Depth real value / virtual value option pricing method based on GARCH family model

Abstract

The invention discloses a depth real value / virtual value option pricing method for a GARCH family model. The method comprises the following steps: step 1, defining a random discount factor; step 2, based on the random discount factor, obtaining a GARCH family model under the physical measure # imgabs0 #; 3, constructing a local risk neutral measure # imgabs 1 # meeting a risk neutral pricing relationship; 4, obtaining a GARCH model under the local risk neutral measure # imgabs 2 #; 5, performing maximum likelihood estimation on parameters in the GARCH family model according to a likelihood function of the GARCH family model under the physical measure # imgabs3 #; step 6, obtaining a parameter value under a local risk neutral measure # imgabs4 # based on the parameter estimation value obtained in the step 5; and step 7, according to the parameter value obtained in the step 6 and the profit agreed by the option contract, determining an option pricing value result under the local risk neutral measure # imgabs5 #.

Description

Technical Field

[0001] The present invention belongs to the technical field of statistics, and in particular relates to a deep in-the-money / out-of-the-money option pricing method based on a GARCH family model. Background Art

[0003] Black and Scholes first proposed a relatively complete European option pricing model, called the BS model. It assumes that the price of the underlying asset follows Brownian motion and the risk-free interest rate remains unchanged, but some of its assumptions are not in line with reality, which affects the accuracy of the results. Many scholars have improved the BS model. Engle proposed the conditional heteroskedasticity model (ARCH). Using the ARCH model, the conditional variance that changes over time can be characterized, and the characteristics of sequence fluctuations can be more accurately fitted. However, when the heteroskedasticity function of the residual sequence has long-term autocorrelation, its fitting accuracy is not ideal. Bollerslev introduced historical information of heteroskedasticity and established the generalized autoregressive conditional heteroskedasticity model (GARCH). The GARCH model can describe volatility changes and the thick tail and volatility aggregation characteristics of the stock market. In the financial market, volatility has a leverage effect. To characterize this effect, Nelson established the EGARCH model; Zakoian proposed the TGARCH model. Both models can be used to describe the leverage effect of the market. Duan first applied the GARCH family model to option pricing. In his paper, he proposed constructing a local risk-neutral measure and derived it based on a relatively simple GARCH family model. However, Duan's method did not take into account the heteroskedasticity of market data and the existence of jump phenomena and leverage effects. Therefore, a more complex GARCH model is needed to price options.

[0004] This shows that domestic research on option pricing is still not mature enough. The models used in existing studies are all very classic models, and there are no newer and more complex models based on actual data. Summary of the invention

[0005] In order to solve the technical problems raised by the above background, the present invention proposes a deep in-the-money / out-of-the-money option pricing method based on the GARCH family model. The GARCH family model used in the present invention is set according to the actual market observation phenomena, and can simultaneously describe more market phenomena, including heteroskedasticity, jump phenomenon, and leverage phenomenon. The core idea is: using GARCH (1,1), GARCH (1,1)-Jump, TGARCH (1,1), and TGARCH (1,1)-Jump models, the daily closing prices of the CSI 300 stock index are modeled and analyzed, and the relevant parameters are estimated by the maximum likelihood method; using measure transformation, the model parameters under the local risk neutral measure are obtained, and based on the Monte Carlo simulation method, the pricing analysis of the screened deep in-the-money and deep out-of-the-money options is performed.

[0006] In order to achieve the above-mentioned design objectives, the technical solutions adopted by the present invention are as follows:

[0007] A deep in-the-money / out-of-the-money option pricing method for a GARCH family model comprises the following steps:

[0008] Step 1: Define the stochastic discount factor;

[0009] Step 2: Establish physical measures based on stochastic discount factors The GARCH family model under

[0010] Step 3: Based on physical measurements Constructing a local risk-neutral measure that satisfies the risk-neutral pricing relation

[0011] Step 4: Build a local risk-neutral measure The GARCH model under

[0012] Step 5: Based on physical measurements The likelihood function of the GARCH family model under , and the maximum likelihood estimation of the parameters in the GARCH family model;

[0013] Step 6: Based on the parameter estimates obtained in step 5, obtain the local risk neutral measure The parameter value below;

[0014] Step 7: Determine the local risk-neutral measure based on the parameter values ​​obtained in step 6 and the payoffs agreed in the option contract Numerical results for option pricing under .

[0015] Furthermore, the stochastic discount factor m t / m t-1 is defined as:

[0016]

[0017] Where a is the mean of the logarithm of the random discount factor, b is the standard deviation of the logarithm of the random discount factor, σ 2 , μ are the variance and mean of the jump amplitude of the compound Poisson part of the random discount factor, respectively, m t is the marginal effect of consumption at time t, are independent standard normal random variables, Medium t It follows a Poisson process with strength λ, is an independent and identically distributed normal distribution and is independent of the Poisson process, λ is the Poisson intensity;

[0018] And, the stochastic discount factor mt / m t-1 It has the following properties:

[0019]

[0020] in, represents the information before time t-1, S t represents the asset price at time t.

[0021] Furthermore, the The GARCH family model under is the TGARCH (1, 1)-Jump model, and its expression is:

[0022]

[0023]

[0024] in, represents the logarithmic uncertainty of the price of the underlying asset from time t-1 to t; For Standardization; represents the standard deviation of the logarithm of the price of the underlying asset from time t-1 to time t; d t-1 Indicates the threshold phenomenon; β0, β1, and β2 respectively represent the impact of each item in the previous time period on the standard deviation of this time period. are the variance and mean of the jump amplitude of the compound Poisson part of the random discount factor, h t is the conditional heteroskedasticity, and ρ is the correlation coefficient.

[0025] Furthermore, the local risk neutral measure Q constructed in step 3 is:

[0026]

[0027] Where T is the final trading time of the option. For physical measurement.

[0028] Furthermore, the The GARCH model under is the TGARCH (1, 1)-Jump model, and its expression is:

[0029]

[0030]

[0031] Furthermore, the specific steps of step 5 include:

[0032] Step 51: Collect historical data of the underlying asset price obtained in the market as samples for the TGARCH (1, 1)-Jump model;

[0033] Step 52: Establish the log-likelihood function of the TGARCH (1, 1)-Jump model:

[0034]

[0035] in,

[0036] ε t =ln(S t / S t-1 )-α t

[0037]

[0038]

[0039]

[0040] In the formula, is the probability density function of the normal distribution;

[0041] Step 53: Observe the historical data of asset prices collected in step 51, and obtain the set that maximizes the log-likelihood function through numerical solution.

[0042] Furthermore, the specific steps of step 6 include:

[0043] Step 61: Let (θ), θ∈Θ′ be the physical measure The parameter estimates under ;

[0044] Step 62: Use Taylor expansion method to find α t approximation:

[0045]

[0046] Step 63: According to equation (32) and equation and α t The estimated values ​​of the remaining parameters b, ρ, γ, and K in are based on The TGARCH (1, 1)-Jump model under the local risk neutral measure The mean and variance equations for the model are:

[0047] 1) The mean equation for the model is:

[0048]

[0049] 2) The variance equation for the model is:

[0050]

[0051] Step 64: Based on the obtained mean equation and variance equation, we can get The estimated values ​​of the parameters in the set Θ′ under .

