KR101172321B1 - Apparatus and method for determining price of asian option based on hestons stochastic volatility model - Google Patents

Abstract

The Asian option pricing apparatus based on the Heston probability variability model collects parameters corresponding to a pricing variable including the type of Asian option for which the expected price is to be calculated and a pricing function defined based on the heston probability variability model. A data collection unit; An option price determination unit which determines the expected price by substituting the pricing variables and parameters into a pricing function corresponding to a selected Asian option type among pricing functions classified according to the type of Asian option; And an output unit configured to output a result including the determined expected price and the selected Asian option type.

Description

Asian Option Pricing Apparatus and Method Based on Heston Probability Variability Model {APPARATUS AND METHOD FOR DETERMINING PRICE OF ASIAN OPTION BASED ON HESTON'S STOCHASTIC VOLATILITY MODEL}

The present invention relates to an Asian option pricing apparatus and method, and more particularly, to an Asian option price for calculating an appropriate price of a discrete-time geometric mean Asian option through a closed form solution based on the Heston probability variability model. A determination device and method.

Securities that generate new cash based on traditional financial instruments, such as stocks and bonds, are called derivatives. Such derivatives have the advantage of being able to avoid risks caused by price changes in underlying assets and to create new financial instruments.

Representative derivatives include forward, forward, options and swaps. Among the derivatives, an option is a contract that purchases or sells the right to buy or sell certain assets at a contracted price at or before the future contracted period for a certain fee. The fixed asset is an underlying asset, the contracted maturity or expiration date, the contracted price is an exercise price or strike price, and the constant fee is premium. ) Is called.

The types of options can be classified into five categories according to the classification method. First, the right to buy a specific asset is called a call option, and the right to sell a specific asset is called a put option. Second, according to the exercise of rights, the option to limit the exercise to the expiration date of the option is called the European option, and the option to exercise the right even before the expiration date is called the American option. Third, it can be divided into physical options and futures options. Fourth, when it is listed and traded on a regular exchange according to the place of exchange, it is called an over-the-counter option when it is issued by an on-site option or an exchange-traded option or by other specialized dealers such as banks or securities firms. do. Fifth, the option that generates profits to the buyer when exercising the rights according to the intrinsic value is an in-the-money (ITM) option, and an option with the same asset price and exercise price. -Money (ATM), and the option that results in loss to the buyer when exercising the right is called Out-The-Money (OTM).

At this time, the value of the option is formed of two elements, an intrinsic value and a time value. Intrinsic value is determined by the difference between the underlying asset price and the exercise price. In the case of a call option, it is in-the-money when the underlying asset price is higher than the exercise price. In the case of a put option, it is in-the-money when the underlying asset price is lower than the exercise price. On the contrary, it is out-of-the-money, and is equal if the underlying asset price and the exercise price are equal. The time value represents the expected value of future changes in the underlying asset price and is relatively large when the current underlying asset price and the exercise price of the option are the same. Thus, the price of an option is determined by the current underlying asset price, the exercise price of the option, the volatility of the target asset price, the period until the option expires, and the interest rate.

A typical option pricing model is the so-called Black-Sholes model, published by Black and Sholes in 1973. The Black-Sholes model is a continuous model in which price formation is continuous, which enables the calculation of theoretical prices. As such, the Black-Sholes model is commonly used to calculate option prices because it provides very useful results that can be used in real markets by using objective inputs.

In this case, it is preferable that only one volatility of the underlying asset input to the Black-Sholes model be input at any one time. However, according to the volatility smile theory, the volatility of the underlying asset appears at different times depending on the volatility of the option, depending on the option's intrinsic volatility, and the volatility of the underlying asset is calculated separately by the exercise price. Should be. As a result, the Black-Sholes model has a weakness in price calculation due to volatility smiles.

To overcome the weaknesses of the Black-Sholes model, the Heston model was developed. According to the Heston model, variability is not a constant but a stochastic process. In addition, volatility and covariance of stock prices are negative, which means that the direction of volatility and the direction of stock prices are reversed. In both cases, the Heston model is a more realistic option pricing model.

