JP2003067581A - Financial derivatives evaluation system - Google Patents

Abstract

(57) [Summary] [Problem] To improve the accuracy of valuation of financial derivatives by Black-Scholes equation. SOLUTION: An extended Black-Scholes equation including a trend μ of an underlying asset of a financial derivative as a parameter is established,
The exact solution is obtained and stored in the storage device 2. Each statistical data obtained from the time-series price data in the financial market is input from the input means 1 and stored in the storage means 2. The price calculation means 3 obtains the price of the financial derivative product from the exact solution based on each statistical data, and outputs the price from the output means 4.
By expressing the trend term, it is possible to evaluate the short-term trend of the financial market.
It can capture the phenomenon of economic physics more accurately than the les equation. It is possible to quantify how the market evaluates asset prices at the current time.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a financial derivative product evaluation system, and more particularly to a financial derivative product evaluation system for obtaining a price of a financial derivative product such as an option by solving an extended Black-Scholes equation.

[0002]

2. Description of the Related Art In financial engineering, actual data on price fluctuations of financial products and financial derivatives in the securities market are analyzed,
Probabilistically seeking prices and risks are being carried out.
Financial products include stocks and bonds, and financial derivatives include stock options.

Options are not securities such as stocks themselves, but underlying assets (securities such as stocks or commodities)
Is a right to buy or sell at a specified price (exercise price) on a specific future date (maturity date). The option buyer does not necessarily have to exercise that right and can choose whether to buy the underlying asset. Futures contracts must meet a sale contract at maturity, whereas option contracts do not impose a sale or purchase obligation.

The options include call options (ca
ll option) and put option. Call options are the right to purchase an asset at a fixed price by a fixed future date. Put options are the reverse and are the right to sell. There are European and American types of options. The European type is an option that can exercise the right only at the maturity date, and the American type is an option that can exercise the right at any time before the maturity date.

The Black-Scholes equation, which was published by Fischer Black and Myron Scholes in 1973, is widely used as an evaluation formula for the price of financial derivative products as a means for obtaining the price of such financial derivative products. Black-Schole
The s equation will be briefly described. For details, see Reference 1:
[Fischer Black and Myron Scholes, "The Pricing of
Options and Corporate Liabilities, "(1973), Journa
l of Political Economy], Reference 2: [Chihiko Minoru's "Understood Black-Scholes Model" (Toyo Keizai Shinposha, March 2000)], Reference 3: [Yun-gen, Takahiko Tanahashi, "Com
See "Putational Market Dynamics Proposal," (Proceedings of Computational Engineering Conference 2000)].

It is assumed that stock price fluctuations follow the Wiener process. The Wiener process is a Marcov stochastic process that assumes that future states do not depend on past processes. It is used to represent the Brownian motion of molecules in the physical world. The Ito process adds the term of drift to this, and also introduces a parameter function of time and variables.

The Black-Scholes equation is based on the assumption that the fluctuation of the underlying securities is Ito's stochastic process. Ito's stochastic process is a stochastic process described by dS / S = μdt + σdW. Where S is the asset price,
μ is the trend of the asset price S, and σ is the volatility of the asset price (price series standard deviation value). W is a Wiener process and satisfies (dW) ² = dt. This Ito process has been well studied and is recognized as a model for describing changes in asset prices.

[0008] Fischer Black and Myron Scholes assume a portfolio Π, which is a combination position in delta hedging, in the hypothesis that arbitrage transactions are executed in the market (hereinafter referred to as arbitration hypothesis). For portfolios, the risk-free safe portfolio increment can be expressed as dΠ = Πrdt. Based on this arbitrage hypothesis, the Black-Scholes equation that determines the dynamics of the derivative product is derived.

An example will be described. Consider a portfolio Π that purchases one unit of a derivative product f of an underlying asset such as a stock (the price is also represented by f; the same applies below) and sells the underlying asset S by Δ units. That is, the portfolio Π is defined as Π = f−ΔS. The fluctuation of this portfolio is dΠ = df−ΔdS, where Δ is constant. Using Ito's lemma, df = [∂f / ∂t + (σS) ² (∂ ² f / ∂S ² ) / 2] dt +
Since it is (∂f / ∂S) dS, substituting this into the formula of dΠ gives dΠ = [∂f / ∂t + (σS) ² (∂ ² f / ∂S ² ) / 2] dt +
It can be transformed into (∂f / ∂S-Δ) dS.

On the right side, there is a definite term [∂f / ∂t + (σS) ² (∂ ² f / ∂S ² ) / 2] dt and a term (∂f / ∂S-Δ) dS that changes by chance. Is appearing. The term that fluctuates by chance is the stock price fluctuation dS
Brought by. This dS varies stochastically with the increment dW of Brownian motion, so it cannot be known in advance. The term exposed to this stochastic change is the risk of the portfolio. In order to eliminate this risk, {(∂f / ∂S) −Δ} dS = 0. That is, Δ = (∂f / ∂S) may be set. In other words, the option value f rises due to the buying of one unit of the call due to the rise of the stock price S, but if the shares are held equal to this rise (∂f / ∂S), this portfolio will be at risk. Will be a secure portfolio without. Of course, this (∂f /
The value of ∂S) changes due to fluctuations in the stock price S, so S
It is necessary to continuously restructure the portfolio in line with changes in the.