[0052] Furthermore, the specific steps of step 7 include:

[0053] Step 71: According to The mean equation of the model below is iteratively calculated:

[0054]

[0055] Among them, S T is the price of the underlying asset on the expiration date (i.e., time T); S t is the price of the underlying asset at any time t between the initial time and the expiration date; is the average level of change in the logarithm of the price of the underlying asset from time (t-1) to time t under the risk-neutral measure Q; is the risk-neutral measure Q measuring time (t-1) to t;

[0056] Step 72: According to the above formula, the predicted price of the underlying asset at time T on the option expiration date is obtained:

[0057]

[0058] Step 72: Based on the predicted price, the pricing formula for the European call option is obtained:

[0059]

[0060] Where K is the strike price of the option;

[0061] Step 73: Obtain the pricing result of the European call option C according to formula (34);

[0062] Step 74: Based on the obtained European call option C, the pricing result of the European put option P is obtained according to the following formula:

[0063] P+S=C+Ke -r(T-t) (35)

[0064] Among them, S is the price of the underlying asset, r is the risk-free interest rate, K is the exercise price, and T is the expiration date.

[0065] Compared with the prior art, the present invention has the following advantages:

[0066] (1) The present invention first verifies the various properties of the underlying asset through hypothesis testing based on the real price data of the underlying asset, and gives the form of the model under the P measure. Secondly, the estimation results of the parameters contained in the model are given by maximizing the log-likelihood function of the model. Thirdly, the relationship between the P measure and the Q measure is established, and the form of the model and the parameter values ​​under the Q measure are given based on the model under P, i.e., the parameter estimation results. Finally, based on the model and parameter values ​​obeyed by the underlying asset under the Q measure, the pricing results of options based on the underlying asset are given.

[0067] (2) The present invention proposes a TGARCH (1,1)-Jump model, which can be used to comprehensively characterize the heteroskedasticity, jump phenomenon and threshold phenomenon of the underlying asset price. By setting some parameters to 0, it can be degenerated into a more simplified model such as a model that only describes the heteroskedasticity of the jump phenomenon. In summary, considering heteroskedasticity, jump phenomenon and threshold phenomenon can more accurately describe the price dynamics of the underlying asset, and can also more accurately price options based on the underlying asset; and considering these phenomena can also more accurately measure the probability of special situations where the price of the underlying asset runs to extremely high or extremely low, and more accurately price deep in-the-money options and deep out-of-the-money options, that is, more accurately measure the gains and losses that may be brought about by investing in options under extreme circumstances, and can accurately predict option pricing, thereby formulating reasonable option pricing. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] Figure 1a-1d is a descriptive statistical chart, where Figure 1a For the closing price trend chart, Figure 1b is the logarithmic return histogram, Figure 1c is the logarithmic rate of return, Figure 1d It is the logarithmic return QQ graph.

[0069] Figure 2a-2b They are the residual plot and residual square plot of the residual sequence respectively.

[0070] Figure 3a-Figure 3b are the Lee-Mykland time-point test results, where: Figure 3a For the jump recognition result, Figure 3b is the jump amplitude.

[0071] Figure 4a-4b They are the closing price trend chart and the change point position chart of the logarithmic rate of return respectively.

[0072] Figure 5a-Figure 5b These are the jump term disturbance scatter plots of the GARCH(1,1)-Jump and TGARCH(1,1)-Jump models respectively.

[0073] Figure 6a-6dThey are the normal term perturbation scatter plots of the GARCH(1,1), GARCH(1,1)-Jump, TGARCH(1,1) and TGARCH(1,1)-Jump models respectively.

[0074] Figure 7a-Figure 7b Line charts of deep in-the-money call option prices in-sample and out-of-sample, respectively.

[0075] Figure 8a-8b Cluster plots of daily absolute relative deviations of deep in-the-money call options in-sample and out-of-sample, respectively.

[0076] Figure 9a-9b Line charts of deep in-the-money put option prices in-sample and out-of-sample, respectively.

[0077] Figure 10a-Figure 10b Cluster plots of daily absolute relative deviations of deep in-the-money put options in-sample and out-of-sample, respectively.

[0078] Figure 11a-Figure 11b Line charts of deep out-of-the-money call option prices in-sample and out-of-sample, respectively.

[0079] Figure 12a-12b Cluster plots of daily absolute relative deviations of deeply out-of-the-money call options in-sample and out-of-sample, respectively.

[0080] Figure 13a-13b Line charts of deep out-of-the-money put option prices in-sample and out-of-sample, respectively.

[0081] Figure 14a-14b Cluster plots of daily absolute relative deviations of deeply out-of-the-money put options in-sample and out-of-sample, respectively.

[0082] Fig.15 It is a histogram of the average absolute relative deviation of real / imaginary depth values. DETAILED DESCRIPTION

[0083] The present invention will be further described below in conjunction with specific embodiments. It should be understood that these embodiments are only used to illustrate the present invention and are not intended to limit the scope of the present invention. In addition, it should be understood that after reading the content taught by the present invention, those skilled in the art will make various changes or modifications to the present invention, and these equivalent forms fall within the scope limited by the appended claims of the application equally.

[0084] 1. GARCH model and measurement transformation

[0085] (1) GARCH model and its extended model

[0086] Research has found that a large amount of financial data presents specific characteristics, such as: (1) peaks and tails; (2) volatility clustering, which means that asset price fluctuations are persistent over a certain period of time; (3) leverage effect, which means that negative returns of the same magnitude will lead to greater volatility intensity relative to positive returns of the same magnitude, while sustained positive returns will lead to reduced volatility intensity. To characterize these characteristics, Engle proposed the autoregressive conditional heteroskedasticity model (ARCH), which uses the moving average of the residual square sequence to fit heteroskedasticity. However, some residual sequences show long-term autocorrelation, resulting in a high moving average order in the final model, making the model more complex and affecting the fitting accuracy. To solve this problem, Bollerslev established the generalized autoregressive conditional heteroskedasticity model (GARCH). The model has the following structure:

[0087]

[0088] Among them, x t represents a process, ε t is the error term of the process, h t is the variance of the error term, λ0 is the constant term of the variance process, e t is the standardized error term, η j is the impact of the variance of the previous periods on the variance of this period, λ i The impact of the errors in the previous periods on the variance of this period;

[0089] The parameters meet the following constraints:

[0090]

[0091] Among them, f(t, x t-1 , …) is the mean equation, that is, using {x t}sequence of known information obtained fitting model; at the same time, the residual sequence {ε t} has conditional heteroskedasticity characteristics; e t It is the residual fluctuation after extracting deterministic information and conditional heteroskedasticity information from the original sequence, and its distribution is standard normal distribution.

