parameter update time step Δtm is completed. If completed, in step 16, the volatility σ is recalculated based on the update formula. In step 17, the trend μ is recalculated based on the update formula. The time progresses by repeating this series of calculations, and the calculation is performed until a specified time.
[0046]
As described above, in the second embodiment of the present invention, the stock price fluctuation numerical analysis apparatus regards stock price fluctuation as an advection-diffusion type phenomenon, derives the Fokker-Planck equation from the stock price model based on the geometric Brownian motion, The trend and volatility is calculated from the stock price for 10 days as the initial value, and the trend and volatility are updated as needed during the numerical calculation, and the Levy distribution is calculated using the shape function using the Galerkin finite element method and the BTD method. Since it is configured to perform numerical analysis on the time evolution of the stock price probability distribution that is reproduced within the numerical calculation, it is possible to capture stock price fluctuations with higher accuracy, and the stock price fluctuation characteristics can be automatically detected only by inputting a small amount of stock price data. Can be estimated, and stock price fluctuations can be analyzed accurately to provide guidelines for stock investment.
[0047]
【The invention's effect】
As is clear from the above description, in the present invention, the stock price fluctuation is regarded as an advection diffusion type phenomenon, the Fokker-Planck equation is derived from the stock price model based on the geometric Brownian motion, and using the Galerkin finite element method and the BTD method, A stock price fluctuation numerical analysis device that numerically analyzes the time evolution of the log stock price probability distribution, a means of setting initial values of trend and volatility from past price strings, and a trend that is an important parameter of stock price fluctuations within numerical calculations And means for updating the volatility, it is possible to analyze stock price fluctuations over a long period of time with high accuracy.
[Brief description of the drawings]
FIG. 1 is a conceptual diagram of a stock price fluctuation numerical analysis device according to a first embodiment of the present invention;
FIG. 2 is a flowchart showing an operation procedure of the stock price fluctuation numerical analysis device according to the first embodiment of the present invention;
FIG. 3 is a conceptual diagram of a stock price fluctuation numerical analysis device according to a second embodiment of the present invention;
FIG. 4 is a flowchart showing an operation procedure of the stock price fluctuation numerical value analysis apparatus according to the second embodiment of the present invention.
[Explanation of symbols]
1 Data input means
2 Data storage means
3 Parameter initial value calculation means
4 Finite element numerical calculation means
5 Parameter update means
6 Data output means
7 Levy distribution finite element method numerical calculation means

Claims (3)

Fluctuating stock price fluctuations as an advection-diffusion phenomenon In the numerical analysis apparatus, data input means for inputting stock price data of a previous past fixed period, data storage means for storing stock price data of a previous past fixed period input from the data input means, and the data input Parameter initial value calculation means for obtaining the initial value by determining the trend μ and volatility σ from the stock price data of the past fixed period immediately before being input from the means, and initializing the trend μ and volatility σ obtained by the parameter initial value calculation means There line a numerical analysis of the time evolution of the stock price probability distribution by using the Galerkin finite element method and the BTD method as a value, further, standardization of probability And line Cormorant FEM numerical calculation means corrects the stock probability distribution by matter, the trend μ in the middle numerical calculations based on the probability distribution log stock obtained by numerical calculation by the finite element method numerical calculation means Parameter updating means for re-evaluating and updating volatility σ, and data output means for outputting expected stock price and volatility σ for a predetermined number of days obtained by numerical calculation by the finite element method numerical calculation means Stock price fluctuation numerical analysis device.   2. The stock price fluctuation numerical analysis apparatus according to claim 1, wherein the parameter initial value calculating means comprises means for calculating an initial value of the trend μ from a stock price in a past fixed period by a least square method.   2. The stock price according to claim 1, further comprising Levy distribution finite element method numerical calculation means for reproducing a Levy distribution in a numerical calculation using a shape function and performing a numerical analysis on a time evolution of a log stock probability distribution. Fluctuation numerical analysis device.

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