On the other hand, the options modified by financial institutions to meet the various needs of customers among the over-the-counter options are called exotic options, in which the benefits of the options are based on the average price over the option period instead of the maturity price of the underlying asset. The option determined by this is called an Asian Option. In such Asian options, the average price is the exercise price, and the gain of the option is determined by the difference between the underlying asset price and the exercise price at maturity.

To date, a reasonable pricing formula for the Vanilla European option has been developed for the Heston model, but the pricing formula for discrete-time geometric mean Asian options is not known.

In addition, in the related art, only a method of calculating a numerical approximation, such as Monte Carlo simulation, exists in relation to the pricing of the discrete time geometric mean Asian option.

As such, in order to obtain an approximation of the appropriate price of the discrete-time geometric mean Asian option, it is necessary to rely on the simulation, which causes a lot of time to estimate the approximation of the price through the simulation, and thus the computing power.

Embodiments of the present invention provide an Asian option pricing apparatus and method based on the Heston probability volatility model that determines Asian option prices close to market prices with low computation time and computing power.

An embodiment of the present invention provides an Asian option pricing apparatus and method based on a Heston probability variability model that provides a pricing function defined for each type of Asian option.

As a technical means for achieving the above-described technical problem, the Asian option pricing apparatus based on the Heston probability variability model according to the first aspect of the present invention includes a pricing variable including a type of Asian option to calculate the expected price; A data collector configured to collect parameters corresponding to a pricing function defined based on the heston probability variability model; An option price determination unit which determines the expected price by substituting the pricing variables and parameters into a pricing function corresponding to a selected Asian option type among pricing functions classified according to the type of Asian option; And an output unit for outputting a result including the determined expected price and the selected Asian option type.

The Asian option pricing method based on the Heston probability variability model according to the second aspect of the present invention includes: receiving a pricing variable including a type of Asian option selected by a user; Selecting a pricing function corresponding to the type of the selected Asian option among pricing functions defined based on the heston probability volatility model based on the price of the underlying option of the Asian option; Parameters such as a volatility variable of the price of the underlying asset corresponding to the selected pricing function, a long-term average constant of the volatility variable, an average regression rate constant, a volatility change constant and a correlation coefficient of the underlying asset price and the volatility variable; Calculating an expected price for the selected Asian option by applying pricing variables such as an expiration time of the selected Asian option, a price of an underlying asset, an exercise price, an observation point, and an interest rate; And providing the calculated expected price to the user.

According to the above-described problem solving means of the present invention, the Asian option pricing apparatus and method based on the Heston probability variability model reduces the time and computing power required to determine the discrete time geometric mean Asian option price, the appropriate Asian option price It has the advantage of providing

In addition, by applying such a configuration to various computer systems for mediating option trading, it is possible to help the market participants to make quick decisions.

DETAILED DESCRIPTION Hereinafter, exemplary embodiments of the present invention will be described in detail with reference to the accompanying drawings so that those skilled in the art may easily implement the present invention. The present invention may, however, be embodied in many different forms and should not be construed as limited to the embodiments set forth herein. In the drawings, parts irrelevant to the description are omitted in order to clearly describe the present invention, and like reference numerals designate like parts throughout the specification.

1 is a block diagram of an Asian option pricing apparatus according to an embodiment of the present invention.

As shown in FIG. 1, the Asian option pricing apparatus 100 according to an exemplary embodiment of the present invention includes a data collector 110, a parameter estimator 120, an option price determiner 130, an output unit 140, and the like. It includes.

 In this case, the data collection unit 110 is based on a pricing parameter and Heston's stochastic volatility model for pricing Asian options input from a user who wants to trade options or a stock exchange brokering options trading. Gather the parameters needed for the pricing functions that are created.

In this case, the pricing variables include information on the type of Asian option for which the expected price is requested from the user or the stock exchange system, the basic asset price (St), the interest rate (r), the exercise price (K), and the expiration date at the observation time t. Variable for (T) and the like.