[0092] In order to broaden the application scope of the GARCH model and improve the fitting accuracy, Zakoian proposed the threshold GARCH model (TGARCH), which combines the threshold autoregression with the GARCH model and introduces dummy variables to measure whether there is an asymmetric effect. The conditional variance of the model is as follows:

[0093]

[0094] Among them, γ i is an asymmetric coefficient, if γ i ≠0 indicates the existence of asymmetry; φt is the standard deviation, α0 is a constant term, β j is the impact of the previous period’s standard deviation on the current period’s standard deviation, d t-i is the threshold indicator.

[0095] In most cases, the volatility of asset return series is small. However, under actual market observation, at certain specific time points, especially when major financial events occur, the return series shows large fluctuations in a short period of time. This short-term sharp fluctuation is called jump phenomenon. At this time, the use of GARCH model has certain limitations and cannot describe the jump phenomenon in the market. For this reason, the jump phenomenon is treated as a random variable and added to the GARCH model. The model form is as follows:

[0096]

[0097] Among them, the rate of return series R t is decomposed into two independent components ε 1，t With ε 2，t and the mean part μ, ε 1，t and ε 2，t They correspond to the residuals under normal fluctuations and the residuals caused by jump phenomena respectively; R t is the asset return rate, μ is the average level of return rate, α0, α1, β1 are the parameters describing the impact of the variance of return rate, P(λ0) is the Poisson distribution, which is used to characterize the number of jumps, μ t , are the mean and variance of the normal distribution that characterizes the amplitude of the jump.

[0098] Different from GARCH, formula (1) adds a random variable ε 2，t ; N(t) represents the number of jumps in adjacent time periods, which obeys the Poisson distribution with Poisson intensity λ0, Y t，k Indicates the amplitude of the jump when it occurs.

[0099] Assume that N(t) and Y t，k Independent of each other, and So, let:

[0100]

[0101] Therefore, by definition, J t It follows a Poisson distribution, and J t The expectation and variance of are:

[0102]

[0103] By introducing the compound Poisson process, the jump phenomenon in the market can be well characterized.

[0104] (2) Measure transformation

[0105] First, remember S t For the asset price at time t, define the probability space on time [0, T] in, Represents the information before time t, and this information is known. Based on the risk-neutral valuation relationship, Duan proposed the Locally Risk-Neutral Valuation Relationship (LRNVR) by improving the valuation relationship, and then constructed a measure that satisfies the relationship, obtained the GARCH family model under the risk-neutral measure, and used it for option pricing. LRNVR is defined as:

[0106] Definition 1: If a measure is equivalent to a physical measure Local risk neutral measure The following conditions are met:

[0107] (1) In the local risk-neutral measure Next, R t =ln(S t / S t-1 ) is a normal distribution;

[0108] (2)

[0109] (3)

[0110] Then the measure Satisfies the local risk-neutral valuation relationship.

[0111] The present invention constructs a local risk neutral measure that satisfies LRNVR by introducing a random discount factor.

[0112] The exponential random discount factor m used in the present invention is t / / m t-1 , which has the following properties:

[0113]

[0114] Definition 2: Let m t is the marginal effect of consumption at time t, and the stochastic discount factor is defined as:

[0115]

[0116] in, are independent standard normal random variables, is the Poisson sum of independent normally distributed random variables with a Poisson strength constant λ.

[0117] According to the asset pricing principle, using the random discount factor, the asset price S at time t can be obtained: t There is a relationship with the asset price at time t-1:

[0118]

[0119] At the same time, considering the risk-free assets, let the single-period continuous compound interest rate be r, that is, the risk-free interest rate. According to equation (5), it can be proved that the random discount factor satisfies equation (3).

[0120] (3) TGARCH (1, 1)-Jump model under physical measurement

[0121] Considering that the model includes thresholds and jump terms, if the model is based on the logarithmic rate of return of asset prices, the parameters cannot be effectively interpreted in a realistic sense. Therefore, the present invention models the rate of return.

[0122] Theorem 1: In physical measure Under this condition, the dynamic change process of asset price return is as follows:

[0123]

[0124] in, are independent standard normal random variables, is the Poisson sum of independent normally distributed random variables, whose Poisson intensity is also λ; h t is conditional heteroskedasticity.

[0125] At the same time, considering that there may be a certain correlation between the standard normal random variables in equation (4) and equation (6), the correlation coefficient ρ is introduced:

[0126]

[0127] Both normal random variables and Poisson random variables will have a certain impact on asset returns. To characterize this impact, the compensation term and risk-free interest rate are set as parameter α t Next, the present invention will use the random discount factor to solve α t specific form of expression.

[0128] According to formula (3), substituting the specific form of the stochastic discount factor, we can get:

[0129]

[0130] Where a is the mean of the logarithm of the random discount factor, b is the standard deviation of the logarithm of the random discount factor, and h ist is the variance of the log return of the underlying asset S;

[0131] and,

[0132] Among them, σ 2 , are the variance and mean of the jump amplitude of the compound Poisson part in the random discount factor respectively; by rearranging the above formula, we can get the expression of the relevant parameters:

[0133]

[0134] Rearranging the above formula, we can get α t expression.

[0135] and:

[0136]

[0137] Complete physical measurements can be obtained The TGARCH (1, 1)-JUMP model under the above formula is as follows:

[0138]

[0139]

[0140] in, represents the logarithmic uncertainty of the price of the underlying asset from time t-1 to t; For Standardization; represents the standard deviation of the logarithm of the price of the underlying asset from time t-1 to time t; d t-1 Indicates the threshold phenomenon; β0, β1, and β2 respectively represent the impact of each item in the previous time period on the standard deviation of this time period.

[0141] (4) Local risk neutrality measure TGARCH (1, 1)-Jump model under

[0142] Due to the existence of risk disturbances in the market, it is impossible to To this end, the present invention constructs a local risk-neutral measure that satisfies LRNVR. And the option pricing is carried out under this measure. To achieve the pricing purpose, assuming that T is the final trading time of the option, the following measure is defined:

[0143] Definition 3: Defining a measure

[0144]

[0145] in, For physical measurement;

[0146] Corollary 1: The form of formula (12) It is a measure.

[0147] Corollary 2: For any A measurable random variable Y t The expectations are:

[0148]

[0149] Corollary 3: For any A measurable random variable Y t ,have:

[0150]

[0151] So far, we have completely constructed the local risk neutral measure (shown in Equation 12), and combined with the random discount factor, we can get the relationship between any random variable in the two measurement transformations, that is, through Equation (13)-Equation (14), we can get the relationship between any random variable in the measurement transformations. and The following transformation.