The change in the portfolio Π at this time is dΠ = [(∂f / ∂t) + (σS) ² (∂ ² f / ∂S ² ) / 2] dt. Since the hedge ratio in this case is Δ, this hedge is called delta hedging. By performing delta hedging, portfolio Π becomes a risk-free and safe portfolio because it does not include random fluctuations.

Fischer Black and Myron Sch is that this safe portfolio Π is equivalent to a payoff equivalent to a bond that is a safe asset because there is no probability fluctuation.
This is the claim of oles. By this arbitration hypothesis dΠ = rΠdt, B1ack-Scholes equation (∂f / ∂t) + (σS) ² (∂ ² f / ∂S ² ) / 2 + rS (∂f
/ ∂S) = rf is derived. The Black-Scholes equation is ultimately reduced to the heat conduction equation by variable transformation.

[0013]

[Problems to be Solved by the Invention] However, the conventional Black-
In the Scholes equation, the element of the trend of the underlying asset is not taken into consideration, and it is difficult to evaluate the short-term trend of the financial market, and there is a problem that it is inaccurate.
Even though the asset price S is described by the trend μ, Black-Scholes for its derivative f (t, S)
Equation (∂f / ∂t) + (σS) ² (∂ ² f / ∂S ² ) / 2 + rS (∂f
The trend μ is not included in / ∂S) = rf. The derivation of the Black-Scholes equation assumes dS = rSdt. The Black-Scholes equation is derived by arbitration theory replacing the effect modeled by the term caused by the trend and the term caused by the stochastic fluctuation by the term equivalent to the interest rate. .

Furthermore, since the arbitrage hypothesis dΠ = Πrdt is inconsistent with the Ito process, the Black-Scholes equation becomes
This is because the behavior of economic physics is not always expressed correctly. This is clear from the fact that the characteristic direction of the Black-Scholes equation in logarithmic space does not match the rate of change of the expected value obtained from Ito's stochastic differential equation.

Further, since the fluctuation characteristic of the asset price has been estimated by the statistical analysis of past time series price data in the financial market, it cannot be quantified how the market evaluates the asset price at the present time. There are also problems. Further, since only one product can be handled, it is difficult to obtain an analytical solution when a plurality of assets are collected and handled collectively.

It is an object of the present invention to solve the above-mentioned conventional problems, improve the accuracy of price evaluation of financial derivative products, and enable evaluation of financial derivative products including a plurality of underlying assets.

[0017]

In order to solve the above problems, the present invention provides a financial derivative product evaluation system that solves a stochastic second-order partial differential equation to evaluate the price of a financial derivative product. Input means for inputting statistical data obtained from time series price data in the financial market concerning the underlying asset, storage means for storing the statistical data, and extended Black-Scholes equation including the trend μ of the underlying asset as a parameter based on the statistical data And a price calculation means for obtaining the price of the financial derivative product from the exact solution of.

Due to such a structure, the conventional Bl
It can capture the phenomenon of economic physics more accurately than the ack-Scholes equation. By introducing a trend term,
It is possible to characterize the tendency of asset price fluctuations within a short period of time, which was difficult to evaluate in the past. By examining the properties of the exact solution obtained from the new equation with this parameter introduced, the solution more realistically expresses the economic dynamics than the conventional one.

Further, based on the statistical data, the multivariable extended Black-Scholes equation including the trends μ _i (i = 1 to n, n is the number of underlying assets) of a plurality of underlying assets as parameters is numerically analyzed to derive financially. It is configured to include a numerical analysis means for obtaining the price of a product. With this configuration,
It is possible to evaluate financial derivative products that combine multiple underlying assets.

[0020]

BEST MODE FOR CARRYING OUT THE INVENTION Embodiments of the present invention will be described in detail below with reference to FIGS.

(First Embodiment) In the first embodiment of the present invention, the price of a financial derivative is evaluated by an exact solution of the extended Black-Scholes equation introducing the trend term of the underlying asset of the financial derivative. This is a financial derivative product evaluation system.

FIG. 1 is a conceptual diagram of a financial derivative product evaluation system according to the first embodiment of the present invention. In FIG. 1, an input unit 1 is a unit for inputting each parameter including statistical data obtained from time-series price data in a financial market regarding an underlying asset of a financial derivative product. Storage means 2
Is a memory that stores each parameter and the exact solution of the extended Black-Scholes equation. The price calculation means 3 is a calculation device that substitutes each parameter into the exact solution of the extended Black-Scholes equation to calculate the price of the financial derivative product. The output unit 4 is a unit that outputs the price of the financial derivative product as the calculation result. FIG. 2 is a flowchart showing an operation flow of the financial derivative product evaluation system in the first embodiment of the present invention.

