European option pricing method based on mixed fractional Brownian motion and jump diffusion model

Abstract

The invention relates to the field of financial derivative pricing, in particular to a mixed fractional Brownian motion and jump diffusion model-based European option pricing method, which comprises the following steps of: introducing a jump diffusion process into mixed fractional Brownian motion to describe an asset price process; solving a mixed fractional jump diffusion process # imgabs0 # formula according to the asset price process; based on the self-financing investment strategy, calculating a partial differential equation of related derivative prices under the condition of mixed score jump diffusion; and deducing European style expansion option pricing in a mixed-order jump diffusion environment in an incomplete market based on the related derivative prices. According to the method, it is assumed that each target asset is influenced by geometric Brownian motion, second-fraction Brownian motion and a Poisson process together, and a pricing formula of European style expansion options is deduced. The invention aims to solve the problems of capturing long memorability and self-similarity of assets and reflecting sudden jump of asset prices.

Description

Technical Field

[0001] The present invention relates to the field of financial derivatives pricing, and in particular to a European option pricing method based on a mixed subfractional Brownian motion and jump diffusion model. Background Art

[0002] In recent years, with the increasing complexity and volatility of the global economic landscape, the demand for risk management in financial markets has significantly increased. Options, as a key risk hedging tool, have once again attracted widespread attention. Unexpected events have exposed financial markets to significant uncertainty, exacerbating the volatility and jump risk of underlying asset prices. Traditional pricing models struggle to fully capture these market characteristics.

[0003] Options are unique in that they not only allow investors to maximize returns through leverage but also offer the option to limit losses in a down market. Proper assessment of option prices is crucial, as mispricing can lead to increased market volatility, which in turn can undermine investor confidence and market liquidity. The transmission effect of deteriorating bank balance sheets has significantly increased market volatility, pushing up the implied volatility of some underlying assets. Therefore, accurate option pricing is not only fundamental to the sound operation of financial markets but also a pillar of the development of various derivatives markets.

[0004] In terms of theoretical development, the classic geometric Brownian motion (GBM) model, while effective under simple conditions, exhibits limitations in complex market environments. The traditional GBM model assumes that market fluctuations follow a Brownian motion with constant variance. However, real-world market data often exhibits characteristics such as "peaked tails," long memory, and price jumps. To address the shortcomings of the GBM model, scholars have introduced more complex models such as fractional Brownian motion (fBM) and subfractional Brownian motion (sfBM) to better reflect the long-range correlations in asset prices. Furthermore, the introduction of jump-diffusion models has provided new research directions for capturing price jumps and sudden changes in the market. The advantages of these extended models are becoming increasingly significant in the context of frequent news events such as Federal Reserve policy adjustments and inflation changes in major global economies.

[0005] With the deepening of academic research and empirical analysis, option pricing models are evolving from static to dynamic, and from a single dimension to a multi-dimensional one. A key challenge in the field of option pricing is how to further integrate the various factors that influence option price fluctuations to develop a more comprehensive and realistic option pricing formula. Summary of the Invention

[0006] In response to the aforementioned technical problems, a European option pricing method based on a hybrid subfractional Brownian motion and jump-diffusion model is provided. This method assumes that each underlying asset is influenced by a combination of geometric Brownian motion, subfractional Brownian motion, and a Poisson process, and derives a pricing formula for European call options. This method aims to address the challenges of capturing the long memory and self-similarity of an asset and reflecting sudden jumps in its price.

[0007] The technical means adopted in the present invention are as follows:

[0008] A European option pricing method based on a hybrid subfractional Brownian motion and jump-diffusion model, including:

[0009] Introducing the jump diffusion process into the mixed fractional Brownian motion to describe the asset price process;

[0010] Solve the mixed fraction jump diffusion process according to the asset price process formula;

[0011] Calculate the partial differential equations of related derivatives prices under mixed fractional jump diffusion conditions based on self-financing investment strategies;

[0012] Based on the prices of the related derivatives, the pricing of European call options in an incomplete market with mixed sub-fractional jump diffusion environment is derived.