[0152] Next, we will give the local risk-neutral measure The form of the TGARCH (1, 1)-Jump model under the local risk neutral measure Next, we still choose to model the rate of return, and record R t =ln(S t / S t-1 ), so:

[0153] Theorem 2: In the local risk-neutral measure Under this condition, the dynamic change process of asset price return is in the following format:

[0154]

[0155] According to formula (12), we can get the measure The random variable R under t The moment generating function of is:

[0156]

[0157] And the moment generating function can be simplified as:

[0158]

[0159] Let c = 0, we have And the exponential function is a monotonically increasing function, so we can get:

[0160]

[0161] Let c = 1, and we can get from formula (14) Combining with formula (6), we can get:

[0162]

[0163] Substituting equations (17)-(18) into equation (16), we can obtain the final moment generating function expression:

[0164]

[0165] According to the one-to-one correspondence between the moment generating function and the distribution function, we can draw Corollary 4.

[0166] Corollary 4: In the local risk-neutral measure Down, It obeys the log-normal distribution.

[0167] Although the form of the moment generating function is obtained, the specific expression of the relevant parameters in formula (14) cannot be obtained. Therefore, the distribution combination is considered:

[0168]

[0169] In the formula,

[0170] Similar to the above, the moment generating function is used to prove that in the local risk neutral measure Next, R t and The same distribution, so far the parameter expression in formula (15) has been obtained.

[0171] Comparing the coefficients gives us the deviations under the two measures: Further we can get:

[0172]

[0173] With the disturbance term relationship, the local risk neutral measure is given The variance equation of the model under In comparison, only one offset occurs, which can be abbreviated as:

[0174]

[0175]

[0176] Using the above offset relationship, we can derive Corollary 5.

[0177] Corollary 5: In the local risk-neutral measure of the form (12) Under these conditions, the variances of log returns under the two measures are equal, namely:

[0178]

[0179] Thus, we can get The TGARCH (1, 1)-Jump model under

[0180]

[0181]

[0182] According to Corollary 3, Corollary 4, and Corollary 5, it can be concluded that the measure defined in the present invention satisfies LRNVR and can be used for option pricing research. Through the model setting, it can be seen that the jump term is included in the compound Poisson process of the model mean equation; the threshold term c is included in the variance equation. The relationship between the GARCH family models under the two measures is shown in Table 1.

[0183] Table 1 Parameter constraint table

[0184]

[0185] From Table 1, we can get the other three models in two measures by simply deleting parameters. Therefore, in the subsequent parameter estimation and option pricing, we only need to change some parameter values ​​of the TGARCH (1, 1)-Jump model to get the other three models.

[0186] 2. Parameter Estimation and Option Pricing

[0187] (1) Physical measurement The parameter estimates under

[0188] By collecting historical data of the underlying assets obtained in the market, the relevant parameters of the TGARCH (1, 1)-Jump model are estimated using the maximum likelihood method. t =ln(S t / S t-1 )-α t , where α t Given by formula (10), at the same time, the variance equation obeys the GARCH process:

[0189]

[0190] Among them, h t and All are given by formula (11). Therefore, the set of parameters to be estimated for this model is:

[0191]

[0192] When the condition N(t) = i is given, combined with the distribution assumption in equation (10), we can know is the sum of several random variables that obey a normal distribution, so It also obeys the normal distribution, and its expectation and variance can be obtained as:

[0193]

[0194]

[0195] Then ε t The probability density function can be obtained from the total probability formula:

[0196]

[0197] in, is the probability density function of the normal distribution.

[0198] The log-likelihood function of the sample is:

[0199]

[0200] After actually observing the asset price series of the CSI 300 stock index, we numerically solve the problem and find the parameter set θ that maximizes equation (29), thus completing the physical measurement. The parameter estimates below.

[0201] Regarding the estimation of the threshold value c, we use the original model without threshold proposed by Jesus Gonzalo to obtain a set of sequences, take the minimum value of the sequence as the initial value, and traverse all the values ​​in the sequence. The threshold c that makes the likelihood function value reach the maximum is the optimal threshold value of the model. Practice shows that h t The sequence value is often very small. The value of is smaller and can be approximately regarded as 0; at the same time, the parameter α in the likelihood function t Since the order of magnitude is too small, it is difficult to estimate, so the Taylor expansion method is used to calculate α t To find the approximate value, first we need to calculate the important term L in the expression. t Perform Taylor expansion, keeping only h in the expansion term t First-order or lower terms of :

[0202]

[0203] Further get α t The approximate value of is:

[0204]

[0205] Using approximate calculation can greatly speed up the solution. Through the study of likelihood function and variance equation, it is found that the parameters λ, It is already included in the likelihood function. On the contrary, some parameters are difficult to separate, such as b, ρ, γ, etc. They only appear in the parameter α formed by their combination t Therefore, according to the actual expression obtained, the set of parameters to be estimated can be further optimized as follows:

[0206]

[0207] (2) Physical measurement and local risk neutral measures Parameter relationship

[0208] Get physical measurements After obtaining the parameter estimates under , we can obtain the local risk neutral measure by replacing some variables. The corresponding parameter values ​​are obtained. First, (θ), θ∈θ′ are taken as the parameter estimates obtained above, where (θ) is the parameter to be estimated in formula (31). By comparing formula (8) and formula (18), we can get:

[0209]

[0210] At the same time, using the equation and the estimated values ​​of other parameters are given by (23)(24)(32)(33) All parameters and The transformation relationship between all the parameters of the model (i.e. the parameters listed in formula (31)) is as follows. Substituting the maximum likelihood estimation results of all parameters into the relationship, we can get The estimated values ​​of the following parameters.

[0211] (3) Local risk neutrality measure Option Pricing and Monte Carlo Simulation under

[0212] The present invention is based on the local risk neutral measurement The option pricing is as follows:

[0213] Depend on The mean equation for the model is:

[0214]

[0215] Take iterative actions:

[0216]

[0217] Therefore, the predicted price of the underlying asset at time T on the option expiration date can be obtained:

[0218]

[0219] Among them, S T is the price of the underlying asset on the expiration date (i.e., time T); S t is the price of the underlying asset at any time t between the initial time and the expiration date; is the average level of change in the logarithm of the price of the underlying asset from time (t-1) to time t under the risk-neutral measure Q; h is the uncertainty of the change in the logarithm of the price of the underlying asset from time (t-1) to time t under the risk-neutral measure Q, which is composed of a normal random variable and a compound Poisson random variable; T is a standard normal random variable under the risk-neutral measure Q, measuring the variance of the change in the logarithm of the price of the underlying asset from time (t-1) to time t;

[0220] Similar to the operation of formula (33), the other three models can also derive the corresponding underlying asset price formula. Therefore, the pricing formula of the European call option is:

[0221]

[0222] Where K is the strike price of the option.

[0223] For a European call option C and a European put option P with exercise price K, when the expiration date T is the same, based on the market's no-arbitrage principle, the following relationship exists:

[0224] P+S=C+Ke -r(T-t) (35)

[0225] Among them, S is the price of the underlying asset and r is the risk-free interest rate.

[0226] The pricing of European call options and European put options can be achieved through equations (33) to (35).

[0227] Since there is no simple analytical expression for formula (35), Monte Carlo simulation is used to calculate the pricing results. Monte Carlo simulation can be roughly divided into three steps: 1) construct a suitable probability model; 2) provide sampling methods for different distributions; 3) statistically process the simulation results.

