JP2010003196A - Volatility estimation device, computer program thereof, and volatility estimation method - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To perform volatility estimation by removing high frequency noise even in such a state in which the high frequency noise exists as in an actual stock price process. <P>SOLUTION: This device for estimating volatility at an observation point t<SB>k</SB>on the basis of observation data X(t<SB>k</SB>) at the discrete observation point t<SB>k</SB>in a random process can smooth the observation data X(t<SB>k</SB>) at the observation point t<SB>k</SB>by auxiliary observation data X(t<SP>i</SP><SB>k</SB>) observed at an auxiliary observation point t<SP>i</SP><SB>k</SB>at a time interval finer than a time interval Δ between respective observation points. The device can estimate volatility at the observation point t<SB>k</SB>by using the smoothed observation data. <P>COPYRIGHT: (C)2010,JPO&INPIT

Description

  The present invention relates to a volatility estimation apparatus, a computer program thereof, and a volatility estimation method.

  Volatility is an indicator of the fluctuation status of a stochastic process (random process) such as stock prices, and is used to appropriately price financial derivatives such as various options, work to evaluate a portfolio, and even a portfolio It is an indispensable parameter for optimization construction work.

ここで、Ｗ（ｔ）は、ブラウン運動であり、σがヴォラティリティである。 The time-series asset price process p (t) such as the stock price fluctuation process is represented by the following Ito process in the Black-Scholes model.

Here, W (t) is Brownian motion, and σ is volatility.

  In the above equation, the volatility σ is theoretically given as a constant. In reality, however, volatility is not a constant, but a function of time. Therefore, estimating the volatility σ from the observation data p (t) becomes an important problem.

  As a conventional method for estimating volatility, a method for obtaining an integrated volatility is generally used. If the integral form of volatility is obtained, the spot volatility can be obtained by differentiating the integral form. However, when spot volatility is obtained by this method, calculation of numerical differentiation is required, and thus the estimation accuracy is lowered.

On the other hand, a method for directly estimating the spot volatility has been proposed mathematically. However, in the conventional method, when the calculation is performed by a computer, the processing time becomes very long and cannot be actually used.
In addition, according to the conventional method, the volatility cannot be estimated unless the observation data is obtained for one period (for example, the period until the expiration of the option), and the real-time property is lacking.

  On the other hand, the present inventors have proposed a scheme capable of estimating volatility (spot volatility) in almost real time at each time point in Non-Patent Document 1. In the proposed scheme, since the amount of calculation is small, the real-time property is excellent, and estimation can be performed with relatively high accuracy.

Here, as shown in FIG. 7, the problem of estimating the spot volatility is the observation data X () at a plurality of observation points t _k (0 ≦ k ≦ N) in one period of the observation interval [0, T]. When t _k ) is given, the volatility at each observation point t _k is obtained.

A volatility estimation formula (estimation scheme) proposed in Non-Patent Document 1 is as follows. In the following formula, volatility is represented by b instead of σ (hereinafter the same).

According to the above equation, as shown in FIG. 8, Δ _k X / Δt at a certain observation point t _k is expressed as Δ _{k + λ} X / Δt (−M ≦ λ ≦ M) in the range of t _kM to t _{k +} M. ) To minimize the volatility estimation error. Non-Patent Document 1 also shows a result of confirming by simulation that the estimation error can be reduced by increasing the value of M (smoothing window size).
S. Ogawa and K. Wakayama "On a real-time scheme for the estimation of volatility", Monte Carlo Methods and Appl., Vol. 13, No. 2 (2007), pp. 99-116

  Although the volatility estimation scheme of Non-Patent Document 1 has been established mathematically (theoretical), further consideration is necessary to apply this scheme to actual observation data. The inventor came to mind.

That is, in the conventional estimation scheme, it is assumed that the observation data X (t _k ) is the stochastic process p (t) itself indicated by the Ito process.
On the other hand, the observation data X (t _k ) as in the actual stock price process is disturbed by high-frequency noise (microstructure noise) Z (t _k ) as shown in the following equation.