[0013] Furthermore, the asset price process specifically includes:

[0014] Introducing the jump process into the mixed fractional Brownian motion, that is, using the mixed fractional jump diffusion process to describe the underlying asset price process, then assuming that stock transactions do not require transaction fees, the underlying asset price process satisfies the stochastic differential equation:

[0015] dS t =(μ-q)S t dt+σS t dW t

[0016] Among them, S t is the underlying asset price, μ is the asset return rate, q is the dividend rate, σ is the volatility of the asset price, t represents time, and W t represents a mixed-time fractional-hop diffusion process.

[0017] Furthermore, the solution of the mixed fraction jump diffusion process Formula, specifically including:

[0018] In the complete probability space (Ω,F t H ,P), assuming that:

[0019]

[0020] in, represents mixed fractional Brownian motion, N t represents a compensated Poisson process, B t Represents the geometric Brownian motion process Represents the fractional Brownian motion process; {B t} t≥0 、 With {N t} t≥0 are independent of each other, H is the Hurst index;

[0021]

[0022] are all square integrable under measure P, N t Indicates that at t i The i-th jump occurs at time t, and f(t,x) represents the mixed-time fractional jump diffusion environment Formula, C 1,2 Indicates that the function has a certain order of continuous partial derivatives on two variables, R + =R indicates that the domain of variable t is non-negative real numbers, R indicates that the domain of variable x is real numbers, →R indicates that the function evaluates to real numbers;

[0023] because Q t represents a Poisson process with strength λ, which represents the jump intensity. When i=1, that is, W t At (0, t) time, only one jump occurs, and the jump time is t1. Then no jump occurs between (0, t1) and (t1, t). The formula is:

[0024]

[0025] in, Indicates approaching W t The state f(0,0) represents the function value in the initial state, x represents the independent variable, represents the mixed fractional-hop diffusion environment at time t1 formula.

[0026] f(t,W t ) at time t1 is have:

[0027]

[0028] The number of jumps i in the interval (0, t) follows a Poisson process,

[0029]

[0030] Let g(φ)∈C 2 (R→R), and (dQ t ,dQ t )=λdt, for g(Q t ) Application in a broad sense Points, there are:

[0031]

[0032] Among them, g(φ) represents a second-order continuous differentiable function from real numbers to real numbers, Q τ It is a Poisson process with strength λ.

[0033] Depend on but

[0034]

[0035] Then we can get the mixed fraction jump diffusion process formula:

[0036]

[0037] Furthermore, the partial differential equation for calculating the price of related derivatives under the mixed-time fractional jump diffusion condition based on the self-financing investment strategy specifically includes:

[0038] Construct a bond P t and stock S t The portfolio consists of bonds P t and stock S t Satisfies the stochastic differential equation:

[0039] dP t =rP t dt

[0040]

[0041] Where r represents the bond yield.

[0042] Self-financing investment strategy Wealth Process satisfy:

[0043]

[0044] Among them, V t represents the total value of the portfolio at time t, θ t 1 represents the number of bonds held, Indicates the number of stocks held.

[0045] According to the self-financing trading strategy, there are:

[0046]

[0047] V t Also meets:

[0048]

[0049] in,

[0050] The value of the option is equal to the value of the constructed portfolio,

[0051]

[0052] Get the stock price S t When the stochastic differential equation is obeyed, the price of the relevant derivative V t =V(t,S t ) is the partial differential equation:

[0053]

[0054] Furthermore, the calculation method for pricing the European call option includes:

[0055] Let V t (t,S t )=C(t,S t )=C, calculate the price C of a common European call option:

[0056]

[0057] Combined with the initial condition C(T,S T )=(S T -K) + ,0 t <+∞Solve, where K is the strike price and T is the expiration date.