The operation of the financial derivative product evaluation system according to the first embodiment of the present invention configured as described above will be described. First, we explain how to derive the extended Black-Scholes equation. The extended Black-Sholes equation is based on the following three points. (1) The price of the underlying asset (stock) follows Ito's stochastic differential equation (the same as the conventional Black-Scholes equation). dS / S = μdt + σdW (2) Financial derivative (optional) f satisfies the following Ito's lemma when μ and σ are constant (conventional Black-Scho
Same as the les equation). df = [∂f / ∂t + (σS) ² (∂ ² f / ∂S ² ) / 2] dt +
(∂f / ∂S) dS (3) Risk-free and safe portfolio Π (= f- △
The increment of S) follows the following process (unlike the traditional Black-Scholes equation). dΠ = rfdt− (∂f / ∂S) μSdt

The partial differential equation of the financial derivative product f is (∂f / ∂t) + (σS) ² (∂ ² f / ∂S ² ) / 2 + μS (∂f
It can be described as / ∂S) = rf. Trend μ is taken into consideration in this extended Black-Sholes equation. If μ = r is set in this equation, it agrees with the conventional Black-Scholes equation.

The derivation process of the extended Black-Scholes equation will be described. When the interest rate is r, the change in the financial derivative product f is df = rfdt. The risk-free rate is applied to the financial derivative product f because its nature is defined as a contract.
If the stochastic variation term is ignored in the Ito process, then dS = μSdt. For the underlying asset S, it is appropriate to set the trend μ as the rate of change in accordance with the idea of the Ito process. It means that the fluctuation of the derivative product corresponds to the interest cost when the interest rate changes, whereas the change of the asset fluctuation characteristic due to the interest rate change is more important for the underlying asset.

Since delta hedging results in Δ = ∂f / ∂S, the safe portfolio Π change is dΠ = df−ΔdS = (rf− (∂f / ∂S) μS) dt. From Ito's lemma, the partial differential equation that financial derivative product f should satisfy is (∂f / ∂t) + (σS) ² (∂ ² f / ∂S ² ) / 2 + μS (∂f
/ ∂S) = rf. We call this the extended Black-Scholes equation.

The rationale for deriving the extended Black-Scholes equation will be described. One of the grounds is that the arbitration hypothesis contradicts the Ito process. That is, ignoring the probability variation term, the Ito process becomes dS / S = μdt. This contradicts the arbitrage hypothesis dS / S = rdt.

Another reason is the characteristic of geometric Brownian motion.
Direction is μ- (1/2) σ²So, on this basis
The characteristic direction of the Black-Scholes equation must also agree with this.
It means that When μ and σ are constant, Ito
The stochastic process of logS (t) = logS₀+ (Μ-σ²/ 2) t + σW (t) Becomes here, S (0) = S₀ Is. When viewed in logarithmic space, the expected value and variance are E [logS (t)] = logS₀+ (Μ-σ²/ 2) t V [logS (t)] = σ²t Given in. When these time derivatives are calculated, dE [logS (t)] / dt = μ−σ²/ 2 dV [logS (t)] / dt = σ² Becomes LogS space of function f that represents the price of financial derivatives
The characteristic speed at is (μ−σ ²/ 2). Therefore, Black
-The characteristic speed of the Scholes equation is (μ−σ²/ 2)
It is

Among the extended Black-Scholes equations improved so that the characteristic direction and the rate of change with time are the same, the conventional Black is included.
-The underlying asset trend μ, which was not included in the Scholes equation, will be included as a parameter. The parameter μ corresponds to the expected rate of return on stocks noted by Paul A Samuelson.

Second, a method of solving the extended Black-Scholes equation will be described. In order to derive the partial differential equation in the logarithmic space, a variable conversion of x = log (S / K) τ = T-tg (τ, x) = f (t, S) / K is performed. Here, K is the exercise price and T is the period. As a result, partial differential equation for the function g (convection-diffusion ^{equation) ∂g / ∂τ = σ 2 (} ∂ 2 g / ∂x 2) / 2 + (μ-σ 2/2)
(∂g / ∂x) -rg is obtained. First term on the right side, the diffusion term of the diffusion coefficient (sigma ^2/2), the second term advection term, characteristic velocity equations, dx / dt = -dx / dτ = μ- (σ 2/2) Given in. This is equal to the rate of change of the expected value in the Ito process.

The Black-Scholes equation is finally reduced to a heat conduction equation by variable transformation. This is because the asset price fluctuation is expressed as a diffusion phenomenon, and therefore has the same form as the equation that describes heat diffusion. Extended Black-Sc
The holes equation captures fluctuations in asset prices as a mixed phenomenon of diffusion and advection phenomena, and is therefore the advection-diffusion equation normally used in fluid dynamics. The advection-diffusion equation is
Unlike the diffusion equation, it contains an advection term that is a nonlinear term, and therefore cannot be solved analytically in general. But extended
The exact solution of the Black-Scholes equation is the traditional Black-Scholes
Like the equation, it can be obtained by using the solution of the heat conduction equation.