As the above formula, stock price process p (t _k) or the like of the observation data X (t _k) is always observed with noise Z (t _k).
This noise Z (t _k ) is a very high frequency random process. That is, for any k, Z (t _k ) is independent (not related to each other).

When such observation data X (t _k ) including noise Z (t _k ) is applied to a conventional estimation scheme, when the number of observation points used for smoothing increases, noise Z (t _k ) is added accordingly. Therefore, the error of the volatility estimation value obtained by the conventional estimation scheme increases.
For this reason, it is difficult to apply the conventional estimation scheme to actual observation data.

  Therefore, an object of the present invention is to eliminate the high-frequency noise and perform the volatility estimation even in a situation where high-frequency noise exists like an actual stock price process.

The present invention is an apparatus for estimating volatility at the observation point (t _k ) based on observation data (X (t _k )) at discrete observation points (t _k ) in a random process, Auxiliary observation data storage means for acquiring and storing auxiliary observation data (X (t ⁱ _k )) observed at auxiliary observation points (t ⁱ _k ) with a time interval finer than the time interval (Δ) between observation points and (31a), said auxiliary observation data storing means acquires the observation point from (31a) (t _k) near the auxiliary observation data _{^{(X (t i k))}} , obtained auxiliary observation data (X (t ⁱ _k )) By the first calculation means (31c) for calculating the smoothed observation data obtained by smoothing the observation data (X (t _k )) at the observation point (t _k ), and the first calculation means (31c). Record the smoothed observation data of each observation point (t _k ) obtained. A smoothing data storage means (32a) for storing and a calculation for estimating the volatility at the observation point (t _k ) using the smoothed observation data stored in the smoothing data storage means (32a) A volatility estimation device comprising: second calculation means (32b, 32c, 32d) for performing

  According to the present invention, since there is auxiliary observation data observed at the auxiliary observation point with a time interval finer than the time interval between the observation points, the auxiliary observation data near the observation point is used to observe the observation point. It is possible to smooth the data. Therefore, even if high frequency noise is included in the observation data, high frequency noise can be reduced.

The second computing means (32b, 32c, 32d) includes the smoothed observation data at the observation point (t _{k + 1} ) next to the target observation point (t _k ) for which the volatility is to be estimated, and the target observation. A calculation value obtained by dividing the difference from the smoothed observation data at the point (t _k ) by the time interval (Δ) between the observation points is obtained for other observation points in the vicinity of the target observation point (t _k ). In addition, it is preferable to calculate an estimated value of volatility by smoothing with the calculated value.

It is preferable that the first calculation means (31c) performs smoothing using only auxiliary observation data (X (t ⁱ _k )) past the observation point (t _k ). In this case, volatility estimation can be performed in real time.

It is preferable that the second calculation means (32b, 32c, 32d) perform smoothing using only the calculated values obtained for the observation points past the target observation point (t _k ). In this case, volatility estimation can be performed in real time.

Smoothing by the first computing means, the observation point and the observed data in the _{(t k) (X (t} k)) and the observation point (t _k) near the auxiliary observation data (X (t ⁱ _k)) of the total sum, the sum, the observation point (t _k) and preferably one that includes an operation of dividing by the total number (2M + 1) of the observation point (t _k) near the auxiliary observation points (t ⁱ _k). In this case, the volatility estimation accuracy can be further increased.

The smoothing by the second arithmetic means, the observation point (t _k) for seeking said calculated value and the observation point (t _k) near the observation point (t k _{+ lL)} for obtained between the calculated value the total sum, the sum, the observation point (t _k) and the observation point (t _k) is preferably one containing an operation of dividing the vicinity of the observation points the total number of _{(t k + lL) (2L} + 1). In this case, the volatility estimation accuracy can be further increased.

  From another viewpoint, the present invention is a computer program for causing a computer to function as the volatility estimation device.