The parameters may include a volatility process variance (vt) of the underlying asset price, a constant long-run average (θ) of the variability variable, and a constant rate of mean reversion (κ). , A constant volatility of volatility (σ), and a correlation coefficient (ρ) of the underlying asset price and the volatility variable.

The parameter estimator 120 obtains parameters collected through the data collector 110 and estimates values of the parameters. In this case, since the values of the parameters cannot be calculated accurately, the estimated values based on the statistics are used.

In addition, the parameter estimator 120 provides the estimated values of the parameters to the data collector 110.

The option pricing unit 130 calculates a price for the corresponding Asian option by substituting the pricing variable and parameter values into the pricing functions defined for determining the Asian option price.

Specifically, the option pricing unit 130 includes a pricing function management module 131, a momentum generation function application module 132 and the option price calculation module 133.

At this time, the pricing function management module 131 manages the pricing basis function based on the differential equation defined based on the heston probability volatility model, and the momentum generation function application module 132 generates conditional moments in the pricing basis function. Manage the pricing function defined by integrating the function.

In an embodiment of the present invention, the pricing function management module 131 and the moment generation function application module 132 perform various operations for deriving the pricing basis function or pricing functions, and store the results of the operation in a database ( Not shown). However, the pricing function management module 131 and the moment generation function application module 132 as described above may be pre-stored or inputted into each module as a result state in which the pricing basic function or pricing functions are calculated. It is also possible to provide a pre-stored or input pricing function according to the request of the option price calculation module 133.

On the other hand, prior to explaining the process of defining the corresponding pricing function for each type of Asian option through the pricing function management module 131 and the moment generation function application module 132 according to an embodiment of the present invention, convenience of understanding For explaining the process of deriving the Asian option pricing function that can be applied to the present invention.

In the following, the variables mentioned are briefly described, where {St} t≥0 is the underlying asset price at observation point t, r is the interest rate, vt is the volatility variable at observation point t, and Wt is related to the price of the underlying asset. Brownian motion, Zt, represents the Brownian motion relative to the variability variable.

Θ is a constant long-run average of the variability variable vt, and σ is a constant volatility of volatility. κ is the rate of mean reversion, ρ represents the correlation coefficient of the underlying asset price and the volatility variable.

The Heston probability volatility model is a model that assumes a probability differential equation as shown in Equation 1 below for the price St of the underlying asset at maturity.

[Equation 1]

In this case, it is assumed in Equation 1 that the two Brownian motions Wt and Zt have a correlation. That is, assume dWtdZt = ρdt.

On the other hand, when calculating option prices for interest rates, risk-neutral conditions should be assumed, not the real world. Thus, there is a need to define a new probability measure by setting the market price of risk. In general, by converting a probability measure of a real state into a risk-neutral probability measure Q, it is possible to shift only the mean without changing other properties of the probability distribution, where the probability measure Q is equivalent to that of the real state probability measure. This is called a martingale measure. These probability measure transformations are useful for pricing derivatives because they can easily eliminate the risk premium of the underlying asset without changing the structure of volatility.

 Under this risk-neutralized martingale measure, it is assumed that the underlying asset price St satisfies a probability equation as shown in Equation 2 below.

[Equation 2]

At this time, the pricing function management module 131 according to the embodiment of the present invention is any time set I ({t1, t2, t3, ..., tn}, 0 ≤ t1 <t2 <… <tn ≤ T) When the geometric mean function of the underlying asset price St is expressed as shown in Equation 3 below.

&Quot; (3) &quot;

이고, 고정형 행사 아시안 풋 옵션(fixed strike Asian put option)의 이득 함수는

이고, 부동형 행사 아시안 콜 옵션(floating strike Asian call option)의 이득 함수는

이며, 부동형 행사 아시안 콜 옵션(floating strike Asian put option)의 이득 함수는

이다.When the geometric mean of the underlying asset price is defined as in Equation 3, the gain function of the fixed strike Asian call option is

The gain function of the fixed strike Asian put option is

The gain function of the floating strike Asian call option is

The gain function of the floating strike Asian put option is

to be.

In addition, the pricing function management module 131 defines pricing basic functions for determining the price of the Asian option using Equation 3 and gain functions for each type of the Asian option.