The exact solution of the diffusion equation ∂u / ∂t = ν∂ ² u / ∂x ² [initial condition u (0, x) = u ₀ (x)] is Fourier
By r conversion, u (x, t) = ∫ _-∞ ^∞ (2πνt) ^-1/2 exp (− (x−y) ² /
(4νt)) u ₀ (y) dy Here, ν is a diffusion coefficient. In order to use this exact solution, variable transformation g (τ, x) = h (τ, x) exp (αx + βτ) is performed on the function g (τ, x). However, α = - (μ- (σ 2/2)) / σ 2 = - is the (k-1) / 2 β = -σ 2 (k-1) 2/8-r k = 2μ / σ 2 .

[0033] As a result, the function h (τ, x) satisfies a simple diffusion equation ^{∂h / ∂τ = (σ 2/} 2) (∂ 2 h / ∂x 2). That is, it is the same as the equation of heat conduction.
When t = τ, τ = 0, so the initial condition is h (0, x) = exp (−αx) g ₀ (x). _{However, g 0 (x) = f} (T, S (T)) / K = (1 / K) max (S-K, 0) = max (S / K-1,0) = max (e x - 1,0). Substituting this result into the above equation, the initial condition is h (0, x) = max [exp ((k + 1) x / 2) -exp ((k-1) x
/ 2), 0].

From the above, the exact solution is obtained as f (t, S) = Kg (τ, x) = Kh (τ, y) exp (αx + βτ). However, h (τ, y) = I ₁ −I ₂ I ₁ = exp {(k + 1) x / 2 + (k + 1) ² σ ² τ / 2} N
(d ₁ ) I ₂ = exp {(k-1) x / 2 + (k-1) ² σ ² τ / 8} N
_{(d 2) N (d 1} ) = a ^{_{∫ -∞ d1 (2π) -1/2 exp}} (-z 2/2) dz k = 2μ / σ 2. N (·) is a standard normal distribution function. _{Here, d 1 = {x + (} μ + σ 2/2)} / (σ√τ) = d 2 + σ√τ d 2 = {x + (μ-σ 2/2)} / (σ√τ) = d 1 −σ√τ x = log (S / K).

Therefore, when the function f (t, S) is arranged, the exact solution f (t, S) = exp (-tr) {SN (d ₁ ) exp (μr) -KN (d
₂ )} is obtained. Here, if μ = r is set, it agrees with the exact solution of the Black-Scholes equation. That is, it shows that in the risk-free safe portfolio, the increase f (t, S) exp (tr) due to the interest rate is equal to {SN (d ₁ ) exp (μr) −KN (d ₂ )}. The first term increases according to the trend μ. So this solution represents a physically correct phenomenon.

European call option c
(τ, S) and the put option is p (τ, S), from the put call parity, c (τ, S) = f (t, S) p (τ, S) = c (τ, S) ) + Kexp (−rτ) −S. Therefore, the call and put option prices are c (τ, S) = exp (−rτ) {SN (d ₁ ) exp (μr) −KN, respectively.
(d ₂ )} p (τ, S) = S {N (d ₁ ) exp ((μ−r) τ) −1 + KN (−
It can be expressed as d ₂ ) exp (−rτ)}. However, the relational expression 1-N (d ₂ ) = N (-d ₂ ) for the normal distribution function was used.

Third, a method of analyzing financial derivative products by the extended Black-Scholes equation will be described. Extended Black-Schole
The s equation includes the underlying asset trend μ, which is not found in the conventional Black-Scholes equation, as a parameter. This parameter μ
The introduction of allows for new interpretations of volatility smiles measured in the derivative market.

In the derivative market, options are usually
Valued by the implied volatility. In general, options with different strike prices are valued higher as the strike price deviates from the current asset price, when viewed using the implied volatility as a measure. This condition is called the volatility smile. The volatility smile is calculated back from the actual option price in the market by the Black-Scholes equation.

The conventional thinking has been interpreted as "the volatility smile indicates a change in volatility when the underlying asset price changes near the exercise price". However, the change in the actual volatility curve is greatly affected by the speed of movement of the underlying asset price, in other words, the speed of price change. The volatility curve converted from the Black-Sholes equation has a larger smile as the exercise price moves away from the underlying asset price. This is because the prices of deep out-of-the-money options are overvalued when converted to volatility. As long as the Black-Scholes equation is used, it is inevitable that this exercise price will be overvalued as it deviates from the underlying price.

Since the extended Black-Sholes equation includes μ indicating the trend of the underlying asset, it is possible to convert the parameter μ as an implied mu like the volatility. The conversion method is a method of calculating the mu converted from the option price by the extended Black-Sholes equation, with the constant volatility at the exercise price of 100 as the reference volatility.