From another viewpoint, the present invention estimates the volatility at the observation point (t _k ) based on the observation data (X (t _k )) at the discrete observation point (t _k ) in the random process. A method in which the computer executes the calculation, and the auxiliary observation data (X (t ⁱ _k ) observed at the auxiliary observation point (t ⁱ _k ) having a time interval finer than the time interval (Δ) between the observation points. storing a)), obtained by smoothing the observed data (X (t _k)) at the observation point (t _k) near the auxiliary observation data (X (t ⁱ _k) the observation point by) (t _k) A step of calculating and storing smoothed observation data, and an operation for estimating the volatility at the observation point (t _k ) using the smoothed observation data stored in the smoothed data storage means (32a). Go and its estimated value And outputting a Vo Rati utility estimation method, which comprises a.

  According to the present invention, high-frequency noise can be reduced even if the observation data includes high-frequency noise.

Hereinafter, embodiments of the present invention will be described with reference to the drawings.
In the following, in order to apply the estimation scheme of Non-Patent Document 1 to actual observation data X (t) including noise Z (t), noise Z (t) included in observation data X (t) A new estimation scheme (estimation formula) for obtaining the volatility estimated value from the observation data X (t) while suppressing the influence of the above will be described.
The new estimation scheme proposed here obtains the volatility estimation value by multi-stage smoothing processing. This multi-stage smoothing process includes a first-stage smoothing process and a second-stage smoothing process. Hereinafter, after describing the noise included in the observation data, the first-stage smoothing process and the second-stage smoothing process will be mathematically described. Then, the volatility estimation apparatus according to the embodiment will be described.

[1. Mathematical description of the invention]
[1.1 Microstructure Noise]
Here, the noise (fine structure noise) Z included in the observation data X (t) of the price process (probability process) p (t) such as a stock price fluctuation process is assumed as follows.

  It should be noted that noise Z (t) is assumed to be independent of the price process p (t) in many documents related to volatility estimation. However, in the present invention, such an assumption is not necessary, and Z (t) and p (t) may or may not be independent.

[1.2 First level smoothing]
Here, in order to suppress the influence of noise in the observation data X (t), as an auxiliary observation point with a finer time interval than the time interval Δ of the original observation point {t _k } for estimating volatility, { t ⁱ _k , 0 ≦ k ≦ N, 0 ≦ i ≦ 2M} are introduced (see FIG. 1).
In the first stage smoothing, 2M (plurality) of auxiliary observation data {X (t ⁱ _k )} at auxiliary observation points {t ⁱ _k } in the vicinity of the original observation point {t _k } are used to The observation data X (t _k ) at the observation point {t _k } is smoothed to obtain observation data in which the influence of noise is suppressed.

Here, the original observation point {t _k } is represented by the following equation.

The auxiliary observation point {t ⁱ _k } having a fine time interval than the time interval Δ is expressed by the following equation. Note that the fine time interval here is Δ / 2M. Further, in the following formulas, auxiliary observation point {t ⁱ _k} is has to include an original observation point {t _k}, may not include the original observation point {t _k}.

Here, the following notation is adopted for the (auxiliary) observation data X (t ⁱ _k ).

Further, the following notation is adopted for the price process p (t).

Furthermore, the following notation is adopted for the noise Z (t).

The first-stage smoothing process (X (t _k ) smoothing) is expressed as follows.

が定義される。 For p and Z, as in the above formula,

Is defined.

以下では、補助観測点｛ｔⁱ _k｝が、本来の観測点ｔ_kについて、対称的となっている場合の平滑化について説明するが、これは、表記上の便宜によるものであり、本発明を限定するものではない。 In the above formula smoothing the first stage, an auxiliary observation point {t ⁱ _k} is the original observation point t _k, has become symmetrical, negative side of the original observation point t _k ( It may be on the past side) or on the plus side (future side).
The smoothing by the auxiliary observation data on the future side and the smoothing by the auxiliary observation data on the past side are expressed as follows.

In the following, smoothing in the case where the auxiliary observation point {t ⁱ _k } is symmetric with respect to the original observation point t _k will be described. It is not intended to limit.