[0228] According to the law of large numbers and the theory of moment estimation, when the sample size is large enough, the average value of the sample converges to the expectation of the distribution it obeys. Therefore, it is reasonable to use Monte Carlo simulation to generate one or more sets of random numbers, and combine equations (33) and (34) to regard the average value of the simulation results as the predicted price of the option. According to the strong law of large numbers, when X1, X2...X n When the random variables X are independent and identically distributed and the expectation exists, the mean value of the random variable X converges to the expectation with probability 1, that is:

[0229]

[0230] According to the central limit theorem, for any δ>0, we have:

[0231]

[0232] Among them, σ is the standard deviation of the random variable, and α is the significance level, which is usually taken as 0.05.

[0233] Therefore, the error of the Monte Carlo simulation result is:

[0234]

[0235] Where σ is the standard deviation of the random variable, δ>0 is an arbitrary constant, and σ, δ and the number of simulations N together constitute the limit of the error of the Monte Carlo simulation results. According to formula (36) and the required order of magnitude of the error, the number of Monte Carlo simulations required to be performed can be determined under the required error order of magnitude.

[0236] The above formula shows that if the error is to be reduced, the number of samples can be increased or the variance can be reduced. However, a larger N will also make the calculation complicated. Taking into account the sampling accuracy and calculation speed, N=1000 is set in the present invention.

[0237] Example

[0238] In this embodiment, the GARCH (1, 1), GARCH (1, 1)-Jump, TGARCH (1, 1), and TGARCH (1, 1)-Jump models are used to model and analyze the daily closing prices of the CSI 300 Index, and the maximum likelihood method is used to estimate the relevant parameters. The model parameters under the local risk-neutral measure are obtained by using measure transformation, and the pricing analysis of the deep in-the-money and deep out-of-the-money options obtained by screening is performed based on the Monte Carlo simulation method.

[0239] 1. Data source and processing

[0240] The underlying asset of the option pricing research of the present invention is the daily closing price sequence of the CSI 300 Index, and the pricing object is the CSI 300 stock index option. According to the relationship between the market price of the option underlying asset and the option strike price, the options are divided into in-the-money options and out-of-the-money options. According to formula (35), there is a relationship between the prices of call options and put options. When three of them are known, the other one can be deduced. Therefore, this embodiment only performs pricing analysis on call options, and obtains the pricing results of the corresponding put options according to formula (36). For call options, when the strike price is lower than the market price of the underlying asset, it is called an in-the-money option; when the strike price is higher than the market price of the underlying asset, it is called an out-of-the-money option; for put options, when the strike price is higher than the market price of the underlying asset, it is called an in-the-money option; when the strike price is lower than the market price of the underlying asset, it is called an out-of-the-money option. If the strike price is too different from the market price of the underlying asset, it is called a deep in-the-money or deep out-of-the-money option.

[0241] As shown in Table 2, the call option classification is defined, where K is the option exercise price and S is the market price of the underlying asset.

[0242] Table 2 Option Definition

[0243]

[0244] According to Table 2, the daily closing price data of 972 trading days from April 1, 2019 to March 30, 2023 are collected for modeling and analysis; since the option prices near the expiration date usually do not change much, this paper selects options with an expiration date of March 17, 2023, and conducts pricing research on deep in-the-money options and deep out-of-the-money options from October 31, 2022 to November 4, 2022. At the same time, the daily closing prices from April 3, 2023 to April 7, 2023 are regarded as out-of-sample data for prediction, and the fitting of the model is compared to study the pricing of the corresponding options with an expiration date of June 16, 2023 during this period. The relevant data are all from the CSMAR database. One year is regarded as 250 trading days, and the SHIBOR annual interest rate is taken as the risk-free interest rate. The risk-free interest rate r used in this embodiment is 1.5% / 250. For the daily closing price sequence, logarithmic differentiation can be performed to obtain the yield sequence. Since the magnitude of the value after differentiation is small, it is further expanded:

[0245]

[0246] 2. Descriptive statistical analysis

[0247] Descriptive statistical analysis of closing prices and yield series is performed, and the following table 3 and Figure 1a-1d . Figure 1a The daily closing price trend chart of the CSI 300 Index is Figure 1aIt can be seen that the closing price in the previous period was stable and had an upward trend, but it showed a gradual downward trend and stabilized from around May 2021, indicating that market prices often fluctuate, but will stabilize over time. The mean of the yield data is 0.0017, and the variance is 1.5580, indicating that the difference between daily yield data is large, and the skewness is -0.53900, that is, a peak feature appears; Figure 1b The histogram of the logarithmic return series reflects these characteristics. Figure 1c This is a time series chart of yields, which is generally stable, but there are large fluctuations at certain time points, which means that there may be jump phenomena and leverage effects. Figure 1d It is the QQ graph of the logarithmic yield sequence. The middle part basically fluctuates around the straight line, and there are partial deviations at both ends. Combined with the KS normality test results, it shows that the sequence does not obey the normal distribution. According to the descriptive statistical results, it can be seen that the present invention uses the GARCH family model to fit the yield. There is rationality.

[0248] Table 3 Descriptive statistics of data series

[0249]

[0250] 3. Stationarity test of GARCH model and ARCH effect test

[0251] In order to establish a GARCH model, it is necessary to test the stability of the model and the ARCH effect. To this end, let the mean term of the model be 0, then the residual sequence is the yield sequence itself:

[0252] ε t =R t

[0253] Depend on Figure 2a It can be seen that the residual sequence shows the characteristics of up and down fluctuations, and the fluctuation range is not consistent. Figure 2b The residual square sequence has significant differences, and the residual square of some time points is significantly higher than that of the surrounding points. Therefore, the residual sequence has heteroscedasticity; when establishing the GARCH model, the ADF test may be biased, so the yield sequence R t A PP stationarity test is performed and the test results in Table 4 are obtained. It can be seen that the PP test results indicate that the sequence is stationary and subsequent modeling can be performed.

[0254] Table 4 PP stability test

[0255]

[0256] To further verify the existence of heteroskedasticity, ARCH effect test is performed on the residuals. Portmanteau Q test statistic and LM test statistic are constructed:

[0257] ①Portmanteau Q test

[0258] 1) Assumptions:

[0259] H0: Homogeneity of variance, H1: Heterogeneity of variance

[0260] 2) Statistics:

[0261]

[0262] Where n is the sequence length, ρ i is the autocorrelation coefficient of the delayed i-order residual square sequence:

[0263]

[0264] ②LM test

[0265] 1) Assumptions:

[0266] H0: Homogeneity of variance, H1: Heterogeneity of variance

[0267] 2) Statistics:

[0268] LM(q)=((SST-SSE) / q) / (SSE / (T-2q-1))

[0269]

[0270] The test results are shown in Tables 5 and 6. The results show that the variance is non-homogeneous, which verifies that the establishment of the GARCH model is reasonable and necessary.