The reference volatility to be evaluated is the implied volatility of the at the money option. In addition, all volatility added to out-of-the-money options, which was measured as a conventional volatility smile, is converted into a trend implied as an implied mu.

Since the mechanism of market is obviously a feedback structure, information influences price formation. Blac
With the k-Scholes model known to all market participants, the arbitrage assumptions claimed by Fischer Black and Myron Scholes have been realized. Specifically, even if the interest rate r and the underlying asset price trend μ are different, if the difference is relatively small, the arbitrage transaction converges the underlying asset price trend μ to the interest rate r. From that point, Blac
The ripple effect of the k-Scholes equation is quite strong.

However, when the underlying asset price trend μ largely deviates from the interest rate r, the convergence of μ to r does not occur. In fact, when a company's stock price changes significantly due to the transmission of information such as business performance, arbitrage traders set prices assuming a considerable margin.
The es equation makes it possible to evaluate the market by evaluating volatility and mu when this dynamic occurs.

The extended Black-Sholes equation is the Black-Schole
The idea of advection was introduced into the diffusion equation called s equation and expanded to the advection diffusion equation. Considering this as a phenomenon, it means that the market price change is not just a stochastic diffusion phenomenon, but a mixed phenomenon of advection phenomenon that means directionality at each moment and diffusion phenomenon expressed as a stochastic element. To do. In other words, extended Black-Shol
The es equation considers the asset price as a flow in phase space,
The price fluctuation is made to correspond to the unsteady flow in the price space.

The stochastic volatility model that has become the mainstream in recent years generally has long-term average volatility, and in addition to that, stochastic volatility fluctuations exist. If this is interpreted as a physical phenomenon, the degree of diffusion becomes stochastic, meaning that the object is moving in a fixed direction. Meanwhile, extended Black-Sholes
The equation is that the diffusion phenomenon is just a diffusion phenomenon,
It is assumed that the subject's independent movement is changing. It is important to consider how to grasp the advection of the market, and to calculate the implied mu in the sense that the market evaluates independent movements.

The extended Black-Sholes equation treats a random variable trend as a variable, as noted by the Ito process. The difference in way of thinking about trend μ makes it possible to translate the overvaluation of deep out-of-the-money options such as volatility smiles into a more realistic valuation.
In addition, the display with the parameter Impolidemu makes it possible to show the point of view for the mean reversion of the market.

Fourthly, the operation of the financial derivative product evaluation system will be described with reference to FIGS. 1 and 2. Extended Black-Schole that includes the trend μ of the underlying asset as a parameter in advance
The s equation is established, its exact solution is obtained, and stored in the storage device 2. In step 11, from the input means 1, each parameter including statistical data obtained from time series price data in the financial market (trend μ of the underlying asset, volatility σ of the underlying asset, short-term interest rate r, strike price K, The period T) to the maturity is input and stored in the storage means 2. Step
At 12, the price calculation means 3 obtains the price of the financial derivative product from the exact solution based on each parameter. Step
At 13, the output means 4 outputs the price of the financial derivative product.

In order to calculate the implied mu, the trend μ can be obtained by not including the trend μ and including the option price in the market in the input. The price calculation means 3 uses the exact solution of the extended Black-Sholes equation to back-calculate the unknown number μ from the option price. This is the same as the method of back-calculating the unknown volatility σ from the option price using the exact solution of the conventional Black-Sholes equation.

As described above, according to the first embodiment of the present invention, the financial derivative product evaluation system uses the exact derivative of the extended Black-Scholes equation in which the trend term of the underlying asset of the financial derivative product is introduced. Since the price is evaluated, the phenomenon of economic physics can be accurately grasped, and the tendency of asset price fluctuation within a short time can be characterized.

(Second Embodiment) The second embodiment of the present invention includes the trend μ _i of a plurality of underlying assets as a parameter based on the statistical data of a financial derivative product consisting of a plurality of underlying assets. This is a financial derivative product evaluation system that numerically analyzes the multivariable extended Black-Scholes equation to obtain the price of financial derivative products.

FIG. 3 is a conceptual diagram of a financial derivative product evaluation system according to the second embodiment of the present invention. In FIG. 3, the input means 1 is means for inputting each parameter including statistical data obtained from time-series price data in the financial market concerning a plurality of underlying assets of a financial derivative product. The storage means 2 is a memory for storing each parameter and the extended Black-Scholes equation. The numerical analysis means 3 is an arithmetic device for numerically analyzing the multivariable extended Black-Scholes equation based on the parameters and calculating the price of the financial derivative product. The output unit 4 is a unit that outputs the price of the financial derivative product as the calculation result. FIG. 4 is a flowchart showing an operation flow of the financial derivative product evaluation system according to the second embodiment of the present invention.