Also, the nature of the smoothing of the first stage, at a certain observation point t _k near the auxiliary observation data X (t ⁱ _k), to smooth the observed data X (t _k) of the observation points t _k Yes, it is not limited to the arithmetic expressions of [Equation 10] and [Equation 12].
For example, in FIG. 1, as auxiliary observation data X for smoothing the observed data X (t _k) of the observation points t _k (t ⁱ _k), observation points t _k-1 and the observation point t _{k + 1} Auxiliary observation data in between was used, but auxiliary observation data on the minus side (past side) from the previous observation point t _{k-1 and} on the positive side (future side) from the next observation point t _{k + 1} ) Auxiliary observation data may be used.

When the observation data X (t _k ) is smoothed based on the first-stage smoothing equation of [Equation 10], the smoothed observation data is obtained from the equation of [Equation 3]: It is expressed as follows.

By performing the first-stage smoothing, the noise included in the observation data is reduced as follows.

Now, taking the change in the section [t _k , t _{k + 1} ] for the equation [Equation 13], it is expressed as follows.

Here, the symbol Δ ⁱ _k represents the difference operation in [t ⁱ _k , t ⁱ _{k + 1} ], and is represented by the following equation, for example.

Therefore, the following equation is obtained from the equation [15].

Moreover, the following formula is obtained from the definition of the price process p (t).

The above formula is expressed as follows by applying the Euler-Maruyama scheme.

Now, in order to evaluate the strength of the error e in the above equation, the following assumptions are made regarding the regularity of the coefficients a (•) and b (•).
(Assumption) The coefficient b (·) is a Helder continuation of the order α∈ [0, 1] in L ² (Ω) −sense, and E | b (t) −b (s) | ² ≦ L ² _B | t-s | constant L _B is present as a ^2α.

なお、上記評価式において、シンボル“α∧１／２”は、２つの値の最小値を表す。 Under the above conditions, the error term e in the equation [Equation 19] satisfies the following evaluation uniformly for i and k.

In the above evaluation formula, the symbol “α∧1 / 2” represents the minimum value of two values.

  As described above, the error (error) can be reduced by performing the first-stage smoothing using the equations [Equation 10] to [Equation 12].

を、非特許文献１記載の推定スキームに適用して、ヴォラティリティ推定値を求める。
第１段目の平滑化処理によって、観測データＸ（ｔ）から高周波雑音Ｚ（ｔ）の影響が除去されているため、観測データＸ（ｔ）に高周波雑音Ｚ（ｔ）が含まれていても非特許文献１記載の推定スキーム（本実施形態における第２段目の平滑化）が適用可能となっている。 [1.3 Second level smoothing]
In the second-stage smoothing process, it was obtained by the first-stage smoothing process.

Is applied to the estimation scheme described in Non-Patent Document 1 to obtain a volatility estimated value.
Since the influence of the high frequency noise Z (t) is removed from the observation data X (t) by the first-stage smoothing process, the observation data X (t) includes the high frequency noise Z (t). Is also applicable to the estimation scheme described in Non-Patent Document 1 (second-stage smoothing in this embodiment).

を、当該観測点ｔ_k近傍の複数（２Ｌ個）の観測点ｔ_k+l-L（ｔ_k-L〜ｔ_k+L）おける

によって、平滑化することである（図２参照）。 The essence of the smoothing process in the second stage, in order to estimate the Vo Rati utility at a certain observation point t _k, at the observation point t _k

The observation points t _{k + lL} plurality of the observation points t _k vicinity (2L _{pieces) (t kL ~t k + L} ) definitive

To smooth (see FIG. 2).

A preferable arithmetic expression for the second-stage smoothing process is as follows.

In the above equation, since L smoothed observation data for the future are required, the real-time property to some extent is obtained, but the complete real-time property is lacking. In order to perform the calculation of the smoothing process in the second stage completely in real time, the past 2L (plurality) of smoothed observation data may be used, and the calculation formula is as follows.

[2. Volatility estimation device]
3-6 has shown the volatility estimation apparatus 1 which concerns on embodiment, and its process sequence. The volatility estimation device 1 estimates the volatility b ² (t _k ) at each observation point t _{k from} the observation data X (t _k ) at each observation point t _k in a price process such as time-series stock price data. Is what you want. This estimation is performed according to the above-mentioned extraordinary explanation.