At this time, the pricing function management module 131 stores a pricing basis function for determining the price of the fixed event Asian call option represented by Equation 4 below.

가 적용된다.Gain function at expiration time T for Equation 4

Is applied.

 &Quot; (4) &quot;

In addition, the pricing function management module 131 defines a pricing basis function for determining the price of the floating exercise Asian call option represented by Equation 5 below.

가 적용된다.Gain function at expiration time T for Equation 5

Is applied.

[Equation 5]

The moment generation function application module 132 integrates the price determination basic function defined as in Equations 4 and 5 with a conditional moment generating function to determine the discrete time geometric mean Asian option type price. Define the decision function.

에 대한 조건부 적률 생성 함수

는 아래 수학식 6과 같이 정의된다.At this time, under probability measure Q

Conditional moment generation function for

Is defined as in Equation 6 below.

 &Quot; (6) &quot;

의 정의역은 모든

에 대해서

일 때 수학식 7과 같이 성공적으로 정의된다.Conditional moment generation functions, such as

The domain of all

about

Is successfully defined as in Equation (7).

 [Equation 7]

를 도출하기 위한

, 및 Tk(t)은 각각 아래 수학식 8과 같이 정의할 수 있다.Meanwhile, conditional moment generation function

To derive

, And Tk (t) may be defined as Equation 8 below.

[Equation 8]

는 아래 수학식 9과 같이 나타낼 수 있다.Applying Equation 8

May be expressed as in Equation 9 below.

[Equation 9]

Where,

를 가격 결정 함수 관리모듈(131)로부터 생성된 가격 결정 기초 함수와 적분하여 하기 수학식 10 내지 13과 같은 가격 결정 함수를 정의한다.At this time, the moment generation function application module 132 is a combined generation function derived from the above

Integrating with the pricing basis function generated from the pricing function management module 131 defines a pricing function as shown in Equations 10 to 13 below.

First, the pricing function of the fixed event Asian call option is shown in Equation 10 below.

 &Quot; (10) &quot;

And, the pricing function of the fixed exercise Asian put option is shown in Equation 11 below.

&Quot; (11) &quot;

In addition, the pricing function of the floating event Asian call option is shown in Equation 12 below.

[Equation 12]

In addition, the pricing function of the floating exercise Asian put option is shown in Equation 13 below.

&Quot; (13) &quot;

The option price calculation module 133 corresponds to the type of option selected by the user among the pricing functions for each Asian option type defined (or stored) through the pricing function management module 131 and the moment generation function application module 132. Select a pricing function.

In addition, the option price calculation module 133 substitutes a pricing parameter and each parameter value collected by the data collector 110 into the selected pricing function to calculate a closed form solution, that is, the option price of the pricing function. Calculate.

The output unit 140 displays or prints information such as option price and type determined from the option price determining unit 130 on an output device such as a screen and a printer and outputs the information for the user to check.

2 is a flowchart illustrating an Asian option pricing method according to an embodiment of the present invention.

3 is a view comparing the Asian option pricing method and the conventional Asian option price estimation method according to an embodiment of the present invention.

First, a pricing function for each Asian option type is defined based on the Heston probability variability model (S210).

In this case, the pricing function for each Asian option type includes a fixed exercise Asian call option, a fixed exercise Asian put option, a floating exercise Asian call option, and a floating exercise Asian put option.

Specifically, the pricing function for each Asian option type is defined as in Equations 10 to 13 above, and the conditional rate factor for the pricing base function obtained by geometrically averaging the price of the underlying asset based on the Heston probability volatility model under the risk neutral measure. Integral with the generation function.

Then, the expected price request for any one of the types of Asian options is input, and a pricing function corresponding to the corresponding type is selected from the above-described pricing functions (S220).

In this case, the expected price request includes data on pricing variables such as the type of Asian option, the underlying asset price, the interest rate, the exercise price, and the expiration date.

In addition, the expected price request may be input from a financial system (not shown) such as a stock exchange, or may be input directly by a user who wants to obtain information about an option through a user interface.

Next, the parameters to be substituted into the selected pricing function are checked (S230).