[0058] Based on the price C of the ordinary European call option, the pricing formula of the European call option in the mixed fractional jump diffusion environment under the incomplete market is derived:

[0059]

[0060] Compared with the prior art, the present invention has the following advantages:

[0061] The European option pricing method based on the mixed fractional Brownian motion and jump diffusion model provided by the present invention introduces the jump diffusion process into the mixed fractional Brownian motion to describe the asset price process; solves the mixed fraction jump diffusion process according to the asset price process​ Formula; Calculate the partial differential equation of the relevant derivative prices under the hybrid sub-fractional jump diffusion condition based on the self-financing investment strategy; Based on the relevant derivative prices, derive the pricing of European call options under the hybrid sub-fractional jump diffusion environment in an incomplete market.

[0062] This paper proposes a European option pricing method based on a hybrid subfractional Brownian motion and jump-diffusion model. It derives a European call option pricing formula that combines a hybrid subfractional Brownian motion with a jump process. This approach aims to address the challenges of capturing the long memory and self-similarity of assets while also reflecting sudden jumps in asset prices. The pricing formula for European call options is derived by assuming that each underlying asset is influenced by a combination of geometric Brownian motion, subfractional Brownian motion, and a Poisson process.

[0063] Based on the above reasons, the present invention can be widely promoted in fields such as financial derivatives pricing. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative labor.

[0065] Figure 1 This is a flow chart of the European option pricing method based on the mixed sub-fractional Brownian motion and jump diffusion model of the present invention.

[0066] Figure 2 The option price changes with the underlying asset price under different jump intensities in the embodiment of the present invention.

[0067] Figure 3 The option price under different Hurst exponents in the embodiment of the present invention changes with the underlying asset price.

[0068] Figure 4 The figure shows the change of option price under different Hurst exponents and expiration times in the embodiment of the present invention.

[0069] Figure 5 represents the option price changes under different risk-free interest rates and volatility in the embodiment of the present invention. DETAILED DESCRIPTION

[0070] It should be noted that, in the absence of conflict, the embodiments and features of the embodiments of the present invention can be combined with each other. The present invention will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0071] In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. The following description of at least one exemplary embodiment is actually only illustrative and is in no way intended to limit the present invention and its application or use. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0072] It should be noted that the terms used herein are only for describing specific embodiments and are not intended to limit the exemplary embodiments according to the present invention. As used herein, unless the context clearly indicates otherwise, the singular form is intended to include the plural form. In addition, it should be understood that when the terms "comprise" and / or "include" are used in this specification, they indicate the presence of features, steps, operations, devices, components and / or combinations thereof.

[0073] Unless otherwise specified, the relative arrangement of the parts and steps, numerical expressions and numerical values ​​set forth in these embodiments do not limit the scope of the present invention. At the same time, it should be clear that, for ease of description, the sizes of the various parts shown in the accompanying drawings are not drawn according to actual proportional relationships. The technology, methods and equipment known to those of ordinary skill in the relevant art may not be discussed in detail, but in appropriate cases, the technology, methods and equipment should be considered as a part of the specification. In all examples shown and discussed here, any specific value should be interpreted as being merely exemplary, rather than as a limitation. Therefore, other examples of the exemplary embodiments can have different values. It should be noted that similar numbers and letters represent similar items in the following drawings, and therefore, once an item is defined in an accompanying drawing, it does not need to be further discussed in subsequent drawings.

[0074] In the description of the present invention, it should be understood that the directions or positional relationships indicated by directional words such as "front, back, up, down, left, right", "horizontal, vertical, vertical, horizontal" and "top, bottom" are usually based on the directions or positional relationships shown in the accompanying drawings. They are only for the convenience of describing the present invention and simplifying the description. Unless otherwise specified, these directional words do not indicate or imply that the device or element referred to must have a specific direction or be constructed and operated in a specific direction. Therefore, they cannot be understood as limiting the scope of protection of the present invention: the directional words "inside and outside" refer to the inside and outside relative to the outline of each component itself.