In the present apparatus 1, in addition to the observation data {X (t _k )} at the original observation point {t _k } whose volatility is to be estimated, the time interval is finer than the time interval Δ of the observation point {t _k }. The observed auxiliary observation data {X (t ⁱ _k )} is handled as observation data.

In the present embodiment, since 2M auxiliary observation data {X (t ⁱ _k )} are required for each observation point t _k , in the observation interval [0, T] (for example, the interval until the option expiration) When the number of observation points t _k is N, the total number of auxiliary observation data {X (t ⁱ _k )} is 2M × N. However, when auxiliary observation data {X (t ⁱ _k )} used for each observation point {t _k } is shared among observation points {t _k }, the total number of auxiliary observation data {X (t ⁱ _k )} May be less than 2M × N.

[2.1 Device configuration]
As shown in FIG. 3, the present apparatus includes a computer 2 that stores a computer program 1 that performs volatility estimation processing in a storage device such as a hard disk.
The computer 2 includes a CPU 21, an internal storage device 22 such as a RAM / ROM, an external storage device 23 such as a hard disk, an input device 24 that accepts data input from the outside, an output device 25 that outputs data to the outside, and the like. ing.

[2.2 Computer program functions]
FIG. 4 is a block diagram illustrating functions of the apparatus. These functions are exhibited when the program 1 is executed in the computer 2.

[2.2.1 Observation Data Smoothing Processing Unit (first level smoothing processing)]
As shown in FIG. 4, the present apparatus receives observation data X (t _k ) such as time-series stock price data and auxiliary observation data X (t ⁱ _k ), and smoothes the observation data X (t _k ). A smoothing processing unit 31 is provided.
The smoothing processing unit 31 performs the above-described first-stage smoothing process, and the high-frequency noise Z (t _k ) included in the observation data (t _k ) is _obtained by the smoothing process. Can be removed.

Here, the observation data X (t _k ) is data (stock price) observed at discrete observation points t _k . The observation data X (t _k ) at each observation point t _k includes a deviation from the true value p (t) by rounding down, rounding up, or rounding. Such rounding down, rounding up, or rounding occurs independently at each observation point t _k independently of each other. This is the high frequency noise Z (t _k ).

When the observation data X (t _k ) at a certain observation point t _k is given to estimate the volatility, the smoothing processing unit 31 uses the auxiliary observation data X (t ⁱ _k ) near the observation point t _k. Then, the observation data X (t _k ) is smoothed.

As shown in FIG. 5, the smoothing processing unit 31 obtains and accumulates observation data X (t _k ) that is sequentially generated at a time interval Δ, and a buffer (storage unit) for the observation data X (t _k ). 31a and observed data X (t _k) auxiliary observation data X (t ⁱ _k) buffers for storing acquired sequentially generating assist observation data X (t ⁱ _k) with a fine time interval than (storage unit ) 31b.
Further, the smoothing processing unit 31 includes a calculation unit 31c that performs a calculation for smoothing the observation data X (t _k ) at the observation point t _k using the data stored in the buffers 31a and 31b. The calculation unit 31c performs calculation according to the calculation formula of [Equation 10] or [Equation 12].

Specifically, as shown in FIG. 1, when the observation data X (t _k ) at a certain observation point t _k is given from the buffer 31 a, the calculation unit 31 c receives the data near the observation point t _k from the buffer 31 b. 2M pieces of auxiliary observation data X (t ⁱ _k ) are acquired and calculation is performed.
Here, M is preferably (sufficiently) larger than the number N of observation points t _{k in} the observation interval [0, T] (for example, the interval until option expiration). By doing in this way, the estimation error of a volatility estimated value can be made small.

When the more complete real-time property is emphasized, only the auxiliary observation data X (t ⁱ _k ) past from the observation point t _k is acquired from the buffer 31b.