[0271] Table 5 Portmanteau Q test

[0272]

[0273] Table 6 LM test

[0274]

[0275] 4. Parameter estimation of GARCH family models

[0276] Using maximum likelihood estimation, combined with the log-likelihood function (Equation 29) given above, we can obtain the parameter estimates of the four models. The following parameter estimation process is based on the physical measurement Next completed.

[0277] (1) GARCH (1, 1) model

[0278] Table 7 shows the parameter estimation results of the GARCH (1, 1) model, and formula (37) is the model expression. According to the results, β1+β2 < 1 and is close to 1, indicating that the model is stable. -bρ = 0.522 in the estimated parameter represents the risk compensation brought by the normal disturbance.

[0279] Table 7 GARCH (1, 1) parameter estimation table

[0280]

[0281]

[0282] (2) GARCH (1, 1)-Jump Model

[0283] To determine whether there is a jump phenomenon, the present invention adopts a time-by-time inspection method. The specific steps are as follows:

[0284] i) Calculate statistics: S(t i ) is t i The closing price at the time, construct the test statistic L(i) to test whether there is a jump:

[0285]

[0286]

[0287] The statistic L(i) asymptotically obeys a normal distribution with a mean of 0 and a standard deviation of 1 / c, where c = 2 / π and K is the window size, which is usually around 3. i-1 , t i ] does not exist, then when △t→0, we have:

[0288]

[0289] Where ψ is the cumulative distribution function, satisfying P(ψ≤x)=exp(-e -x ),and:

[0290]

[0291] ii) Select the significance level α and calculate the threshold β * , and determine whether there is a jump phenomenon based on the threshold:

[0292] P(ψ≤β * ) = exp(-e -β* )=1-α

[0293] The test results are as follows Figure 3a-Figure 3b , Figure 3aThe identification results show that there is a jump phenomenon in the market, where 1 represents a positive jump and -1 represents a negative jump; Figure 3b The jump amplitude diagram shows that the amplitude fluctuates around 0, and there are extreme large jumps, so the impact of the jump phenomenon cannot be ignored. This test further shows that it is necessary to establish a GARCH model with jump terms to fit the series.

[0294] Table 8 shows the parameter estimation results of the GARCH (1, 1)-Jump model, and Equation (38) is the model expression. Compared with GARCH (1, 1), this model has the following differences in addition to the normal disturbance term: In addition, a normal random variable is introduced to represent the jump intensity, namely And the number of occurrences follows the Poisson distribution P(λ), and the two constitute a composite Poisson process, which is used to measure the disturbance caused by the jump phenomenon. The estimated value of λ is about 0.498, indicating that under market observation, a jump will occur approximately every two trading days, and the intensity of each jump follows a normal distribution with a mean of 0.020 and a standard deviation of 0.028. It can be seen that although the jump phenomenon exists, the amplitude of the jump is small and most of them are positive jumps. Due to the introduction of the jump term, the risk compensation brought by the normal disturbance and the jump disturbance becomes difficult to separate, but it is already included in the model expression.

[0295] Table 8 GARCH (1, 1)-Jump parameter estimation table

[0296]

[0297]

[0298] (3) TGARCH (1, 1) model

[0299] For a set of financial time series data, its variance usually remains approximately unchanged for a period of time, but there may be a moment when the variance suddenly changes and returns to the original level after a period of time. The moment when the variance suddenly changes is called the variance structure change point. When a mutation point appears, it means that there is often a leverage effect in a certain period of time. The IT statistic can be used to test whether a variance mutation point occurs, denoted by e t =100log(S t / S t-1 ), the inspection steps are as follows:

[0300] i) H0: the sequence has no significant change point; H1: the sequence has a significant change point;

[0301] ii) Construct IT statistics:

[0302]

[0303] The significance level is set to α = 0.05, where D k The distribution of is more complex, and its critical value IT * It can be obtained through simulation and approximate calculation;

[0304] iii) Determine the rejection region: IT>IT * , where k that maximizes the statistic is the most likely mutation point.

[0305] The results obtained by calculation are shown in Table 9. The change point positions are shown in Figure 4a-4b . The results show that there is a change point in the sequence. The most likely change point occurs around March 25, 2021. Before the change point, the closing price reaches the maximum value and then drops sharply. The closing price and yield fluctuations of the CSI 300 Index are abnormal. The test shows that there may be a leverage effect within a specific period of time. Therefore, it is reasonable to establish a TGARCH (1, 1) model for the sequence.

[0306] Table 9 IT test results

[0307]

[0308] Table 10 is the parameter estimation table of the TGARCH (1, 1) model, and equation (40) is the expression. Compared with the GARCH (1, 1) model, this model introduces the indicator function d t-1 , setting conditional heteroskedasticity as the threshold value, and obtaining the threshold value of 1.070 through parameter estimation, overcoming the defect of the GARCH (1, 1) model that responds equally to positive and negative disturbances, and describing the leverage effect. The coefficient β3 = 0.010 ≠ 0, indicating that there is a leverage effect in the market, further verifying the conclusion of the IT test. When the conditional heteroskedasticity is large, the impact coefficient brought by the disturbance is 0.057, which is larger than the disturbance coefficient when the heteroskedasticity is small, indicating that the larger the conditional heteroskedasticity, the greater the impact of the disturbance.

[0309] Table 10 TGARCH (1, 1) parameter estimation table

[0310]

[0311]

[0312] (4) TGARCH (1, 1)-Jump Model

[0313] Both the jump term and the threshold term are added to the GARCH (1, 1) model to observe the model fitting effect. Table 11 is the parameter estimation table of the TGARCH (1, 1)-Jump model, and formula (41) is the expression. The model adds jump terms and threshold terms. The number of jump phenomena follows a Poisson distribution with a parameter of 0.502, that is, a jump occurs approximately every two trading days, and the amplitude of each jump follows a normal distribution with a mean of 0.067 and a standard deviation of 0.010. Similar to the second model, there are more positive jumps and it is difficult to have negative jumps; the threshold value is 1.010, and the coefficient of the threshold term is β3 = 0.055 ≠ 0, indicating that there is a leverage effect in the market. Similar to the GARCH (1, 1)-Jump model, the risk compensation brought by the normal disturbance and the jump disturbance becomes difficult to separate, but is included in the model expression.

[0314] Table 11 TGARCH (1, 1)-Jump parameter estimation table

[0315]

[0316]

[0317] 5. Comparison of GARCH family models

[0318] (1) Residual ARCH effect test

[0319] Whether the GARCH family model is reasonable depends on whether the established model fully extracts heteroscedasticity information. Therefore, the ARCH effect test is performed on the residual sequence of the variance equation of the four GARCH family models, and the LM statistic constructed above is used for the test. The results are shown in Table 12. The test shows that the four models have effectively extracted heteroscedasticity information, and the residual sequence has no ARCH effect.

[0320] Table 12 ARCH effect test

[0321]

[0322] (2) Residual normality test

[0323] According to the residual distribution setting of the model, the model residual needs to be tested for KS normality.