[0075] For ease of description, spatially relative terms such as "above", "above", "on the upper surface of", "above", etc. may be used herein to describe the spatial positional relationship of a device or feature to other devices or features as shown in the figures. It should be understood that spatially relative terms are intended to include different orientations of the device in use or operation in addition to the orientation described in the figures. For example, if the device in the drawings is inverted, the device described as "above other devices or structures" or "above other devices or structures" will be positioned as "below other devices or structures" or "below their position devices or structures". Thus, the exemplary term "above" can include both "above" and "below". The device can also be positioned in other different ways (rotated 90 degrees or in other orientations), and the spatially relative descriptions used here are interpreted accordingly.

[0076] In addition, it should be noted that the use of terms such as "first" and "second" to limit components is only for the convenience of distinguishing the corresponding components. Unless otherwise stated, the above terms have no special meaning and therefore cannot be understood as limiting the scope of protection of the present invention.

[0077] like Figure 1 As shown, the present invention provides a European option pricing method based on a mixed sub-fractional Brownian motion and jump diffusion model, comprising:

[0078] Introducing the jump diffusion process into the mixed fractional Brownian motion to describe the asset price process;

[0079] In specific implementation, as a preferred embodiment of the present invention, the asset price process specifically includes:

[0080] Consider a financial market where stock prices exhibit both continuous and discontinuous characteristics. The continuous portion exhibits "peaked peaks and thick tails" and "long-term correlation," while the discontinuous portion exhibits abnormal price fluctuations. By introducing jump processes into mixed fractional Brownian motion, using a mixed fractional jump-diffusion process to describe the underlying asset price process, we obtain the stochastic differential equation satisfying the underlying asset price process, assuming that no transaction fees are incurred for stock trading:

[0081] dS t =(μ-q)S t dt+σS t dW t

[0082] Among them, S t is the underlying asset price, μ is the asset return rate, q is the dividend rate, σ is the volatility of the asset price, t represents time, and W t represents a mixed-time fractional-hop diffusion process.

[0083] Solve the mixed fraction jump diffusion process according to the asset price process formula;

[0084] In specific implementation, as a preferred embodiment of the present invention, the solution of the mixed fraction jump diffusion process Formula, specifically including:

[0085] In the complete probability space (Ω,F t H ,P), assuming that:

[0086]

[0087] in, represents mixed fractional Brownian motion, N t represents a compensated Poisson process, B t represents the geometric Brownian motion process, Represents the fractional Brownian motion process. t} t≥0 、 With {N t} t≥0 are independent of each other, H is the Hurst index;

[0088]

[0089] Both are square-integrable under measure P. 2 (P), N t Indicates that at t i The i-th jump occurs at time t, and f(t,x) represents the mixed-time fractional jump diffusion environment Formula, C 1,2 Indicates that the function has a certain order of continuous partial derivatives on two variables, R + Indicates that the domain of the first variable is non-negative real numbers, R indicates that the domain of the second variable is real numbers → R indicates that the function's value is a real number represents the first-order partial derivative of the function, represents the second-order partial derivative of the function, Represents the second-order partial derivative of the function.

[0090] because Q t It represents a Poisson process with strength λ, and λ represents the jump intensity. When i=1, that is, W t At (0, t) time, only one jump occurs, and the jump time is t1. Then no jump occurs between (0, t1) and (t1, t). The formula is:

[0091]

[0092] in, Indicates approaching W t The state f(0,0) represents the function value in the initial state, x represents the independent variable, represents the mixed fractional-hop diffusion environment at time t1 formula.

[0093] f(t,W t ) at time t1 is have:

[0094]

[0095] The number of jumps i in the interval (0, t) follows a Poisson process,

[0096]

[0097] Let g(φ)∈C 2 (R→R), and (dQ t ,dQ t )=λdt, for g(Q t ) Application in a broad sense Points, there are:

[0098]

[0099] Among them, g(φ) represents a second-order continuous differentiable function from real numbers to real numbers, Q τ It is a Poisson process with strength λ.