When the real-time property may be somewhat impaired, smoothing may be performed using not only the past auxiliary observation data X (t ⁱ _k ) but also the future auxiliary observation data X (t ⁱ _k ). In this case, the calculation unit 31c waits until auxiliary observation data X (t ⁱ _k ) generated in the future from the observation point t _k is accumulated in the buffer 31b, and necessary auxiliary observation data X (t ⁱ _k ) is stored. If accumulated, it is obtained from the buffer 31b and smoothed.

[2.2.2 Volatility estimation based on smoothed observation data (second level smoothing)
As shown in FIG. 4, the apparatus includes a volatility estimation unit 32 that performs volatility estimation (second level smoothing process) based on the smoothed observation data.

The volatility estimation unit 32 includes a smoothed observation data buffer (storage unit) 32 a that accumulates the smoothed observation data output from the smoothing processing unit 31. The buffer 32a stores smoothed observation data at each past observation point t _k .

Moreover, Vo Rati utility estimation unit 32 includes a smoothed observation data at observation points t _k should estimate Vo Rati utility, the next observation point t _{k +} and smoothed observation data in _one, the difference calculating unit 32b for obtaining a difference between It has.

  The difference value calculated by the difference calculation unit 32 is accumulated in the difference buffer (storage unit) 32c. The buffer 32c stores the difference value calculated for each observation point.

Furthermore, the volatility estimation unit 32 includes an estimated value calculation unit 32d that calculates a volatility estimated value using the difference value according to the calculation formula of [Equation 24] or [Equation 25].
Specifically, the estimated value calculation unit 32d obtains, from the buffer 32c, difference values at 2L (plural) observation points in the vicinity in addition to the difference value at the observation point t _k for which the volatility is to be estimated. Then, using the obtained difference value, an operation according to the equation of [Equation 24] or [Equation 25] is performed, and an estimated volatility value is output.
Here, L is preferably (sufficiently) smaller than the number N of observation points t _k in the observation interval [0, T]. By doing in this way, the estimation error of a volatility estimated value can be made small.

When obtaining a more complete real-time property, the estimated value calculation unit 32d acquires a difference value at an observation point in the past (minus side) from the observation point t _k from the buffer 32c according to the calculation formula of [Equation 25]. The operation may be performed.
On the other hand, when the real-time property may be somewhat impaired, smoothing may be performed using not only the difference value at the past observation point but also the difference value at the future observation point. In this case, the estimated value calculation unit 32d waits until the difference value at the future observation point is accumulated in the buffer 32c from the observation point t _k, and when the necessary difference value is accumulated, obtains it from the buffer 32c. Estimate volatility.

  When the estimated value calculation unit 32d calculates the estimated value of volatility, it outputs it. The output volatility estimation value is highly accurate, and appropriate pricing of financial derivatives such as various options, portfolio evaluation work, and portfolio optimization construction work can be appropriately performed.

In addition, this invention is not limited to the said embodiment, A various deformation | transformation is possible.
For example, in the above embodiment, the time interval between the original observation points and the time interval between the auxiliary observation points are set to be equal intervals for convenience of description, but they need not be equal intervals and may be irregular intervals. Good.

In the above embodiment, in the calculation for smoothing in the first stage, the sum of 2M + 1 X (t ⁱ _k ) (sum of 2M auxiliary observation points and one original observation point) is calculated. The simple average of 2M + 1 X (t ⁱ _k ) is obtained by dividing by the total number (2M + 1), but the smoothing is not limited to this. For example, by calculating the weighted average Smoothing may be used. Further, in the second-stage smoothing, smoothing may be performed by obtaining a weighted average instead of a simple average.

  Further, in the above embodiment, volatility regarding one random process (time series data of stock price of one brand) is obtained, but the present invention is applied to an n-dimensional process composed of a plurality (n) of random processes. It is also possible to do.

Is a diagram showing the original and observation points t _k and the auxiliary observation point t ⁱ _k. It is a figure which shows a 2nd smoothing area. It is a figure which shows the hardware constitutions of a computer. It is a functional block diagram of a volatility estimation apparatus. It is a block diagram of a smoothing process part. It is a block diagram of a volatility estimation part. It is a figure which shows the observation area and observation point in the conventional estimation scheme. It is a figure which shows the smoothing area in the conventional estimation scheme.