[0324] For the GARCH (1, 1)-Jump and TGARCH (1, 1)-Jump models, the total residual also contains jump term disturbances. The jump term disturbance is set as a composite Poisson process of a combination of independent normal distributions. According to the additivity of independent normal distributions, the jump term disturbance also obeys the normal distribution. In order to test whether the jump disturbance obeys the normal distribution, the present invention adopts the filtering separation method proposed by Christoffersen, using the standard deviation (sd) as the weight, to separate the two disturbances, and the normal term disturbance part is recorded as Z t (norm), the jump term perturbation part is Z t (jump), there are:

[0325]

[0326] The jump term disturbance is separated by equation (42) and the KS test is performed on it. The jump term disturbance scatter plot is shown in Figure 5a-Figure 5b ,It can be seen from the figure that whether or not to add the threshold term has little effect on the amplitude of the jump term disturbance. The KS test results are shown in Table 13. The jump term disturbances of the two models obey the normal distribution and meet the model assumptions.

[0327] Table 13 KS test table of jump disturbance term

[0328]

[0329] In the residual settings of the four models, normal term disturbances are included The results of the KS test are shown in Table 14, where the significance level is α = 0.05; the normal term perturbation scatter plots of the four models are Figure 6a-6d , it can be seen from the figure that the two groups of graphs after adding the jump term ( Figure 6b and Figure 6d ) compared with the other two groups of graphs ( Figure 6a and Figure 6c ) has a smaller and more concentrated fluctuation range, fewer outliers, and is more consistent with the characteristics of a normal distribution scatter plot. As shown in Table 14, the models with jump terms all pass the test, further verifying the characteristics shown in the figure. The results show that compared with the traditional GARCH family model, the model with jump terms can meet the model's assumptions about distribution and can better describe some phenomena in the financial market.

[0330] Table 14 KS test table of normal disturbance term

[0331]

[0332] (3) Model comparison

[0333] In order to measure the fitting accuracy of the model, the present invention selects the average absolute relative deviation (AARD) to compare the absolute size of the fitting errors of the four groups of models. The sample size is recorded as N, and the market observation results are recorded as Y. i (i=1,2,....N), the model prediction result is recorded as

[0334]

[0335] The calculated AARDs of the four models are shown in Table 15. The results show that TGARCH (1, 1)-Jump has the best fitting accuracy both in-sample and out-of-sample.

[0336] Table 15 Average absolute relative deviation of the four groups of models

[0337]

[0338] According to the parameter results of the above four groups of GARCH family models, the log-likelihood function values ​​of the four groups of models are listed in Table 16:

[0339] Table 16 Log-likelihood function values

[0340]

[0341] Among the four models, the TGARCH (1, 1)-Jump model has the largest log-likelihood function value, and the GARCH (1, 1) model has the smallest log-likelihood function value. The numerical values ​​show that adding jump terms or threshold terms can improve the log-likelihood function value. Preliminary results show that adding parameters is helpful to improve the model. To verify this hypothesis, a likelihood ratio test was performed on the four models. The likelihood ratio test is often used to compare the goodness of fit of models. It constructs statistics through the log-likelihood function values ​​of two models to compare the advantages and disadvantages of the models. Construct statistics:

[0342] Λ=-2(l A -l B ) (44)

[0343] Among them, l A is the log-likelihood function value of model A, l B is the log-likelihood function value of model B, and Λ follows a chi-square distribution with degrees of freedom equal to the difference in the number of model parameters.

[0344] Next, likelihood ratio tests were performed on the four models, and the results are shown in Table 17. The test shows that adding jump terms or threshold terms is significantly effective, indicating that adding jump terms or threshold terms is necessary on the basis of the GARCH model, and can improve the fitting effect of the model.

[0345] Table 17 Likelihood ratio test table

[0346]

[0347] 6. Analysis of option pricing results

[0348] Based on Monte Carlo simulation and formula (34), the predicted prices of various call options in and out of sample can be obtained, and the predicted prices of put options can be obtained by formula (35). The absolute relative deviation is calculated using formula (43) to measure the prediction error. The pricing results are shown below using options with exercise prices of 3100 and 4900 as examples.

[0349] (1) Deep In-the-Money Options

[0350] a) Call option

[0351] The pricing result of the deep in-the-money call option with an exercise price of 3100 is as follows Figure 7a and Figure 7b As can be seen from the figure, the pricing results obtained by the in-sample TGARCH (1,1) show huge fluctuations. Although the price trend presented by the GARCH (1,1) model is consistent with the actual situation, there is a significant underestimation. The out-of-sample GARCH (1,1) has a large deviation and is not included. TGARCH (1,1) has a slightly higher pricing. In comparison, GARCH (1,1)-Jump and TGARCH (1,1)-Jump are closer to the actual price. In order to further compare the advantages and disadvantages of the pricing results of each model, the daily absolute relative deviation of each model is calculated, and the cluster diagram is shown in Figure 8a and Figure 8b As can be seen from the figure, the daily absolute relative deviation of the TGARCH (1,1)-Jump model is significantly lower than that of other models, indicating that the TGARCH (1,1)-Jump model performs better in the application of deep in-the-money call option pricing.

[0352] b) Put option

[0353] For deep in the money put options, the pricing results when the strike price is 4900 are shown in Figure 9a-9b The pricing results of the four models in the sample show the same trend as the actual price. Due to the small difference in values, some lines almost overlap. The same situation occurs outside the sample, but the errors of GARCH (1,1) and TGARCH (1,1) pricing results are large and are no longer compared. This result shows that the pricing accuracy of the four models for deep in-the-money put options is relatively similar. Figure 10a-Figure 10bThis is a cluster diagram of daily absolute relative deviations, which also shows that the accuracy of the three improved GARCH family models is better than the ordinary GARCH (1,1) model; and in terms of pricing results both inside and outside the sample, the pricing results of the two models with jump terms are better.

[0354] (2) Deep out-of-the-money options

[0355] a) Call option

[0356] The pricing result of the deep out-of-the-money call option with an exercise price of 4900 is as follows Figure 11a-Figure 11b Unlike deep in-the-money call options, the GARCH (1,1) model shows abnormal deviations in pricing deep out-of-the-money call options, indicating that the model is not applicable to deep out-of-the-money call options. Therefore, the GARCH (1,1) predicted price line is not included in the figure; the pricing results of GARCH (1,1)-JUMP and TGARCH (1,1)-JUMP are still relatively accurate. See the daily absolute relative deviation cluster diagram Figure 12a-12b , except for the large error on October 31, 2022, the pricing results of the TGARCH (1,1)-JUMP model are all optimal. At the same time, the error of the TGARCH (1,1) model in pricing deep out-of-the-money call options is smaller than that of deep in-the-money options.