[0100] Depend on but

[0101]

[0102] Then we can get the mixed fraction jump diffusion process formula:

[0103]

[0104] Calculate the partial differential equations of related derivatives prices under mixed fractional jump diffusion conditions based on self-financing investment strategies;

[0105] In specific implementation, as a preferred embodiment of the present invention, the partial differential equation for calculating the price of related derivatives under the mixed-time fractional jump diffusion condition based on the self-financing investment strategy specifically includes:

[0106] Construct a bond P t and stock S t The portfolio consists of bonds Pt and stock S t Satisfies the stochastic differential equation:

[0107] dP t =rP t dt

[0108]

[0109] Where r represents the bond yield.

[0110] Self-financing investment strategy Wealth Process satisfy:

[0111]

[0112] Among them, V t represents the total value of the portfolio at time t, θ t 1 represents the number of bonds held, Indicates the number of stocks held.

[0113] According to the self-financing trading strategy, there are:

[0114]

[0115] V t Also meets:

[0116]

[0117] in,

[0118] The value of the option is equal to the value of the constructed portfolio,

[0119]

[0120] Get the stock price S t When the stochastic differential equation is obeyed, the price of the relevant derivative V t =V(t,S t ) is the partial differential equation:

[0121]

[0122] Based on the prices of the related derivatives, the pricing of European call options in an incomplete market with mixed sub-fractional jump diffusion environment is derived.

[0123] In specific implementation, as a preferred embodiment of the present invention, the calculation method for pricing a European call option includes:

[0124] Let V t(t,S t )=C(t,S t )=C, calculate the price C of a common European call option:

[0125]

[0126] Combined with the initial condition C(T,S T )=(S T -K) + ,0 t <+∞ Solve, where K is the strike price and T is the expiration date;

[0127] Let X = lnS t ,have:

[0128]

[0129] Perform the second transformation, let τ=γ(t),η τ =X+α(t),W(τ,η τ )=C(X,t)e β(t) . Among them, α(t), β(t), γ(t) are unknown functions, and we have:

[0130]

[0131] After finishing, we can get:

[0132] in,

[0133] make

[0134]

[0135] According to the final value condition α(T)=β(T)=γ(T)=0, we can get:

[0136] β(t)=r(Tt),

[0137]

[0138] get:

[0139]

[0140] Solve the heat conduction equation:

[0141]

[0142] make:

[0143]

[0144] but: ​

[0145]

[0146] make:

[0147]

[0148] So

[0149] The solution is:

[0150]

[0151] After the variables are restored, there are:

[0152]

[0153] Based on the price C of the ordinary European call option, the pricing formula of the European call option in the mixed fractional jump diffusion environment under the incomplete market is derived:

[0154] C(t,S t )=e -β(t) W(τ,η τ )=S t e -q(T-t) N(d1)-Ke -r(T-t) N(d2).

[0155] Example

[0156] like Figure 1 This paper presents a European option pricing method based on a hybrid subfractional Brownian motion and jump-diffusion model. In this example, a series of numerical experiments are presented to evaluate the effectiveness and accuracy of European option pricing based on the hybrid subfractional Brownian motion model under a jump-diffusion environment. The experimental design consists of two parts: accuracy testing and sensitivity analysis. All numerical examples are implemented using the Python programming language on a personal computer equipped with an Intel Core i5 CPU.

[0157] (1) Accuracy analysis

[0158] The accuracy is tested by comparing the results obtained from the analytical pricing formula with those obtained from Monte Carlo simulation (hereinafter referred to as MC). To evaluate the accuracy of the option pricing method, the relative error is defined as follows:

[0159]

[0160] Among them, P MC and P C represent the call option prices obtained by Monte Carlo simulation and hybrid subfractional Brownian motion method, respectively.

[0161] The option prices provided by the approximate formulas are systematically compared with those obtained through Monte Carlo simulations in Tables 1, 2, 3, and 4 below. In these tables, the number of Monte Carlo iterations of the option payoffs is denoted by the symbol N.

[0162] Table 1 considers the case without jumps. In this scenario, we set the jump intensity λ = 0, the Hurst exponent H = 0.75, the strike price K = 10, the initial asset price S0 = 10, the continuous dividend rate q = 0.01, and the time t = 0.