Explanation of symbols

31 smoothing section 31a observation data buffer 31b auxiliary observation data buffer 31c calculation unit 32 Vo Rati utility estimator 32a smoothed observation data buffer 32b difference calculation unit 32c differential buffer 33d estimation value calculation unit t _k original observation point t ⁱ _k auxiliary observation point X (t _k ) original observation data X (t ⁱ _k ) auxiliary observation data Δ time interval between original observation points

Claims (8)

An apparatus for estimating volatility at the observation point (t _k ) based on observation data (X (t _k )) at discrete observation points (t _k ) in a random process,
Auxiliary observation data storage for acquiring and storing auxiliary observation data (X (t ⁱ _k )) observed at auxiliary observation points (t ⁱ _k ) with a time interval finer than the time interval (Δ) between each observation point Means (31a);
The auxiliary observation data (X (t ⁱ _k )) in the vicinity of the observation point (t _k ) is acquired from the auxiliary observation data storage means (31a), and the observation is performed using the acquired auxiliary observation data (X (t ⁱ _k )). First computing means (31c) for calculating smoothed observation data obtained by smoothing the observation data (X (t _k )) at the point (t _k );
Smoothed data storage means (32a) for storing smoothed observation data of each observation point (t _k ) obtained by the first calculation means (31c);
Using the smoothed observation data stored in the smoothed data storage means (32a), the second computing means (32b, 32c, 32, 32c, 32c, 32c, 32c) performs a calculation for estimating the volatility at the observation point (t _k ). 32d)
A volatility estimation device characterized by comprising: The second calculation means (32b, 32c, 32d)
The difference between the smoothed observation data at the next observation point (t _{k + 1} ) of the target observation point (t _k ) whose volatility is to be estimated and the smoothed observation data at the target observation point (t _k ) By smoothing the calculated value divided by the time interval (Δ) between the observation points with the calculated value obtained for other observation points in the vicinity of the target observation point (t _k ), estimation of volatility is performed. The volatility estimation apparatus according to claim 1, which calculates a value. The said 1st calculating means (31c) is smoothed using only the auxiliary observation data (X (t ⁱ _k )) past from the said observation point (t _k ). Volatility estimation device. 3. The second computing means (32 b, 32 c, 32 d) performs smoothing using only the computed values obtained for observation points past the target observation point (t _k ). Volatility estimation device. Smoothing by the first computing means, the observation point and the observed data in the _{(t k) (X (t} k)) and the observation point (t _k) near the auxiliary observation data (X (t ⁱ _k)) seeking summation claim 1 the sum, it is intended to include an operation of dividing by the total number (2M + 1) of the observation point (t _k) and the observation point (t _k) near the auxiliary observation point (t ⁱ _k) 5. The volatility estimation apparatus according to any one of 4 above. The smoothing by the second arithmetic means, the observation point (t _k) for seeking said calculated value and the observation point (t _k) near the observation point (t k _{+ lL)} for obtained between the calculated value the total sum, the sum, the observation point (t _k) and the observation point (t _k) near the observation point (t k _{+ lL)} total (2L + 1) is intended to include an operation of dividing the claims 2 or 4 The volatility estimation device described.   The computer program for functioning a computer as a volatility estimation apparatus in any one of Claims 1-6. This is a method in which a computer executes an operation for estimating volatility at the observation point (t _k ) based on observation data (X (t _k )) at discrete observation points (t _k ) in a random process. And
Storing auxiliary observation data (X (t ⁱ _k )) observed at auxiliary observation points (t ⁱ _k ) having a time interval finer than the time interval (Δ) between the observation points;
The smoothed observation data obtained by smoothing the observation data (X (t _k )) at the observation point (t _k ) by the auxiliary observation data (X (t ⁱ _k )) near the observation point (t _k ) is calculated. Memorizing step;
Using the smoothed observation data stored in the smoothed data storage means (32a), performing an operation for estimating the volatility at the observation point (t _k ), and outputting the estimated value;
A volatility estimation method comprising:

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