In this case, the parameters are parameters for differential equations based on the Heston probability variability model, and may be estimated based on statistics previously stored in financial institutions, securities trading systems, and the like.

Specifically, the parameters include a volatility process variance (vt) of the underlying asset price, a constant long-run average (θ) of the variability variable, and a constant rate of mean reversion (κ). , A constant volatility of volatility (σ), and a correlation coefficient (ρ) of the underlying asset price and the volatility variable.

Thereafter, the data collected in steps S220 and S230 and the estimated parameters are substituted into the selected pricing function to calculate an expected price for the corresponding Asian option (S240).

Next, a screen or printing is output such that the calculated Asian option price and option type information are readable by the user (S250).

As such, by using the Heston probability variability model to generate a function for determining the Asian option price, and using the pricing function to determine the price, the required computation time and computing power consumption are reduced compared to the conventional price estimation method.

In detail, FIG. 3 compares a result of calculating an estimated price for an Asian option according to a price estimation method through a conventional simulation under the same condition and a price of an Asian option determined using a pricing function defined according to the present invention. A list is shown.

3, a price and an estimated time calculated using a pricing function according to the present invention may be compared with an estimated price and estimated time according to a conventional price estimating method for a fixed event Asian call option. In this case, T means an expiration time, and CPU means a computing operation time.

In addition, in Fig. 3, the exercise price K is 90, the underlying asset S is 100, the volatility variable v of the underlying asset price is 0.09, the observation time t is 0, the interest rate is 0.05, the average regression rate constant κ is 1.15, As an example, the long-term average constant θ of the variability variable is 0.348, the correlation coefficient ρ of the underlying asset price and the variability variable is −0.64, and the variability change constant σ is 0.39.

For example, in FIG. 3, when the expiration time T is 0.2 and n is 100, the price of the option determined through the pricing function defined in accordance with the present invention has a value of 10.6731, and the computation processing time, that is, the CPU has a value of 2.516. I can confirm that I have. At this time, under the same conditions, the price of the option estimated through the conventional price estimation method is estimated to be 10.6670, and it can be seen that the values have the values of 10.6207 and 10.7134 as confidence intervals. In addition, it can be seen that the simulation computing time is 100.375 consumed in the conventional price estimation method.

In the above, the pricing according to the conventional price estimating method and the pricing method of the present invention was compared on the basis of the same expiration time and the n value. In this case, as shown in FIG. 3, the processing speed when calculating the option price using the pricing method according to the present invention may be remarkably faster than the conventional price estimation method.

In addition, in determining the option price, in the conventional price estimating method, the estimate of the option price and the confidence interval values are calculated, but in the pricing method according to the present invention, the most suitable expected price is calculated so that the option price can be efficiently determined. can confirm.

While the apparatus and method of the present invention have been described in connection with specific embodiments, some or all of their components or operations may be implemented using a computer system having a general hardware architecture.

The option pricing method performed by the Asian option pricing apparatus based on the Heston probability variability model according to the present invention may also be implemented on a computer capable of performing data processing and operations. That is, the computer-readable recording medium may be embodied as program code that can be read by the computer. The computer-readable recording medium includes all kinds of recording devices for storing data that can be read by a computer system. Examples of the computer-readable recording medium include ROM, RAM, CD-ROM, magnetic tape, floppy disk, optical data storage device, and the like, and are implemented in the form of a carrier wave (for example, transmission through the Internet). It includes being. In addition, the computer-readable recording medium may be stored in a computer-readable code distributed in a networked computer system and executed in a distributed manner.

In addition, the option pricing method performed by the Asian option pricing apparatus based on the Heston probability variability model according to the present invention includes a microprocessor for controlling the computer in a general computer system including a network interface, an input / output interface, and a storage device. Can be done through For example, an Asian option determining apparatus based on the Heston probability variability model according to the present invention may be implemented in a system-on-chip form in the microprocessor.