[0163] Table 1 Comparison between analytical pricing formula and Monte Carlo simulation when λ = 0, H = 0.75

[0164]

[0165] Table 2 considers the case of jumps, setting the jump intensity λ to 1, while keeping all other parameters consistent with Table 1. Comparing the option pricing results in these two cases reveals the specific impact of the jump process on option prices, providing strong support for further analysis.

[0166] Table 2 Comparison between analytical pricing formula and Monte Carlo simulation when λ = 1 and H = 0.75

[0167]

[0168] By observing Table 1 and Table 2, it can be clearly seen that when the number of repetitions of the Monte Carlo simulation is N=10 6 When N=10, the option prices calculated using the analytical formula are very close to those obtained by Monte Carlo simulation. Under this condition, the maximum relative error is 0.1346%, while the minimum relative error is as low as 0.0084% (both results are listed in Table 1). 6 In this case, Monte Carlo simulation can accurately estimate the price of real options. The results in Table 1 and Table 2 prove the reliability and effectiveness of the analytical formula obtained in Chapter 2.

[0169] In addition, it can be observed from the results that for some specific parameter values, the Monte Carlo simulation shows slow convergence. 3 The maximum relative error obtained in the Monte Carlo simulation is 0.6091% (this result appears in the second case of Table 2). This finding highlights the usefulness of the analytical formula because, under the same conditions, the analytical solution can provide more stable and faster results.

[0170] Further time analysis showed that 6The computational time required for the Monte Carlo simulation is a minimum of 356.68 seconds (see the second case in Table 1) and a maximum of 469.93 seconds (see the second case in Table 2). In contrast, the computational time using the analytical formula is only 0.001 seconds. These results demonstrate that the hybrid fractional Brownian motion and jump process-based option pricing method not only offers significant advantages in accuracy but also demonstrates excellent computational efficiency, achieving a dual improvement in both accuracy and efficiency.

[0171] While keeping other parameters unchanged, the Hurst exponent was adjusted to 0.9, yielding the results in Tables 3 and 4. This modification aims to explore the impact of mixed subfractional Brownian motion combined with jump processes on option pricing under different Hurst exponents.

[0172] Table 3 Comparison between analytical pricing formula and Monte Carlo simulation when λ = 0, H = 0.9

[0173]

[0174] Table 4 Comparison between analytical pricing formula and Monte Carlo simulation when λ = 1 and H = 0.9

[0175]

[0176]

[0177] After detailed calculations and analysis, the results in Tables 3 and 4 also show similar trends and phenomena as in the previous experiments. This finding further verifies the robustness of the proposed analytical formula and demonstrates its ability to maintain efficient pricing under various market conditions.

[0178] (2) Sensitivity analysis:

[0179] Different jump intensities are simulated, and the following assumptions are made about other parameters:

[0180] T=1, t=0, r=0.05, q=0.01, K=10, H=0.75, σ=0.2

[0181] Figure 2 This chart shows how the value of a European option changes under different jump strengths and stock prices. It's clear from the chart that, for a fixed strike price, the value of a call option increases with increasing jump strength. This phenomenon suggests a positive correlation between option price and jump strength.

[0182] This positive correlation stems from the fact that, in addition to the impact of regular, continuous fluctuations on asset prices, jump intensity also reflects the potential for irregular and sudden price fluctuations. When market jumps occur, asset prices can deviate from their original trend for a short period of time, significantly increasing the potential returns of options. Specifically, jump intensity represents the frequency or magnitude of sudden price fluctuations in the underlying asset. As the magnitude or frequency of jumps increases, the underlying asset's price is more likely to move away from its current level, creating more profit opportunities for option holders. This uncertainty not only increases the potential returns of options but also increases their time value, particularly before expiration, when the likelihood of drastic price fluctuations is greater. Therefore, investors tend to have higher expectations for the value of options when faced with higher jump intensity, reflecting the market's reassessment of risk and potential returns.