The operation of the financial derivative product evaluation system according to the second embodiment of the present invention configured as described above will be described. The equation used in this financial derivative product evaluation system is a multivariable extended Black-Sholes equation obtained by extending the extended Black-Sholes equation into n variables. An n-variable option pricing equation that takes into account the trend is used for each stock price that has a geometric Brownian motion independently and has a strong correlation with each other. Using standardized normal variables, this pricing equation reduces to a diffusion equation with a Cross term. Therefore, an exact solution is obtained by using the n-dimensional Fourier transform. However, the exact solution is found in Cr
Only when there is no oss term, that is, when there is no correlation.

In order to derive the solution by deriving the Black-Scloles equation of n variables, the property of the geometric Brownian motion which is the basis of the fluctuation of the stock price is used. Ito's lemma is also extended to n variables. When variables become multivariable, the correlation of stock prices cannot be ignored. In case of strong correlation (always dW _i dW _j = d
t) and the case of no correlation (dW _i dW _j = 0, i ≠ j).

The derivation process of the multivariable Black-Scloles equation will be described. The Ito process is also called the geometric Brownian motion, and the stochastic process of the stock price S follows the Ito process, dS / S = μdt +
It can be described as σdW. The first term on the right side is a deterministic term, and μ means drift. The second term is the stochastic term, σ is the standard deviation of the normal distribution, W (t) is the expected value 0, and the standard Wie of variance σ ²
It is a ner process. Next, this is integrated. It has
Ito's lemma (Ito's integral formula) is necessary.

For an arbitrary function f (t, S) of t and S, Ito's integral formula is f (t, S (t))-f (0, S (0)) = ∫ ₀ ^t {( ∂f / ∂t) +
(1/2) (σS) ² (∂ ² f / ∂S ² )} dt + ∫ ₀ ^t (∂f / ∂
It can be described as S) dS. Here, dS = S (μdt + σdW).

In particular, if f (t, S) = logS is selected, then ∂f / ∂t = 0 ∂f / ∂S = 1 / S ∂ ² f / ∂S ² =-(1 / S ² ). Therefore, if Ito's integral formula is applied to d (logS), logS−logS (0) = ∫ ₀ ^t (μ− (1/2) σ ² ) dt + ∫ ₀ ^t σ
Get dW. Note that logS−logS (0) = ∫ ₀ ^t μdt + ∫ ₀ ^t σdW.

Particularly, when μ and σ are constant, logS = logS (0) + (μ− (1/2) σ ² ) t + σ (W (t) −
W (0)). However, W is a Wiener process, and W (0) = 0. Where S (0) = K, where K is the strike price. Therefore, if we introduce a new variable and put x = log (S / K) μ ~ = μ- (1/2) σ ² τ = T-t, the integration result will be simply x = μ ~ (T It can be displayed as −τ) + σW (τ).

The stochastic differential equation for this integral expression is dx = -μ to dτ + σdW. That is, x represents a geometric Brownian motion, and its expected value and variance are E (x) =-μ ~ τ V (x) = σ ² τ. This means that the probability density function is p (x, τ), and the probability in the dx section is p (x, τ) dx = (2πτσ) ^-1/2 × exp {( ^-1/2 ) ((x +
It means that it can be described as μ ~ τ) / (σ√τ)) ² } dx. Therefore, if the variable conversion is y = (x + μ ~ τ) / (σ√τ) dy = dx / σ√τ, then p (x, τ) dx = (1 / (√ (2π))) exp { -y obtain ^{2/2} dy = q (} y) dy.

The probability density function q is a normal distribution density function that does not include time and has an expected value of 0 and a variance of 1. That is,
The Ito process is normalized by using the variables y and τ. This means that y becomes a standard variable of Ito process. The complex movement showing the fluctuation of the option price is converted into a simple movement by using the standard variable.

[0060] When the component are summarized above variable transformation, τ = T-t x = log (S / K) y = (1 / σ√τ) becomes ^{{x + (μ-σ 2} /2) τ}. So far, it is the same as the conventional Black-Scloles equation.

In the case of multivariables, it is assumed that each stock price moves in a geometric Brownian motion independently and explicitly. That is, dS _i / S _i = μ _i dt + σ _i dW _i (i = 1, 2, ...,
n) is established. Next, the formula corresponding to the Black-Scholes equation for n variables is derived. Here, the correlation between dW _i and dW _j is written as dW _i dW _j = P _ij dt. If i = j, then dW _i dW _i = dt, so P _ii = 1. That is, it is autocorrelation. When i ≠ j, the value of the correlation coefficient P _ij is generally unknown. Here, independent of each other, that is, uncorrelated,
Consider the case of P _ij = δ _{ij and} the case of strong correlation and P _ij = 1.