The foregoing description of the present invention is intended for illustration, and it will be understood by those skilled in the art that the present invention may be easily modified in other specific forms without changing the technical spirit or essential features of the present invention. will be. It is therefore to be understood that the above-described embodiments are illustrative in all aspects and not restrictive. For example, each component described as a single entity may be distributed and implemented, and components described as being distributed may also be implemented in a combined form.

The scope of the present invention is shown by the following claims rather than the above description, and all changes or modifications derived from the meaning and scope of the claims and their equivalents should be construed as being included in the scope of the present invention. do.

1 is a block diagram of an Asian option pricing apparatus according to an embodiment of the present invention.

2 is a flowchart illustrating an Asian option pricing method according to an embodiment of the present invention.

3 is a view comparing Asian option pricing method and conventional Asian option price estimation method according to an embodiment of the present invention.

Claims (10)

In the Asian option pricing system based on the Heston ’s stochastic volatility model, A data collection unit for collecting a parameter corresponding to a pricing variable including a type of an Asian option for calculating an expected price and a pricing function defined based on the heston probability variability model; An option price determination unit which determines the expected price by substituting the pricing variables and parameters into a pricing function corresponding to a selected Asian option type among pricing functions classified according to the type of Asian option; And And an output unit configured to output a result including the determined expected price and the selected Asian option type. The option pricing unit, When the type of the selected Asian option is a fixed strike Asian call option, the expected price is determined through a pricing function according to Equation 1 below. &Quot; (1) &quot;

 When the type of the selected Asian option is a fixed strike Asian put option, the expected price is determined through a pricing function according to Equation 2 below. <Equation 2>

When the type of the selected Asian option is a floating strike Asian call option, the expected price is determined through a pricing function according to Equation 3 below. &Quot; (3) &quot;

And when the type of the selected Asian option is a floating strike Asian put option, determining the expected price through a pricing function according to Equation 4 below. &Quot; (4) &quot;

The method of claim 1, The pricing variable is Including the expiration date of the Asian option, the price of the underlying asset, the exercise price, the observation point and the interest rate; The parameters are And a volatility variable of the price of the underlying asset, a long-term average constant of the volatility variable, an average regression rate constant, a volatility change constant, and a correlation coefficient of the underlying asset price and the volatility variable. The method of claim 1, The option pricing unit, A pricing function management module configured to manage a pricing basis function defined by discrete time geometric mean of the underlying asset price of the Asian option based on a histone probability variability model; A probability generation function application module for managing a pricing function for each type of the Asian option defined by integrating the pricing basis function with a conditional moment generating function under a risk-neutralized measure; And And an option price calculation module that calculates the expected price by substituting the pricing variable and the parameters into a pricing function corresponding to the selected Asian option type. delete In the Asian option pricing device to determine the Asian option price based on the Heston ’s stochastic volatility model, Receiving a pricing variable including a type of the selected Asian option; Selecting a pricing function corresponding to the type of the selected Asian option among pricing functions for the expected price of the Asian option based on the heston probability variability model; A parameter comprising a volatility variable of an underlying asset price corresponding to the selected pricing function, a long term average constant of the volatility variable, an average regression rate constant, a volatility change constant, and a correlation coefficient of the underlying asset price and the volatility variable; Calculating an expected price for the selected Asian option by applying pricing variables including expiration time, underlying asset price, exercise price, observation point and interest rate of the selected Asian option; And Outputting the calculated expected price, In selecting the pricing function, If the type of the selected Asian option is a fixed strike Asian call option, the pricing function according to Equation 7 is selected. <Equation 7>

When the type of the selected Asian option is a fixed strike Asian put option, a pricing function according to Equation 8 is selected. <Equation 8>

If the type of the selected Asian option is a floating strike Asian call option, select a pricing function according to Equation 9 below. &Quot; (9) &quot;

And selecting a pricing function according to Equation 10 when the selected type of the Asian option is a floating strike Asian put option. <Equation 10>

6. The method of claim 5, The pricing function is A conditional moment generating function is defined by integrating a conditional moment generating function based on a difference between a discrete time geometric mean of the underlying asset and the exercise price of the Asian option, The conditional moment generation function is the Asian option pricing method, characterized in that defined as in Equation 6. <Equation 6>

Where,

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