[0183] Different Hurst exponents are simulated, and the following assumptions are made about other parameters:

[0184] T=1, t=0, r=0.05, q=0.01, K=10, λ=1, σ=0.2

[0185] Figure 3 This figure shows the evolution of European option values ​​under different Hurst exponents and stock prices. It can be seen that, at a fixed strike price, the value of a call option decreases as the Hurst exponent increases, indicating a negative correlation between option prices and the Hurst exponent. This trend stems from the fact that, in the Gaussian mixture model, a larger Hurst exponent indicates a stronger long-term memory of asset price fluctuations. That is, when asset prices rise (or fall), there is a higher probability that future prices will continue in the same trend. This persistence enhances market autocorrelation, making future price movements more predictable. Because the value of an option depends heavily on the volatility and uncertainty of the underlying asset price, when the market becomes more predictable, price uncertainty decreases, market volatility weakens, and investors' demand for risk hedging using options decreases. Consequently, the time value of an option decreases due to increased market predictability, leading to a decrease in option value. This relationship between reduced risk and decreased option value is consistent with actual market conditions.

[0186] Figure 4 The influence of different Hurst exponents and expiration time on option prices under the same parameter assumptions is shown. It can be observed that as the expiration time increases, the option price increases accordingly, reflecting the role of option time value and in line with market laws. At the same time, by combining Figure 3 and Figure 4The results further reveal that when the time to expiration is long (e.g., T = 2), the downward trend in option value becomes more pronounced as the Hurst exponent increases. This further demonstrates that a larger Hurst exponent leads to greater predictability in asset prices, ultimately depressing option prices. Furthermore, options with longer expiration times are more sensitive to changes in the Hurst exponent, resulting in more significant declines in option value. This phenomenon reveals the close connection between option time value and market predictability, further validating the model's rationale and its alignment with actual market conditions.

[0187] Finally, let’s look at the impact of volatility and risk-free interest rates on option value, with the same parameter assumptions as before:

[0188] T=1, t=0, q=0.01, St=10, K=10, λ=1, H=0.75

[0189] Figure 5 This figure illustrates the impact of changes in the risk-free interest rate and volatility on the value of European call options. As can be seen from the figure, both the risk-free interest rate and volatility are positively correlated with option value. Option value generally increases with rising risk-free interest rates. This is because higher risk-free rates increase the opportunity cost of holding the underlying asset, leading investors to pay higher premiums for options to secure the right to purchase the asset at a predetermined price in the future. Furthermore, rising volatility also drives up the value of call options. This is primarily because greater volatility creates greater price uncertainty, increasing the likelihood of potential gains for option holders. In a volatile market environment, asset prices fluctuate more widely, which increases the time value of options. Investors anticipate profits from larger future price fluctuations and are therefore willing to pay higher prices.

[0190] Therefore, both a rise in the risk-free interest rate and an increase in volatility will directly lead to an increase in the value of call options. This phenomenon reflects the market's sensitivity to changes in risk and uncertainty, consistent with a fundamental principle of financial markets: investors tend to pay more for options in the face of greater uncertainty and higher opportunity costs, seeking higher potential returns. This result further validates the effectiveness and rationality of option pricing models in analyzing changes in market variables. Participants in the options market must closely monitor the changes in these key factors when considering investment decisions, so as to make more informed choices in a complex market environment.

[0191] Numerical experiments validate the accuracy and efficiency of the new formula and discuss the impact of various parameters on option prices. We observe a positive correlation between jump intensity and option prices, while a negative correlation between the Hurst exponent and option prices. This is primarily due to the long memory of asset prices, which makes the underlying asset price predictable and reduces option prices. The combination of mixed subfractional Brownian motion and jump-diffusion processes for option pricing is rare, and this represents a key contribution of this paper.

[0192] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the above embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A European option pricing method based on a hybrid subfractional Brownian motion and jump diffusion model, characterized in that: include: Introducing the jump diffusion process into the mixed fractional Brownian motion to describe the asset price process; Solve the mixed fraction jump diffusion process according to the asset price process formula; Calculate the partial differential equations of related derivatives prices under mixed fractional jump diffusion conditions based on self-financing investment strategies; Based on the prices of the related derivatives, the pricing of European call options in an incomplete market with mixed sub-fractional jump diffusion environment is derived.