First, consider the general case of dW _i dW _j = P _ij dt. When the option f is composed of n stock prices, it can be written as f = f (t, S ₁ , S ₂ , ..., S _n ). Ito's lemma for the option considering the correlation is df = {(∂f / ∂t) + (1/2) Σ _{i = 1} ⁿ Σ _{j = 1} ⁿ P for dW _i dW _j = P _ij dt. _ij σ _i σ _j
S _i S _j (∂ ² f / ∂S _i ∂S _j )} dt + Σ _{i = 1} ⁿ (∂f / ∂S _i )
It becomes dS _i .

Therefore, for delta hedging,
Consider the portfolio _{π = f-Σ i = 1} n Δ i S i. In order to create a risk-free and safe portfolio, if Δ _i is fixed and the total differentiation is performed, dπ = df−Σ _{i = 1} ⁿ Δ _i dS _i . Therefore, by substituting df in the above equation, the riskless condition can be written as the coefficient of dS _i being 0. From this, Δ _i = (∂f / ∂S _i ). At this time, dπ = {(∂f / ∂t) + (1/2) Σ _{i = 1} ⁿ Σ _{j = 1} ⁿ P _ij σ _i σ _j
S _i S _j (∂ ² f / ∂S _i ∂S _j )} dt is derived.

On the other hand, when considering the trend, the interest rate is r
Then, since df = rfdt dS _i = μ _i S _i dt, dπ = rfdt−Σ _{i = 1} ⁿ μ _i S _i (∂f / ∂S _i ) dt is obtained as the total derivative of π. If these dπ are set to be equal, an n-variable equation (∂f / ∂t) + (1/2) Σ _{i = 1} ⁿ Σ _{j = 1} ⁿ P _ij σ _i σ _j S _i S _j considering the correlation
(∂ ² f / ∂S _i ∂S _j ) ＋ Σ _{i = 1} ⁿ μ _i S _i (∂f / ∂S _i ) dt
= Rf is obtained.

Here, the variable transformation x _i = log (S _i / K _i ) τ = T-t is performed. That is, the conversion of (t, S _i ) → (τ, x _i ) is performed. As a result, ∂f / ∂τ = (1/2) Σ _{i = 1} ⁿ Σ _{j = 1} ⁿ P _ij σ _i σ _j (∂ ² f /
_{∂x i ∂x j) + Σ i} = 1 n (μ i -σ i 2/2) (∂f / ∂x i) -
Get rf. That is, the first term on the right side is the diffusion term including Cross section, the second _{term, μ i ~ = (μ i} -σ i 2/2) trend term with directions, the third term, the sink term according to interest Is represented.

Now, consider the case where P _ij = 1. Since the coefficient of the trend term μ _i ~ represents the characteristic direction of the hyperbolic PDE, a new variable transformation f (τ, x _i ) = exp (−rτ) g (τ, y _i ) y _i = By performing (x _i + μ _i ~ τ) / σ _i dy _i = dx _i / σ _i , the terms f and (∂f / ∂x _i ) are eliminated and ∂g / ∂τ = (1/2) Σ _{i = 1} ⁿ Σ _{j = 1} ⁿ (∂ ² g / ∂y _i ∂
y _j ) can be derived. It is an n-dimensional heat conduction equation with a Cross term having a diffusion coefficient of 1/2. When there is no correlation, P _ij = for i ≠ j
Since it becomes 0, (∂ ² g / ∂y _i ∂y _j ) = 0, that is, the Cross term becomes 0. Therefore, ∂g / ∂τ = (1/2) ∇ ² g. Where ∇ ² is n-dimensional Laplacian ∇ ² = (∂ ² g / ∂y ₁ ² ) + (∂ ² g / ∂y ₂ ² ) + ・ ・ ・ + (∂
² g / ∂y _n ² ). The case of one variable corresponds to this special case.

[0067] Fourier transform and inverse transform of the n-dimensional, G = F (g) = ∫gexp {-ik · y} dy g = F -1 (G) = (1 / 2π) n ∫Gexp {ik · y } dk is defined. _{Here, k · y = k 1 y} 1 + k 2 y 2 + ··· + k n y n ∫ = ∫ -∞ ∞ ∫ -∞ ∞ ··· ∫ -∞ ∞ dy = dy 1 + dy 2 + ··· a _{+ dy n dk = dk 1 +} dk 2 + ··· + dk n.

Solve the heat conduction equation using the n-dimensional Fourier transform. If the initial condition of τ = 0 is g (0, y) = g ₀ (y), the exact solution is g (τ, y) = ∫ _-∞ ^∞ (2πτ) ^{-n / 2} exp {-( y−x) ² / (2
τ)} g ₀ (y) dy If the variable transformation (y _i −x _i ) / √τ = z _i dy _i / √τ = dy _i is used, the exact solution of the heat conduction equation is g (τ, y) = ∫ using the normal distribution function. ^{(2πτ) -n / 2 exp {} -z 2/2} g 0 (x-z
√τ) dz. The only remaining problem is the determination of initial conditions.