[0357] b) Put option

[0358] For the deep out-of-the-money put option, the pricing results when the strike price is 3100 are shown in Figure 13a-13b . At this time, TGARCH(1,1) shows too large deviation both in-sample and out-of-sample, so it is no longer included in the line chart. For in-sample pricing, the TGARCH(1,1)-Jump model is still the closest to the actual price, and it is almost the same as the actual price in the last two trading days in the sample; while out-of-sample, the errors of the two sets of jump term models are similar. The daily absolute relative deviation cluster diagram is shown in Figure 14a-14b , it can be seen that the TGARCH(1,1)-Jump model is much better than other models.

[0359] 5. Conclusion

[0360] In order to further test the quality of the model prediction results, the average absolute relative deviation of the results obtained by each model for pricing deep in-the-money and deep out-of-the-money options in and out of sample is calculated. The specific numerical results are shown in Table 18 and Fig.15 .

[0361] Table 18 Depth real value / imaginary value average absolute relative deviation

[0362]

[0363] (1) For the pricing of deep in-the-money options, the GARCH(1,1) model shows a large deviation, followed by the TGARCH(1,1) model. The average absolute relative deviations of the two groups of models with jump terms are both small, indicating that the pricing results of the two groups of models are more accurate and that the addition of jump terms can reduce pricing errors. Compared with the model with only jump terms, the TGARCH(1,1)-Jump with threshold terms is more complex in form but has smaller errors and is more suitable for option pricing research.

[0364] (2) For the pricing of deeply out-of-the-money options, the average absolute relative deviations of the GARCH (1,1) and TGARCH (1,1) models are much larger than those of other models. By adding jump terms, the pricing error is greatly reduced, and the TGARCH (1,1)-Jump model still performs best.

[0365] 3) In summary, the TGARCH (1,1)-Jump model is the optimal model and can reasonably price options.

[0366] The above contents are further detailed descriptions of the present invention in combination with specific preferred embodiments, so that those skilled in the art can understand and apply the present invention. It cannot be determined that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art of the present invention, some simple deductions or substitutions can be made without departing from the concept of the present invention, without creative labor. Therefore, simple improvements made to the present invention by those skilled in the art based on the disclosure of the present invention should be within the scope of protection of the present invention.

Claims

1. A deep in-the-money / out-of-the-money option pricing method for a GARCH family model, characterized in that: The following steps are involved: Step 1: Define the stochastic discount factor; Step 2: Establish physical measures based on stochastic discount factors The GARCH family model under Step 3: Based on physical measurements Constructing a local risk-neutral measure that satisfies the risk-neutral pricing relation Step 4: Build a local risk-neutral measure The GARCH model under Step 5: Based on physical measurements The likelihood function of the GARCH family model under , and the maximum likelihood estimation of the parameters in the GARCH family model; Step 6: Based on the parameter estimates obtained in step 5, obtain the local risk neutral measure The parameter value below; Step 7: Determine the local risk-neutral measure based on the parameter values ​​obtained in step 6 and the payoffs agreed in the option contract Numerical results for option pricing under .

2. The deep in-the-money / out-of-the-money option pricing method of the GARCH family model according to claim 1, characterized in that: Stochastic discount factor m t / m t-1 is defined as: Where a is the mean of the logarithm of the random discount factor, b is the standard deviation of the logarithm of the random discount factor, σ 2 , μ are the variance and mean of the jump amplitude of the compound Poisson part of the random discount factor, respectively, m t is the marginal effect of consumption at time t, are independent standard normal random variables, Medium t It follows a Poisson process with strength λ, is an independent and identically distributed normal distribution and is independent of the Poisson process, λ is the Poisson intensity; And, the stochastic discount factor m t / m t-1 It has the following properties: in, represents the information before time t-1, S t represents the asset price at time t.

3. The deep in-the-money / out-of-the-money option pricing method of the GARCH family model according to claim 2, characterized in that: Step 2 Created The GARCH family model under is the TGARCH (1,1)-Jump model, and its expression is: in, represents the logarithmic uncertainty of the price of the underlying asset from time t-1 to t; For Standardization; represents the standard deviation of the logarithm of the price of the underlying asset from time t-1 to time t; d t-1 Indicates the threshold phenomenon; β0, β1, and β2 respectively represent the impact of each item in the previous time period on the standard deviation of this time period. are the variance and mean of the jump amplitude of the compound Poisson part of the random discount factor, h i is the conditional heteroskedasticity, and ρ is the correlation coefficient.

4. The deep in-the-money / out-of-the-money option pricing method of the GARCH family model according to claim 1, characterized in that: The local risk-neutral measure constructed in step 3 for: Where T is the final trading time of the option. For physical measurement.

5. The deep in-the-money / out-of-the-money option pricing method of the GARCH family model according to claim 1, characterized in that: Step 4 build The GARCH model under is the TGARCH (1,1)-Jump model, and its expression is:

6. The deep in-the-money / out-of-the-money option pricing method of the GARCH family model according to claim 5, characterized in that: The specific steps of step 5 include: Step 51: Collect historical data of the underlying asset price obtained in the market as samples for the TGARCH (1,1)-Jump model; Step 52: Establish the log-likelihood function of the TGARCH(1,1)-Jump model: in, e t =ln(S t / S t-1 )-a t In the formula, is the probability density function of the normal distribution; Step 53: Observe the historical data of asset prices collected in step 51, and obtain the set that maximizes the log-likelihood function through numerical solution.

7. The deep in-the-money / out-of-the-money option pricing method of the GARCH family model according to claim 6, characterized in that: The specific steps of step 6 include: Step 61: Let (θ), θ∈Θ′ be the physical measure The parameter estimates under ; Step 62: Use Taylor expansion method to find α t approximation: Step 63: According to equation (32) and equation and α t The estimated values ​​of the remaining parameters b, ρ, γ, and K in are based on The TGARCH(1,1)-Jump model under the local risk neutral measure is obtained The mean and variance equations for the model are: 1) The mean equation for the model is: 2) The variance equation for the model is: Step 64: Based on the obtained mean equation and variance equation, we can get The estimated values ​​of the parameters in the set Θ′ under .

8. The deep in-the-money / out-of-the-money option pricing method of the GARCH family model according to claim 7, characterized in that: The specific steps of step 7 include: Step 71: According to The mean equation of the model below is iteratively calculated: Among them, S T is the price of the underlying asset on the expiration date (i.e., time T); S t is the price of the underlying asset at any time t between the initial time and the expiration date; is the average level of change in the logarithm of the price of the underlying asset from time (t-1) to time t under the risk-neutral measure Q; is the risk-neutral measure Q measuring time (t-1) to t; Step 72: According to the above formula, the predicted price of the underlying asset at time T on the option expiration date is obtained: Step 72: Based on the predicted price, the pricing formula for the European call option is obtained: Where K is the strike price of the option; Step 73: Obtain the pricing result of the European call option C according to formula (34); Step 74: Based on the obtained European call option C, the pricing result of the European put option P is obtained according to the following formula: P+S=C+Ke -r(T-t) (35) Among them, S is the price of the underlying asset, r is the risk-free interest rate, K is the exercise price, and T is the expiration date.

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