2. The European option pricing method based on the mixed sub-fractional Brownian motion and jump-diffusion model according to claim 1, characterized in that: The asset price process specifically includes: Introducing the jump process into the mixed fractional Brownian motion, that is, using the mixed fractional jump diffusion process to describe the underlying asset price process, then assuming that stock transactions do not require transaction fees, the underlying asset price process satisfies the stochastic differential equation: dS t =(μ-q)S t dt+σS t dW t Among them, S t is the underlying asset price, μ is the asset return rate, q is the dividend rate, σ is the volatility of the asset price, t represents time, and W t represents a mixed-time fractional-hop diffusion process.

3. The European option pricing method based on the mixed sub-fractional Brownian motion and jump diffusion model according to claim 1, characterized in that: The solution of the mixed fraction jump diffusion process Formula, specifically including: In the complete probability space (Ω,F t H ,P), assuming that: in, represents mixed fractional Brownian motion, N t represents a compensated Poisson process, B t represents the geometric Brownian motion process, Represents the fractional Brownian motion process; {B t } t≥0 、 With {N t } t≥0 are independent of each other, H is the Hurst index; and are all square integrable under measure P, N t Indicates that at t i The i-th jump occurs at time t, and f(t,x) represents the mixed-time fractional jump diffusion environment Formula, C 1,2 Indicates that the function has a certain order of continuous partial derivatives on two variables, R + =R indicates that the domain of variable t is non-negative real numbers, R indicates that the domain of variable x is real numbers, →R indicates that the function evaluates to real numbers; because Q t represents a Poisson process with strength λ, where λ represents the jump intensity. When i = 1, that is, W t At (0, t) time, only one jump occurs, and the jump time is t1. Then no jump occurs between (0, t1) and (t1, t). The formula is: in, Indicates approaching W t The state, f(0,0) represents the function value in the initial state, x represents the independent variable, represents the mixed fractional-hop diffusion environment at time t1 formula; f(t,W t ) at time t1 is have: The number of jumps i in the interval (0, t) follows a Poisson process, Let g(φ)∈C 2 (R→R), and (dQ t ,dQ t )=λdt, for g(Q t ) Application in a broad sense Points, there are: Among them, g(φ) represents a second-order continuous differentiable function from real numbers to real numbers, Q τ It represents a Poisson process with strength λ; Depend on but Then we can get the mixed fraction jump diffusion process formula:

4. The European option pricing method based on a hybrid subfractional Brownian motion and jump diffusion model according to claim 1, characterized in that: The partial differential equation for calculating the price of related derivatives under the mixed-time fractional jump diffusion condition based on the self-financing investment strategy specifically includes: Construct a bond P t and stock S t The portfolio consists of bonds P t and stock S t Satisfies the stochastic differential equation: dP t =rP t dt Where r represents the bond yield; Self-financing investment strategy Wealth Process satisfy: Among them, V t represents the total value of the portfolio at time t, θ t 1 represents the number of bonds held, Indicates the number of stocks held; According to the self-financing trading strategy, there are: V t Also meets: in, The value of the option is equal to the value of the constructed portfolio, Get the stock price S t When the stochastic differential equation is obeyed, the price of the relevant derivative V t =V(t,S t ) is the partial differential equation:

5. The European option pricing method based on the hybrid sub-fractional Brownian motion and jump diffusion model according to claim 1, characterized in that: The calculation method for pricing the European call option includes: Let V t (t,S t )=C(t,S t )=C, calculate the price C of a common European call option: Combined with the initial condition C(T,S T )=(S T -K) + ,0 t <+∞Solve, where K is the strike price and T is the expiration date;​ Based on the price C of the ordinary European call option, the pricing formula of the European call option in the mixed fractional jump diffusion environment under the incomplete market is derived: C(t,S t )=S t e -q(T-t) N(d1)-Ke -r(T-t) N(d2)

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