Since it is difficult to analytically solve the decision equation of the option price including the trend μ _{i in} consideration of the correlation, the solution is obtained by numerical analysis. In the case of one variable, ∂f / ∂τ = (1/2) σ ² (∂ ² f / ∂x ² ) + (μ−σ ² /
2) Reduce to (∂f / ∂x) -rf. Therefore, μ, σ, r, K, T and initial conditions,
If the boundary condition is given, the solution f (t, S) can be obtained. The numerical solution for μ = r corresponds to the exact solution of the Black-Scholes equation. In the case of n variables, numerical analysis is performed directly.

The operation of the financial derivative product evaluation system will be described with reference to FIGS. 3 and 4. Multivariable extended Black-Scholes containing the trend μ _i of multiple underlyings as a parameter
An equation is set up and stored in the storage device 2. Step 21
Then, each statistical data obtained from the time-series price data in the financial market regarding a plurality of underlying assets is input from the input means 1 and stored in the storage means 2. In step 22, the numerical analysis means 5 calculates the multivariable expanded Blac based on each statistical data.
Numerical analysis of the k-Scholes equation is performed to determine the price of financial derivatives. In step 23, the output means 4 outputs the price of the financial derivative product.

As described above, in the second embodiment of the present invention, the financial derivative evaluation system uses the trend derivative μ _i of a plurality of underlying assets based on the statistical data of the financial derivative products consisting of a plurality of underlying assets. Multi-variable expansion Bla that contains as a parameter
Since the ck-Scholes equation is numerically analyzed to obtain the price of the derivative financial product, it is possible to collect multiple assets and evaluate the derivative financial products collectively.

[0072]

As is apparent from the above description, in the present invention, a financial derivative product evaluation system that evaluates the price of a derivative financial product by solving a stochastic second-order partial differential equation is used to provide financial information related to the underlying asset of the derivative financial product. Input means for inputting statistical data obtained from time-series price data in the market, storage means for storing statistical data, and extended Black-Schole containing the trend μ of the underlying asset as a parameter based on the statistical data
Since it is configured to include a price calculation means for obtaining the price of a financial derivative product from the exact solution of the s equation, the conventional Black-Sc
The effect that the phenomenon of economic physics can be captured more accurately than the holes equation can be obtained. By expressing the trend term, it becomes possible to evaluate the short-term trend of the financial market. It becomes possible to quantify how the market evaluates the asset price at the present time without depending on the past asset price fluctuation characteristics.

Further, a multivariable expansion Black including the trend μ _i of a plurality of underlying assets as a parameter based on the statistical data
-Since it is equipped with a numerical analysis means for numerically analyzing the Scholes equation to obtain the price of financial derivative products, it is possible to evaluate multiple assets by a series of numerical calculations when handling multiple assets collectively. The effect is obtained.

[Brief description of drawings]

FIG. 1 is a conceptual diagram of a financial derivative product evaluation system according to a first embodiment of the present invention,

FIG. 2 is a flowchart showing a flow of operations of the financial derivative product evaluation system according to the first embodiment of the present invention;

FIG. 3 is a conceptual diagram of a financial derivative product evaluation system according to a second embodiment of the present invention,

FIG. 4 is a flowchart showing an operation flow of a financial derivative product evaluation system in the second embodiment of the present invention.

[Explanation of symbols]

1 Input means 2 storage means 3 Price calculation means 4 output means 5 Numerical analysis means

   ─────────────────────────────────────────────────── ─── Continued front page    (72) Inventor Shinichi Misato             3-14-1 Hiyoshi, Kohoku Ward, Yokohama City, Kanagawa Prefecture             Keio University Faculty of Science and Engineering F term (reference) 5B056 BB01 BB04 HH00

Claims (3)

[Claims] 1. In a financial derivative product evaluation system that evaluates the price of a derivative financial product by solving a stochastic second-order partial differential equation, input statistical data obtained from time series price data in the financial market regarding the underlying asset of the derivative financial product. Input means, storage means for storing the statistical data, and price calculation means for obtaining the price of the financial derivative from the exact solution of the extended Black-Scholes equation including the trend μ of the underlying asset as a parameter based on the statistical data. A financial derivative product evaluation system comprising: 2. The input means is provided with means for inputting a price of a financial derivative product in a financial market, and the price calculation means is provided with the expansion based on the statistical data and the price.
From the exact solution of the Black-Scholes equation, the trend of the underlying asset μ
2. The financial derivative product evaluation system according to claim 1, further comprising means for determining. 3. A financial derivative product evaluation system for evaluating a price of a derivative financial instrument by solving a stochastic second-order partial differential equation, and a statistic obtained from time series price data in a financial market regarding a plurality of underlying assets included in the derivative derivative product. Input means for inputting data, storage means for storing the statistical data, and trends μ _i (i = 1 to n, n is the number of the underlying assets) of a plurality of underlying assets as parameters based on the statistical data A financial derivative product evaluation system comprising: a numerical analysis means for numerically analyzing a multivariable extended Black-Scholes equation to obtain the price of a financial derivative product.